Quatroids and rational plane cubics

Taylor Brysiewicz Department of Mathematics, University of Western Ontario, London, Canada (ORCID: 0000-0003-4272-5934) [email protected] , Fulvio Gesmundo Saarland Informatics Campus, Universität des Saarlandes, Saarbrücken, Germany (ORCID: 0000-0001-6402-021X) [email protected] and Avi Steiner Fakultät für Mathematik, Technische Universität Chemnitz, Chemnitz, Germany (ORCID: 0000-0003-2095-9203) [email protected]

Abstract.

It is a classical result that there are $12$ (irreducible) rational cubic curves through $8$ generic points in $\mathbb{P}_{\mathbb{C}}^{2}$ , but little is known about the non-generic cases. The space of $8$ -point configurations is partitioned into strata depending on combinatorial objects we call quatroids, a higher-order version of representable matroids. We compute all $779777$ quatroids on eight distinct points in the plane, which produces a full description of the stratification. For each stratum, we generate several invariants, including the number of rational cubics through a generic configuration. As a byproduct of our investigation, we obtain a collection of results regarding the base loci of pencils of cubics and positive certificates for non-rationality.

Key words and phrases:

quatroid, rational cubic, matroid, stratification

2020 Mathematics Subject Classification:

(primary) 14N10, (secondary) 14E08, 55R80, 14H50, 05B35, 14Q05

1. Introduction

In this article, we address the following planar interpolation problem:

(Problem 1)

\textit{How many rational cubic curves pass through eight distinct points in }\mathbb{P}_{\mathbb{C}}^{2}\textit{?}

Famously, the answer to this enumerative problem is $12$ whenever the eight points are in generic position (see Figure 1). A modern proof of this classical result follows by an evaluation of Kontsevich’s Formula [21, Claim 5.2.1]. We consider the non-generic cases of Problem 1.

Refer to caption — Figure 1. Twelve rational cubics through eight generic points in the plane. Any appearances of reducibility or points coinciding with nodes are visual illusions.

For almost all enumerative problems of interest, the count is known only for generic parameters. Determining the conditions of non-genericity for an enumerative problem is a hefty task; the challenge of describing the solution set over non-generic parameters is more difficult still. Such a full analysis has been achieved for few enumerative problems (e.g. [24]). We advance the understanding of non-generic instances of Problem 1 through combinatorial objects we call quatroids.

The parameter space of Problem 1 is the space ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathcal{P}}\subseteq(\mathbb{P}_{\mathbb{C}}^{2})^{8}$ of configurations of eight distinct points. We partition $\mathcal{P}$ into $779777$ quatroid strata. Each stratum is a locally closed subset ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathcal{S}_{\mathcal{Q}}}\subset\mathcal{P}$ indexed by a pair ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathcal{Q}}=({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathcal{I}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathcal{J}})$ of triples $\mathcal{I}$ and sextuples $\mathcal{J}$ of $\{1,2,\ldots,8\}$ . Those pairs $\mathcal{Q}$ for which $\mathcal{S}_{\mathcal{Q}}$ is nonempty are called (representable) quatroids. A configuration ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textbf{p}}=(p_{1},\ldots,p_{8})$ is in $\mathcal{S}_{\mathcal{Q}}$ if the triples in $\mathcal{I}$ index the triples of points in $\mathbf{p}$ lying on lines, and the sextuples in $\mathcal{J}$ do so for sextuples of points lying on (irreducible) conics. A similar construction was explored in [13] in the context of M-curves and singular cubic curves interpolating convex configurations of points in $\mathbb{P}_{\mathbb{R}}^{2}$ .

The $779777$ quatroids appear in $125$ orbits under the natural action of the symmetric group $\mathfrak{S}_{8}$ . We compute a representative quatroid ${\mathcal{Q}}$ of each orbit, along with several of its invariants (see Figure 2). Included in this list is the number ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}d_{\mathcal{Q}}}$ of rational cubics through a generic configuration in $\mathcal{S}_{\mathcal{Q}}$ .

The main results about the stratification are summarized in the data contained in the Appendix:

•

Figure 10 and Figure 11 illustrate each quatroid orbit via a real representative if available.
•

Table 4 and Table 5 enumerate each quatroid orbit. Included in these tables are their orbit sizes, combinatorial descriptions, descriptions of the reducible cubics through generic representatives, and the counts of rational cubics through a generic representatives.
•

Table 3 lists and describes the auxiliary files used to obtain the results in the paper.

The code and auxiliary files may be found at

https://mathrepo.mis.mpg.de/QuatroidsAndRationalPlaneCubics

Figure 10/ Figure 11 and Table 4/Table 5 contain $126$ entries, rather than $125$ . These correspond to the $126$ orbits of candidate quatroids computed in Section 3. The $63^{\text{rd}}$ entry in each describes a collection of linear and quadratic dependencies which cannot be realized by any configuration of eight distinct points in $\mathbb{P}_{\mathbb{C}}^{2}$ .

An enumerative problem such as Problem 1 can be viewed as a branched cover, in the sense of [3, Section 2.2]. Specifically, let ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathbb{P}S^{3}\mathbb{C}^{3}}$ be the projectivization of the space of homogeneous ternary cubics and consider the following two subsets:

	$\displaystyle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathcal{C}}$	$\displaystyle=\{C\in\mathbb{P}S^{3}\mathbb{C}^{3}\mid C\text{ is rational and irreducible}\},$
	$\displaystyle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathcal{D}}$	$\displaystyle=\{C\in\mathbb{P}S^{3}\mathbb{C}^{3}\mid C\text{ is singular}\}.$

It is a classical fact that $\mathcal{C}$ is an open subvariety of the degree $12$ irreducible hypersurface $\mathcal{D}\subseteq\mathbb{P}S^{3}\mathbb{C}^{3}$ called the discriminant (see, e.g., [16, Example I.4.15]).

Problem 1 pertains to rational cubics, but it is useful to consider the analogous problem for singular cubics. Hence, we consider incidence correspondences

	$\displaystyle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathcal{X}}$	$\displaystyle=\{(\textbf{p},C)\in\mathcal{P}\times\mathcal{C}\mid C(p_{i})=0,i=1,\ldots,8\},$
	$\displaystyle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\overline{\mathcal{X}}}$	$\displaystyle=\{(\textbf{p},C)\in\mathcal{P}\times\mathcal{D}\mid C(p_{i})=0,i=1,\ldots,8\}.$

The natural projection maps $\pi:\mathcal{X}\to\mathcal{P}$ and $\overline{\pi}:\overline{\mathcal{X}}\to\mathcal{P}$ define two branched covers over $\mathcal{P}$ ; clearly $\overline{\mathcal{X}}$ is the closure of $\mathcal{X}$ in $\mathcal{P}\times\mathbb{P}S^{3}\mathbb{C}^{3}$ and $\pi$ is the composition of the open inclusion map $\mathcal{C}\hookrightarrow\mathcal{D}$ with $\overline{\pi}$ . With this setup, solving an instance of either enumerative problem over a configuration $\textbf{p}\in\mathcal{P}$ is the same as computing the fibers $\pi^{-1}(\textbf{p})$ or $\overline{\pi}^{-1}(\textbf{p})$ of the respective branched cover.

For fixed $\textbf{p}\in\mathcal{P}$ , the equations $C(p_{i})=0$ define a linear space ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}L_{\textbf{p}}}\subset\mathbb{P}S^{3}\mathbb{C}^{3}$ on the space of ternary cubics; explicitly, $L_{\mathbf{p}}$ is the projectivization of the homogeneous component of degree $3$ of the ideal of the points $\mathbf{p}$ . If the conditions $C(p_{i})=0$ are linearly independent, then the dimension of $L_{\textbf{p}}$ is $\dim(\mathbb{P}S^{3}\mathbb{C}^{10})-8=1$ : in this case $L_{\mathbf{p}}$ is a pencil. Hence, singular cubics through $\textbf{p}\in\mathcal{P}$ correspond to points in $L_{\textbf{p}}\cap\mathcal{D}$ and the analysis of $\overline{\pi}$ is equivalent to the analysis of how pencils of cubics in $\mathbb{P}S^{3}\mathbb{C}^{3}$ meet the discriminant hypersurface $\mathcal{D}$ .

In Section 2 we outline the relevant algebro-geometric geometric facts underlying our analysis of the branched covers $\pi$ and $\overline{\pi}$ . We focus on the geometry of cubics. We characterize the point configurations $\textbf{p}\in\mathcal{P}$ for which there are infinitely many singular cubics through p (2.1) as well as those configurations for which there are fewer than $12$ rational cubics through p (Theorem 2.5).

Theorem 2.5 forms the main idea behind the stratification of $\mathcal{P}$ . It states that there are fewer than $12$ rational cubics through $\textbf{p}\in\mathcal{P}$ if and only if p represents a non-uniform quatroid. In Section 3, we define (representable) quatroids (3.1) of eight distinct points in $\mathbb{P}_{\mathbb{C}}^{2}$ and discuss their relationship to matroids. We compute the set of candidate quatroids ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathfrak{Q}}$ , a superset of all quatroids, using algorithm 1. We achieve this by making use of two necessary criteria for $\mathcal{Q}$ to be a quatroid: the underlying matroid is representable and $\mathcal{Q}$ satisfies Bézout’s weak criteria (3.9). For each candidate quatroid $\mathcal{Q}$ , we determine if $\mathcal{Q}$ is representable over $\mathbb{Q}$ , $\mathbb{R}$ , or $\mathbb{C}$ . We give explicit representations in the auxiliary file RationalRepresentatives.txt. We show that candidate $\mathcal{Q}_{63}$ is not representable over $\mathbb{C}$ , and as a consequence, we obtain the true list of $125$ orbits of quatroids.

In Section 4 we show that all quatroids have irreducible realization spaces except for $\mathcal{Q}_{41}$ , which has two irreducible components. This result justifies our use of the word generic when considering points on these strata.

Since the number ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}d_{\textbf{p}}}$ of rational cubics through a configuration p is upper semicontinuous, counting the rational cubics interpolating any representative configuration in $\mathcal{S}_{\mathcal{Q}}$ provides a lower bound on the number $d_{\mathcal{Q}}$ of rational cubics through a generic point of $\mathcal{S}_{\mathcal{Q}}$ . This is done in Section 5; specifically, we compute these bounds in Theorem 5.5 using our rational representatives.

In Section 6 we turn toward showing that our lower bounds on each $d_{\mathcal{Q}}$ are tight. We compute upper bounds for each $d_{\mathcal{Q}}$ by bounding the number of reducible cubics through any configuration $\textbf{p}\in\mathcal{S}_{\mathcal{Q}}$ , using the multiplicity of each reducible cubic as a point on $\mathcal{D}$ . Our lower bounds agree with our upper bounds in all but $24$ cases (see Theorem 6.3). In these remaining $24$ cases, the bound is off by one. We account for this discrepency by showing that configurations which represent any of these $24$ quatroid orbits must correspond to lines $L_{\textbf{p}}\subset\mathbb{P}S^{3}\mathbb{C}^{3}$ which are tangent to a branch of $\mathcal{D}$ through a reducible cubic. Doing so, we achieve our main result Theorem 6.13 of determining $d_{\mathcal{Q}}$ .

In Section 7 we collect several immediate consequences of our computations and outline challenges to refining our stratification. We explain how quatroids offer positive certificates for the non-rationality of cubics. In this section, we also compute the number of rational quartic curves through two special configurations using numerical algebraic geometry. Finally, we discuss examples which show that a complete stratification may be within reach, which would fully answer Problem 1.

Remark 1.1.

The computations supporting several of our main theorems were first performed numerically using HomotopyContinuation.jl [7] in julia [4]. This includes

•

The computation of generic points on each (nonempty) quatroid stratum.
•

The computation of real points on each stratum other than $\mathcal{S}_{\mathcal{Q}_{41}}$ and $\mathcal{S}_{\mathcal{Q}_{63}}$ .
•

The computation of $d_{\mathcal{Q}}$ for each candidate quatroid $\mathcal{Q}$

For the sake of accessibility, brevity, and clarity, we decided to list explicit $\mathbb{Q}$ -representatives instead of relying on subtle claims which depend upon numerical certification methods. Ultimately, despite its important role in developing our intuition about the present problem, no proof in this manuscript comes from a numerical computation. We showcase the power of numerical methods in Section 7 when discussing the possibility of extending our results to quartics. ∎

Acknowledgements

We are grateful for the helpful conversations with Lukas Kühne during the early stages of this project. The first author is partially supported by an NSERC Discovery Grant (Canada). The third author is partially supported by DFG Emmy-Noether-Fellowship RE 3567/1-1.

2. Background on Algebraic Geometry and Cubics

In this section, we collect some standard facts about the geometry of points in the plane and cubic curves through them. Let $\textbf{p}\in\mathcal{P}$ be a configuration of eight distinct points in $\mathbb{P}_{\mathbb{C}}^{2}$ and $L_{\textbf{p}}\subseteq\mathbb{P}S^{3}\mathbb{C}^{3}$ be the linear space of cubics vanishing at p. Write $Z(L_{\textbf{p}})$ for the scheme in $\mathbb{P}_{\mathbb{C}}^{2}$ defined by the ideal generated by $L_{\textbf{p}}$ , called the base locus of $L_{\textbf{p}}$ . A $1$ -dimensional subspace of $\mathbb{P}S^{3}\mathbb{C}^{3}$ is called a pencil of cubics. Plane curves of degree one, two, and three are called lines, quadrics, and cubics, respectively. An irreducible quadric is a conic and a rational cubic is a singular irreducible cubic.

A classical consequence of Bézout’s Theorem [14, Section 5] guarantees that in the cases of interest for Problem 1 the linear space $L_{\mathbf{p}}$ is $1$ -dimensional. We include a proof for completeness.

Proposition 2.1.

Let $\textbf{p}\in\mathcal{P}$ . The following are equivalent.

(1)

The set $\overline{\pi}^{-1}(\textbf{p})=L_{\textbf{p}}\cap\mathcal{D}$ is finite,
(2)

There is an irreducible cubic through p,
(3)

No four points of p are on a line and no seven points of p are on a conic,
(4)

The residue of the scheme $Z(L_{\mathbf{p}})$ with respect to $\mathbf{p}$ is a reduced point $p_{9}$ , called the Cayley–Bacharach point of p.

Proof.

Observe preliminarily that by Bézout’s Theorem, the intersection of two cubics is either of positive dimension or it is a $0$ -dimensional scheme of degree $9$ in $\mathbb{P}_{\mathbb{C}}^{2}$ . We show part (2) is equivalent to all others.

Part (1) $\implies$ Part (2): The linear space $L_{\textbf{p}}$ has dimension at least $1$ because it is defined by eight linear conditions on $\mathbb{P}S^{3}\mathbb{C}^{3}\simeq\mathbb{P}_{\mathbb{C}}^{9}$ . Hence, if $L_{\textbf{p}}\cap\mathcal{D}$ is finite, there exists some $C\in L_{\textbf{p}}\backslash\mathcal{D}$ , that is, an irreducible cubic through p.

$\neg$ Part (1) $\implies\neg$ Part (2): If the intersection $L_{\textbf{p}}\cap\mathcal{D}$ is infinite, then either $L_{\mathbf{p}}\subseteq\mathcal{D}$ or $\dim L_{\mathbf{p}}\geq 2$ .

In the first case, $L_{\mathbf{p}}$ is a pencil of singular cubics. By Bertini’s Theorem [15, p.137] the generic element of $L_{\mathbf{p}}$ is smooth away from the base locus of $L_{\mathbf{p}}$ , so there is a point $p\in\mathbb{P}_{\mathbb{C}}^{2}$ such that all elements of $L_{\mathbf{p}}$ are singular at $p$ . Hence, the fat point $2p$ supported at $p$ is contained in $Z(L_{\textbf{p}})$ . Since $\deg(2p)=3$ , and the base locus of $L_{\mathbf{p}}$ contains at least $7$ additional distinct points, this implies $Z(L_{\textbf{p}})$ is not a $0$ -dimensional scheme of degree $9$ and therefore must contain a positive dimensional component. This component is a curve in $\mathbb{P}_{\mathbb{C}}^{2}$ and all elements of $L_{\mathbf{p}}$ are multiples of the polynomial $f$ defining it. The polynomial $f$ has degree smaller than three since otherwise $L_{\textbf{p}}$ is just a point. Hence, $L_{\mathbf{p}}$ contains no irreducible element.

Now suppose $\dim L_{\mathbf{p}}\geq 2$ . If the points of $\mathbf{p}$ lie on a line or on a conic, then $L_{\mathbf{p}}$ is generated by the corresponding equation of degree $1$ or $2$ so it does not contain irreducible elements. Indeed, if $L_{\mathbf{p}}$ contains an equation not generated by the ones of lower degree, then $\mathbf{p}$ would be contained in a complete intersection of type $(1,3)$ or $(2,3)$ , which contains at most $6$ points. In particular, we may assume the ideal of $\mathbf{p}$ has no elements of degree $1$ or $2$ . We apply [5, Thm. 3.6], and we refer to [8, Thm. 2.3] for a simpler statement: if $\dim L_{\mathbf{p}}\geq 2$ , and the ideal of $\mathbf{p}$ contains no elements of degree $2$ , then the h-vector of $\mathbf{p}$ is $(1,2,3,1,1)$ ; in this case $\mathbf{p}$ has five points on a line $\ell=0$ . Let $q_{1},q_{2},q_{3}$ be three linearly independent quadrics vanishing on the (at most) three points of $\mathbf{p}$ not lying on $\ell$ , then $L_{\mathbf{p}}=\langle\ell q_{1},\ell q_{2},\ell q_{3}\rangle$ . As before, this shows that all elements of $L_{\mathbf{p}}$ share a common factor.

Part (2) $\implies$ Part (3): By Bézout’s Theorem, an irreducible cubic contains no four points on a line or seven on a conic.

Part (3) $\implies$ Part (2): We showed above that if $\dim L_{\mathbf{p}}\geq 2$ then either $\mathbf{p}$ lies on a line or a conic, or at least five elements of $\mathbf{p}$ lie on a line. Therefore, assume $\dim L_{\mathbf{p}}=1$ and suppose all of its elements are reducible. We will show that $\mathbf{p}$ has at least four points on a line or at least seven on a conic. As before, by Bertini’s Theorem, all elements of $L_{\mathbf{p}}$ have at least one common singular point, and we deduce that either a line or a conic is contained in the base locus of $L_{\mathbf{p}}$ .

Suppose the line $\{\ell=0\}$ is a line in the base locus so that $L_{\mathbf{p}}=\langle\ell q_{1},\ell q_{2}\rangle$ . We consider two cases. If $\{q_{1}=q_{2}=0\}$ is $0$ -dimensional, then it contains at most four points of $\mathbf{p}$ , and at least four points of $\mathbf{p}$ lie on $\{\ell=0\}$ . If $\{q_{1}=q_{2}=0\}$ is positive dimensional then $q_{1}=\ell^{\prime}\ell_{1}$ and $q_{2}=\ell^{\prime}\ell_{2}$ : then $\mathbf{p}\subseteq\{\ell=0\}\cup\{\ell^{\prime}=0\}\cup\{\ell_{1}=\ell_{2}=0\}$ , so that at least four points lie on a line.

Suppose the conic $\{q=0\}$ is a conic in the base locus, so that $L_{\mathbf{p}}=\langle q\ell_{1},q\ell_{2}\rangle$ . Then $\mathbf{p}\subseteq\{q=0\}\cup\{\ell_{1}=\ell_{2}=0\}$ , showing that at least seven points lie on a conic.

Part (2) $\implies$ Part (4): This is the classical Cayley–Bacharach Theorem. If $L_{\mathbf{p}}$ contains at least one irreducible element, then the base locus of $L_{\mathbf{p}}$ is a $0$ -dimensional scheme of degree $9$ . The residue is, by definition, cut out by the ideal $(L_{\mathbf{p}}):I(\mathbf{p})$ , where $(L_{\mathbf{p}})$ denotes the ideal generated by the cubics of $L_{\mathbf{p}}$ . Since $\deg(Z(L_{\mathbf{p}}))=9$ , and $\mathbf{p}$ consists of $8$ points with $\mathbf{p}\subseteq Z(L_{\mathbf{p}})$ , we have that the degree of the residue is $1$ . This shows that it is a reduced point $p_{9}$ .

Part (4) $\implies$ Part (2): If the residue is a reduced point, then $Z(L_{\mathbf{p}})$ is $0$ -dimensional. In this case, the same arguments used above show that the generic element of $L_{\mathbf{p}}$ is an irreducible cubic. ∎

A consequence of 2.1 is that if either $L_{\mathbf{p}}$ is not a pencil, or if $L_{\mathbf{p}}\subseteq\mathcal{D}$ , then there are no rational cubics through $\mathbf{p}$ . When $L_{\textbf{p}}$ is a pencil intersecting $\mathcal{D}$ in finitely many points, write ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textrm{imult}_{C}(L_{\textbf{p}},\mathcal{D})}$ for the intersection multiplicity of $L_{\mathbf{p}}$ and $\mathcal{D}$ at a point $C\in L_{\textbf{p}}\cap\mathcal{D}$ (see [14, Section 3]): in other words, ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textrm{imult}_{C}(L_{\textbf{p}},\mathcal{D})}$ is the degree of the component supported at $C$ in the $0$ -dimensional scheme $L_{\mathbf{p}}\cap\mathcal{D}$ . The related concept of the multiplicity of a point $C\in\mathcal{D}$ is defined as follows:

(1)

{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textrm{mult}_{\mathcal{D}}(C)}=\min\{\mathrm{imult}_{C}(L,\mathcal{D})\mid L\in\mathrm{Gr}(2,S^{3}\mathbb{C}^{3})\text{ with }C\in L\},

where ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathrm{Gr}(2,S^{3}\mathbb{C}^{3})}$ is the Grassmannian of projective lines in $\mathbb{P}S^{3}\mathbb{C}^{3}$ . By a semicontinuity argument, a generic line $L$ through $C$ realizes $\mathrm{imult}_{C}(L,\mathcal{D})=\mathrm{mult}_{\mathcal{D}}(C)$ .

Our main interest for this work is the following quantity. Recall that $\mathcal{C}$ is the subset of $\mathcal{D}$ consisting of irreducible cubics.

Definition 2.2.

For $\textbf{p}\in\mathcal{P}$ , define

{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}d_{\textbf{p}}}=\sum_{C\in L_{\textbf{p}}\cap\mathcal{C}}\textrm{imult}_{C}(L_{\textbf{p}},\mathcal{D})

to be the number of rational cubics through p counted with multiplicity.

The following is a direct consequence of 2.1.

Corollary 2.3.

For $\textbf{p}\in\mathcal{P}$ the value $d_{\textbf{p}}$ is finite.

The set-difference of $L_{\textbf{p}}\cap\mathcal{D}$ and $L_{\textbf{p}}\cap\mathcal{C}$ is comprised of the reducible cubics through p. Taking intersection multiplicity into account allows for a stronger correspondence. Hence, for any $\textbf{p}\in\mathcal{P}$ satisfying the conditions of 2.1, we define

{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}r_{\textbf{p}}}=\sum_{\text{reducible }C\in L_{\textbf{p}}\cap\mathcal{D}}\textrm{imult}_{C}(L_{\textbf{p}},\mathcal{D})=\sum_{C\in L_{\textbf{p}}\cap\mathcal{D}\backslash\mathcal{C}}\textrm{imult}_{C}(L_{\textbf{p}},\mathcal{D}).

If $\mathbf{p}$ fails to satisfy the conditions in 2.1, we set $r_{\mathbf{p}}=\infty$ . The following proposition justifies this convention and expresses how $r_{\textbf{p}}$ measures the difference of $L_{\textbf{p}}\cap\mathcal{D}$ and $L_{\textbf{p}}\cap\mathcal{C}$ .

Proposition 2.4.

Let $\textbf{p}\in\mathcal{P}$ satisfy any of the equivalent conditions in 2.1. Then

12=\sum_{C\in L_{\textbf{p}}\cap\mathcal{D}}\mathrm{imult}_{C}(L_{\textbf{p}},\mathcal{D})=r_{\textbf{p}}+\sum_{C\in L_{\textbf{p}}\cap\mathcal{C}}\mathrm{imult}_{C}(L_{\textbf{p}},\mathcal{D})=r_{\textbf{p}}+d_{\textbf{p}}.

If $\mathbf{p}$ does not satisfy the condition of 2.1, then there are infinitely many reducible cubics through p and moreover $d_{\textbf{p}}=0$ .

Proof.

If p satisfies 2.1, then $L_{\textbf{p}}$ is a line and $L_{\textbf{p}}\cap\mathcal{D}$ is finite. Since $\mathcal{D}$ is a projective hypersurface of degree $12$ , the intersection $L_{\mathbf{p}}\cap\mathcal{D}$ is a $0$ -dimensional scheme of degree $12$ . This shows the left equality. The middle equality follows from the definition of $r_{\textbf{p}}$ . The last equality is the definition of $d_{\textbf{p}}$ . The final statement follows from 2.1. ∎

2.4 answers the problem of counting (with multiplicity) the singular cubics through eight distinct points: when finite, this number is $12$ . The main insight of 2.4 is that the number $d_{\textbf{p}}$ is determined as the difference $12-r_{\mathbf{p}}$ , and $r_{\textbf{p}}$ is the number of reducible cubics through p counted with multiplicity. This main idea anchors our story.

Theorem 2.5.

Let $\textbf{p}\in\mathcal{P}$ . The following are equivalent:

(1)

Either p has three points on a line or six points on a conic,
(2)

There is a reducible cubic through p, i.e. $r_{\textbf{p}}>0$ ,
(3)

The inclusion $\pi^{-1}(\textbf{p})\subsetneq\overline{\pi}^{-1}(\textbf{p})$ is strict,
(4)

Counted with multiplicity, there are fewer than $12$ rational cubics through p, i.e. $d_{\textbf{p}}<12$ .

Proof.

If $L_{\mathbf{p}}\cap\mathcal{D}$ is infinite, the result follows directly from 2.1. Hence, suppose that $L_{\mathbf{p}}$ is a pencil and $L_{\mathbf{p}}\cap\mathcal{D}$ is finite.

Part (1) $\iff$ Part (2): If p has three points on a line $\ell$ then the union of $\ell$ and a conic through the remaining five points is a reducible cubic through p. Similarly, if six points are on a conic, then the union of that conic and the line containing the other two is a reducible cubic through p. Conversely, if there is a reducible cubic through p, the pigeonhole principle implies that either three are on a line or six are on a conic since reducible cubics are either unions of three lines or a line-conic union.

Part (2) $\iff$ Part (3): This is immediate from the definitions. Part (4) $\implies$ Part (3): Since the fibre $\overline{\pi}^{-1}(\textbf{p})$ is finite, by 2.4 it consists of $12$ cubics counting multiplicity. From the hypothesis, $\pi^{-1}(\textbf{p})$ does not, so part (3) is true.

Part (2) $\implies$ Part (4): Since the fibre is finite, $\overline{\pi}^{-1}(\textbf{p})$ consists of $12$ points counting multiplicity by 2.4. Since part (2) is true, at least one cubic in $\overline{\pi}^{-1}(\mathbf{p})$ is reducible, and so there are fewer than $12$ rational cubics through p counting multiplicity. ∎

3. Quatroids

As a consequence of the results in the previous section, any configuration $\textbf{p}\in\mathcal{P}$ admitting fewer than $12$ rational cubics must lie on at least one of the following two types of hypersurfaces in $\mathcal{P}$ :

{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}X_{i_{1}i_{2}i_{3}}}=\{\textbf{p}\in\mathcal{P}\mid p_{i_{1}}\ldots p_{i_{3}}\text{ are on a line}\},\quad\quad{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}Y_{i_{1}i_{2}\cdots i_{6}}}=\{\textbf{p}\in\mathcal{P}\mid p_{i_{1}}\ldots p_{i_{6}}\text{ are on a conic}\}.

Elements of the intersection lattice of these ${{8}\choose{3}}+{{8}\choose{6}}=84$ hypersurfaces are indexed by pairs $\mathcal{Q}=(\mathcal{I},\mathcal{J})$ of triples $\mathcal{I}$ and sextuples $\mathcal{J}$ of $\{1,2,\ldots,8\}$ . Define

{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathcal{Z}_{\mathcal{Q}}}=\biggl{(}\bigcap_{I\in\mathcal{I}}X_{I}\biggr{)}\cap\biggl{(}\bigcap_{J\in\mathcal{J}}Y_{J}\biggr{)}\quad\text{ and }\quad{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathcal{S}_{\mathcal{Q}}}=\mathcal{Z}_{\mathcal{Q}}\setminus\bigcup_{\mathcal{Q}^{\prime}}\mathcal{Z}_{\mathcal{Q}^{\prime}}

where the union ranges over all $\mathcal{Q}^{\prime}$ such that $\mathcal{Z}_{\mathcal{Q}^{\prime}}\subsetneq\mathcal{Z}_{\mathcal{Q}}$ .

By construction, the stratification ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathcal{S}}$ of nonempty loci $\mathcal{S}_{\mathcal{Q}}$ partitions the parameter space $\mathcal{P}$ . The strata, and thus their indices, are naturally ordered by $\mathcal{Q}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\leq}\mathcal{Q}^{\prime}$ if and only if $\mathcal{S}_{\mathcal{Q}}\subseteq\overline{\mathcal{S}_{\mathcal{Q}^{\prime}}}$ .

With this setup, we introduce the main combinatorial object of interest. Although we focus on fields of characteristic zero, we give the definition for any field ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathbb{K}}$ . Write ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathcal{P}_{\mathbb{K}}}$ for the open subset $8$ -tuples of distinct points in $(\mathbb{P}_{\mathbb{K}}^{2})^{8}$ .

Definition 3.1.

Let $\mathbb{K}$ be a field and let $\mathcal{I}$ and $\mathcal{J}$ be collections of triples and sextuples of $\{1,2,\ldots,8\}$ , respectively. The pair $\mathcal{Q}=(\mathcal{I},\mathcal{J})$ is a ( $\mathbb{K}$ -representable) quatroid if there exists a configuration $\textbf{p}\in\mathcal{P}_{\mathbb{K}}$ such that $\mathbf{p}\in\mathcal{S}_{\mathcal{Q}}$ . In this case, we say $\mathcal{Q}$ is representable over $\mathbb{K}$ . Equivalently, there exists $\textbf{p}\in\mathcal{P}_{\mathbb{K}}$ such that

•

Every triple of points in p indexed by an element of $\mathcal{I}$ lies on a line,
•

Every sextuple of points in p indexed by an element of $\mathcal{J}$ lies on a conic,
•

No other triple of points in p lies on a line,
•

No other sextuple of points in p lies on a conic.

Such a configuration p is said to represent $\mathcal{Q}$ over $\mathbb{K}$ . The set $\mathcal{S}_{\mathcal{Q}}$ is called the realization space of $\mathcal{Q}$ over $\mathbb{K}$ .

For any $\textbf{p}\in\mathcal{P}$ and $S\subseteq\{1,2,\ldots,8\}$ we write ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\ell_{S}}$ for the line through $\{p_{s}\}_{s\in S}$ and ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}q_{S}}$ for the conic through $\{p_{s}\}_{s\in S}$ when such a curve exists and is unique. Given a configuration p satisfying any condition in 2.1, the quatroid represented by p exactly tracks the reducible cubics through p, as detailed in the following result.

Lemma 3.2.

Let p be a configuration satisfying 2.1 and representing a quatroid $\mathcal{Q}=(\mathcal{I},\mathcal{J})$ . A reducible cubic through p has one of the forms:

(a)

$\ell_{I}\cdot q_{I^{c}}$ for $I\in\mathcal{I}$ ,
(b)

$\ell_{I_{1}}\cdot\ell_{I_{2}}\cdot\ell_{(I_{1}\cup I_{2})^{c}}$ for disjoint $I_{1},I_{2}\in\mathcal{I}$ ,
(c)

$q_{J}\cdot\ell_{J^{c}}$ for $J\in\mathcal{J}$ .

Proof.

If there is a reducible cubic through p, by the pigeonhole principle, and 2.1, either exactly six points of $\mathbf{p}$ lie on a conic or exactly three points of $\mathbf{p}$ lie on a line.

In the first case, the reducible cubic must be the union of the conic containing six points and the line containing the remaining two. The six points on the conic must be indexed by some $J\in\mathcal{J}$ , so the reducible cubic has the form (c).

In the second case, the reducible cubic must be the union of the line containing three points and the quadric containing the remaining five. Following a similar argument as in the first case, if the quadric is a irreducible, then the cubic has the form (a) and otherwise it has the form (b). ∎

Remark 3.3.

An consequence of 3.2 is that the ordering $(\mathcal{I},\mathcal{J})\leq(\mathcal{I}^{\prime},\mathcal{J}^{\prime})$ may be expressed combinatorially. Given our conventions, the full characterization is rather technical. However, we list two simple sufficient conditions:

(a)

$(\mathcal{I},\mathcal{J})\leq(\mathcal{I}^{\prime},\mathcal{J}^{\prime})$ if $\mathcal{I}^{\prime}\subseteq\mathcal{I}$ and $\mathcal{J}^{\prime}\subseteq\mathcal{J}$ ,
(b)

$(\mathcal{I},\mathcal{J})\leq(\mathcal{I}^{\prime},\mathcal{J}^{\prime})$ if $\mathcal{I}^{\prime}\subseteq\mathcal{I}$ and $\mathcal{J}^{\prime}\subseteq\mathcal{J}\cup\mathcal{I}^{2}$ , where $\mathcal{I}^{2}$ denotes the set of subsets obtained as union of two elements of $\mathcal{I}$ .

Condition (b) completely characterizes $\leq$ restricted to a class of quatroids called Bézoutian quatroids defined in 3.15. ∎

We now digress into several remarks on the choice of the term quatroid.

Remark 3.4 (Inspiration for the term).

The term (representable) quatroid is inspired by the term representable matroid. Quatroids partially extend the concept of matroids; representable matroids encode the affine linear dependencies among points, whereas representable quatroids track linear and quadratic dependencies. There are several standard references for matroids. We suggest [28]. ∎

Remark 3.5 (Abstraction of quatroids).

There are several natural ways to extend 3.1 to include point configurations with more points or in higher dimensional spaces. We use the specific definition in 3.1 for the sake of brevity since this is all we need for the purpose of the paper. We leave the challenge of defining a non-representable quatroid for future work. If $\mathcal{Q}=(\mathcal{I},\mathcal{J})$ is not a quatroid, we merely call it a pair. ∎

Remark 3.6 (Pascal’s Theorem to linearize conic conditions).

Pascal’s Theorem states that six points are on a conic if and only if the three auxiliary intersection points of a hexagram as shown in Figure 3 lie on a line, called the Pascal line. One may attempt to convert conic dependencies into linear dependencies using Pascal’s Theorem. This result may be applied to $60$ orderings of the six points, introducing $15$ hexagram lines, $45$ auxiliary points, and $60$ Pascal lines. The right-hand image in Figure 3 illustrates a fraction of how complicated the entire arrangement gets, and thus, the benefit of recording the conic conditions directly. ∎

Remark 3.7 (Simplicity).

Our definition of a quatroid requires that a representative point configuration p consists of distinct points. Such a restriction for matroids means that the matroid is simple: the matroid has no parallel elements (repeated points) and the matroid has no loops (consists of points in $\mathbb{P}_{\mathbb{C}}^{2}$ ). We suspect that an effective definition of quatroid which covers the case of repeated points would need to involve additional information about this repetition. Possibly, there is a way to combinatorially track some information encoded in the Hilbert scheme of eight points in $\mathbb{P}_{\mathbb{C}}^{2}$ . ∎

Remark 3.8.

For the remainder of this article, we assume that $\mathbb{K}$ is $\mathbb{Q}$ , $\mathbb{R}$ , or $\mathbb{C}$ . ∎

Given 3.1, a natural question emerges:

Which pairs $\mathcal{Q}=(\mathcal{I},\mathcal{J})$ are representable quatroids?

We address this problem by providing two necessary conditions on $\mathcal{Q}$ to be a quatroid, finding all $\mathcal{Q}$ satisfying those conditions, and subsequently identifying which of those candidates are quatroids by either exhibiting an element $\mathbf{p}\in\mathcal{P}$ representing it or proving that no such $\mathbf{p}$ exists.

3.1. Necessary conditions for being a quatroid

The underlying matroid of a quatroid $\mathcal{Q}=(\mathcal{I},\mathcal{J})$ is the matroid ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathcal{M}_{\mathcal{I}}}$ whose nonbases of size three are given by $\mathcal{I}$ . The realization space of $\mathcal{M}_{\mathcal{I}}$ is ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathcal{S}_{\mathcal{I}}}=\bigcup_{\mathcal{J}}\mathcal{S}_{(\mathcal{I},\mathcal{J})}$ because if $\mathbf{p}$ represents $\mathcal{Q}$ , then it represents $\mathcal{M}_{\mathcal{I}}$ as well. The matroid $\mathcal{M}_{\mathcal{I}}$ is representable if $\mathcal{S}_{\mathcal{I}}$ is nonempty. We call a pair $\mathcal{Q}=(\mathcal{I},\mathcal{J})$ matroidal whenever $\mathcal{M}_{\mathcal{I}}$ is representable. A representable quatroid $\mathcal{Q}$ is necessarily matroidal. Additional necessary conditions for $\mathcal{Q}$ to be representable rely upon Bézout’s Theorem.

Definition 3.9.

A pair $\mathcal{Q}=(\mathcal{I},\mathcal{J})$ satisfies Bézout’s weak criteria if

(1)

“A line and conic meet in at most two points”:

$|I\cap J|\leq 2\text{ for all }(I,J)\in\mathcal{I}\times\mathcal{J}.$

(2)

“Two lines meet in at most one point”:

|I_{1}\cap I_{2}|>1\text{ for some $I_{1},I_{2}\in\mathcal{I}$}\implies I\in\mathcal{I}\text{ for every }3\text{-subset of }I_{1}\cup I_{2}.

(3)

“Two conics meet in at most four points”:

|J_{1}\cap J_{2}|>4\text{ for some $J_{1},J_{2}\in\mathcal{J}$}\implies J\in\mathcal{J}\text{ for every }6\text{-subset of }J_{1}\cup J_{2}.

Example 3.10.

The pair $\mathcal{Q}=(\{123,145\},\{234678\})$ satisfies all of Bézout’s weak criteria. The pair $\mathcal{Q}^{\prime}=(\{123,234\},\emptyset)$ satisfies the first and (vacuously) the third of Bézout’s weak criteria, but not the second. The pair $\mathcal{Q}^{\prime\prime}=(\{123\},\{123456\})$ does not satisfy the first of Bézout’s weak criteria. By adding additional triples to $\mathcal{Q}^{\prime}$ , one may obtain the pair $(\{123,124,134,234\},\emptyset)$ which does satisfy all of Bézout’s weak criteria. This procedure is impossible for pairs such as $\mathcal{Q}^{\prime\prime}$ which fail Bézout’s first weak criterion. ∎

Lemma 3.11.

If $\mathcal{Q}$ is representable, then $\mathcal{Q}$ is matroidal and satisfies Bézout’s weak criteria.

Proof.

A representable quatroid is, by definition, matroidal. The proof that a representable quatroid satisfies Bézout’s weak criteria is suggested by the quotations preceding each criterion in 3.9 as detailed below.

Suppose $\textbf{p}\in\mathcal{P}$ represents $\mathcal{Q}$ . For every $I\in\mathcal{I}$ , $J\in\mathcal{J}$ , the line $\ell_{I}$ and the conic $q_{J}$ meet in at most two points of $\mathbf{p}$ by Bézout’s Theorem. Therefore, $\mathcal{Q}$ satisfies the first weak criterion. Similarly, if $I_{1},I_{2}\in\mathcal{I}$ , then $\ell_{I_{1}},\ell_{I_{2}}$ meet in at most one point of $\mathbf{p}$ , unless $\ell_{I_{1}}=\ell_{I_{2}}$ . If $\ell_{I_{1}}=\ell_{I_{2}}$ , then any three points of $\mathbf{p}$ lying on such line is indexed by a triple in $\mathcal{I}$ . This shows that $\mathcal{Q}$ must satisfy the second weak criterion. The proof for the third criterion is similar. ∎

Motivated by our goal of computing all quatroids, and inspired by Bézout’s weak criteria, we introduce the following operation on pairs.

Definition 3.12.

Let $\mathcal{I}$ be a set of triples of $\{1,2,\ldots,8\}$ . Define ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\overline{\mathcal{I}}}$ to be the smallest set of triples containing $\mathcal{I}$ which satisfies Bézout’s second weak criterion. Similarly, for a set $\mathcal{J}$ of sextuples of $\{1,2,\ldots,8\}$ , define ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\overline{\mathcal{J}}}$ to be the smallest set of sextuples containing $\mathcal{J}$ which satisfies Bézout’s third weak criterion. For a pair $\mathcal{Q}=(\mathcal{I},\mathcal{J})$ , define ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\overline{\mathcal{Q}}}=(\overline{\mathcal{I}},\overline{\mathcal{J}})$ .

Lemma 3.13.

The sets $\overline{\mathcal{I}}$ and $\overline{\mathcal{J}}$ are well-defined, and thus so is $\overline{\mathcal{Q}}$ .

Proof.

Let $\mathcal{I}$ be a set of triples of $\{1,2,\ldots,8\}$ . The definition of $\overline{\mathcal{I}}$ is the smallest set of triples containing $\mathcal{I}$ which satisfies Bézout’s second weak criterion. This set is not empty since the set of all triples of $\{1,2,\ldots,8\}$ contains $\mathcal{I}$ and satisfies the second criterion. Moreover, if $\mathcal{I}_{1}$ and $\mathcal{I}_{2}$ both satisfy Bézout’s second weak criterion, then so does their intersection, proving that the smallest such set of triples is well-defined. The argument for sextuples is exactly the same. ∎

Example 3.14.

Consider the pair $\mathcal{Q}=(\{123,124,345\},\{245678\})$ . Then

\overline{\mathcal{Q}}=(\{123,124,125,134,135,145,234,235,245,345\},\{245678\}).

Note that $\mathcal{Q}$ satisfies the first and third weak criteria, but not the second. On the other hand $\overline{\mathcal{Q}}$ satisfies the second and third weak criteria, but not the first. ∎

When a configuration $\mathbf{p}$ is contained in an irreducible cubic, Bézout’s Theorem applied to such a cubic restricts which lines and conics pass through subsets of $\mathbf{p}$ . This motivates the following definition.

Definition 3.15.

A pair $\mathcal{Q}=(\mathcal{I},\mathcal{J})$ satisfies Bézout’s strong criteria if

(1)

“An irreducible cubic meets a line in at most three points”:

$I_{1},I_{2}\in\mathcal{I}\implies|I_{1}\cap I_{2}|\leq 1.$
(2)

“An irreducible cubic meets a conic in at most six points”:

$J_{1},J_{2}\in\mathcal{J}\implies|J_{1}\cap J_{2}|\leq 4.$

Pairs which satisfy both Bézout’s weak and strong criteria are called Bézoutian.

As with the weak criteria, the quotes suggest the proof of an important result.

Lemma 3.16.

A configuration $\textbf{p}\in\mathcal{P}$ satisfies the conditions of 2.1 if and only if p represents a Bézoutian quatroid. Hence, there are no rational cubics through a configuration p that represents a non-Bézoutian quatroid.

Proof.

Let $\mathcal{Q}$ denote the quatroid represented by $\mathbf{p}\in\mathcal{P}$ . By 3.11, $\mathcal{Q}$ satisfies Bézout’s weak criteria. Suppose p satisfies the conditions of 2.1. Then there is an irreducible cubic through p and so no four points of p lie on a line and no seven lie on a conic. As suggested by the quotations in 3.15, this implies the strong criteria are satisfied: if $|I_{1}\cap I_{2}|>1$ for some $I_{1},I_{2}$ then at least four points of $\mathbf{p}$ lie on the line $\ell_{I_{1}}=\ell_{I_{2}}$ which is a contradiction. The proof for the second criterion is similar.

Conversely, suppose p represents a Bézoutian quatroid $\mathcal{Q}=(\mathcal{I},\mathcal{J})$ . Then no four points of p lie on a line and no seven lie on a conic and so p satisfies the conditions in 2.1. To see this, suppose towards a contradiction that $p_{1},\ldots,p_{4}$ lie on a line: then $123,124,134,234\in\mathcal{I}$ so Bézout’s first strong criterion is violated. A similar conclusion holds if seven points lie on a conic. ∎

3.2. Computing all candidate quatroids

We approach the task of computing all quatroids by first computing a list ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathfrak{Q}}$ of candidate quatroids. A pair $\mathcal{Q}$ is a candidate quatroid if it is matroidal and satisfies Bézout’s weak criteria. By 3.11, all (representable) quatroids are in $\mathfrak{Q}$ .

Remark 3.17.

Our goal is to count rational cubics through $\mathbf{p}\in\mathcal{P}$ , so by 3.16 it is enough to determine all Bézoutian quatroids. However, for completeness, we compute all quatroids. ∎

Since matroidal pairs have underlying representable matroids, we take advantage of existing databases of matroids to begin our computation. We use the database [23], to compile a list ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathfrak{M}}$ of matroids representable by eight distinct points in $\mathbb{P}_{\mathbb{C}}^{2}$ . These are the representable simple matroids of rank at most three. There are $67$ orbits of such matroids, under the action of $\mathfrak{S}_{8}$ : $66$ of rank three and $1$ of rank two. In this work, we record matroids in terms of their nonbases of size three. The original list from [23] is in the auxiliary file SimpleMatroids38.txt.

We now give an algorithm which produces all candidate quatroids with a given underlying matroid by greedily extending its the conic conditions.

Input: A pair

\mathcal{Q}=(\mathcal{I},\emptyset)

where

{\mathcal{I}}\in\mathfrak{M}

Output: A list of all pairs of the form

\mathcal{Q}^{\prime}=(\mathcal{I},\mathcal{J})

which satisfy Bézout’s weak criteria.

1 initialize PairsFound

=\{\mathcal{Q}\}

2 initialize PairsToExtend

=\{\mathcal{Q}\}

3 (Compute the stabilizer

H

\mathcal{I}

\mathfrak{S}_{8}

)

4 while PairsToExtend $\neq\emptyset$ do

5 Choose

\mathcal{Q}^{\prime}=(\mathcal{I},\mathcal{J})

from PairsToExtend and delete

\mathcal{Q}^{\prime}

from PairsToExtend

6 Set

\texttt{Q'\_Extensions}=\{\overline{(\mathcal{I},\mathcal{J}\cup\{J\})}\}_{J\text{ a }6\text{-subset of }\{1,\ldots,8\}}

7 for $\mathcal{Q}^{\prime\prime}\in\texttt{Q'\_Extensions}$ do

8 if $\mathcal{Q}^{\prime\prime}$ satisfies Bézout’s weak criteria then

9 (Let

\mathcal{Q}^{*}

be a canonical representative of the

H

-orbit of

\mathcal{Q}^{\prime\prime}

)

10 if $\mathcal{Q}^{*}\not\in$ QuatroidsFound then

11 QuatroidsFound

\leftarrow\mathcal{Q}^{*}

12 PairsToExtend

\leftarrow\mathcal{Q}^{*}

13return PairsFound

Algorithm 1 AllConicExtensions

Theorem 3.18.

The set of pairs produced by algorithm 1 on the input $(\mathcal{I},\emptyset)$ coincides with the set of candidate quatroids whose underlying matroid is $\mathcal{M}_{\mathcal{I}}$ . Thus, applying algorithm 1 to every matroid in $\mathfrak{M}$ produces $\mathfrak{Q}$ .

Proof.

Clearly all pairs in the output of algorithm 1 on the input $(\mathcal{I},\emptyset)$ are candidate quatroids whose underlying matroid is $\mathcal{M}_{\mathcal{I}}$ . We need to show that all candidate quatroids arise in this way. We ignore the parenthetical steps of algorithm 1, which are used only to speed up the process by working modulo the $\mathfrak{S}_{8}$ -action.

By contradiction, assume $\mathcal{Q}=(\mathcal{I},\mathcal{J})$ is a candidate quatroid which is not in the output of algorithm 1 applied to $(\mathcal{I},\emptyset)$ . Let $\mathcal{Q}^{\prime}=(\mathcal{I},\mathcal{J}^{\prime})$ be a maximal candidate quatroid which is generated by algorithm 1 with $\mathcal{J}^{\prime}\subsetneq\mathcal{J}$ . Let $J\in\mathcal{J}\setminus\mathcal{J}^{\prime}$ .

The first time $\mathcal{Q}^{\prime}$ is found in algorithm 1, it is placed in PairsToExtend at step 12. Therefore, it is then chosen in step 5 and the pair $\mathcal{Q}^{\prime\prime}=(\mathcal{I},\overline{\mathcal{J}^{\prime}\cup\{J\}})$ is one of the elements of Q’_Extensions at step 6. When $\mathcal{Q}^{\prime\prime}$ is chosen in the for loop at step 7, it satisfies the if condition at step 8, because $\overline{\mathcal{J}^{\prime}\cup\{J\}}\subseteq\mathcal{J}$ guarantees $\mathcal{Q}^{\prime\prime}$ satisfies Bézout’s weak criteria. Hence, $\mathcal{Q}^{\prime\prime}$ is a candidate quatroid and it is produced by algorithm 1. This contradicts the maximality of $\mathcal{Q}^{\prime}$ . ∎

Theorem 3.19.

The list $\mathfrak{Q}$ of candidate quatroids consists of $780617$ pairs. Up to the symmetry of $\mathfrak{S}_{8}$ , these occur in $126$ distinct orbits. The numbers of orbits of each size are tallied below:

Orbit Size	1	8	28	35	56	70	105	168	210	280
# Orbits	3	2	2	1	3	1	1	3	2	3
Orbit Size	420	560	840	1680	2520	3360	5040	6720	10080	20160
# Orbits	2	1	13	4	10	13	17	6	22	17

Proof.

The candidate quatroids are computed by calling algorithm 1 on all representable matroids $\mathfrak{M}$ . This computation may be performed by calling the function AllConicExtensions(M) in the Quatroids.jl package (see Table 3 for details). All candidate quatroids are produced by the function GenerateAllCandidateQuatroids() and the results are automatically stored in a text file. The function OrbitSizes() then produces a text file listing the corresponding orbit sizes. These functions use features from GAP [17] and OSCAR [26] to work modulo the $\mathfrak{S}_{8}$ -symmetry. ∎

3.3. Representability of Quatroids

A candidate quatroid $\mathcal{Q}\in\mathfrak{Q}$ is a quatroid if and only if it is representable. In order to compile a list of quatroids, for every candidate quatroid $\mathcal{Q}$ , we either exhibit an element $\mathbf{p}\in(\mathbb{P}_{\mathbb{C}}^{2})^{8}$ representing it, or we prove that the realization space $\mathcal{S}_{\mathcal{Q}}$ is empty.

Theorem 3.20.

Every $\mathcal{Q}\in\mathfrak{Q}$ is representable over $\mathbb{Q}$ except for the ones in the $\mathfrak{S}_{8}$ -orbits of

	$\displaystyle\mathcal{Q}_{41}$	$\displaystyle=(\{123,145,167,246,258,357,368,478\},\{\}),$
	$\displaystyle\mathcal{Q}_{63}$	$\displaystyle=(\{123,145,246,356\},\{125678,134678,234578\}).$

Those in the orbit of $\mathcal{Q}_{41}$ are representable over $\mathbb{C}$ but not over $\mathbb{R}$ . Those in the orbit of $\mathcal{Q}_{63}$ are not representable over $\mathbb{C}$ .

Proof.

An explicit $\mathbb{Q}$ -realization is given in the auxiliary files for each of the $124$ orbits different from the ones of $\mathcal{Q}_{41}$ and $\mathcal{Q}_{63}$ . A Macaulay2 [18] script verifying that they are representatives is provided as well. The claims regarding $\mathcal{Q}_{41}$ and $\mathcal{Q}_{63}$ are proved in 3.22 and 3.26. ∎

The following corollary combines Theorem 3.19 and Theorem 3.20.

Corollary 3.21.

Of the $780617$ pairs ( $126$ orbits) in $\mathfrak{Q}$ , exactly $779777$ ( $125$ orbits) are representable over $\mathbb{C}$ and $778937$ ( $124$ orbits) are representable over $\mathbb{R}$ and $\mathbb{Q}$ . In particular, the stratification $\mathcal{S}$ of $\mathcal{P}$ consists of $779777$ strata, $544748$ ( $76$ orbits) of which are Bézoutain.

To complete the proof of Theorem 3.20, we show that $\mathcal{Q}_{41}$ is representable over $\mathbb{C}$ and not over $\mathbb{R}$ , and that $\mathcal{Q}_{63}$ is not representable. To analyze $\mathcal{S}_{\mathcal{Q}_{41}}$ we extend a standard matroid procedure to put coordinates on the realization space of a quatroid.

We represent an element $\mathbf{p}\in(\mathbb{P}_{\mathbb{C}}^{2})^{8}$ as a $3\times 8$ matrix, whose columns correspond to elements of $\mathbb{P}_{\mathbb{C}}^{2}$ . We are free to work modulo the action of $\mathrm{PGL}_{2}$ on $\mathbb{P}_{\mathbb{C}}^{2}$ ; it is convenient to normalize either some of the points of $\mathbf{p}$ or some of the conic conditions of $\mathcal{Q}$ , leaving some free parameters ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}z_{1},\ldots,z_{r}}$ . These parameters are free to vary in a quasi-projective variety ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}S_{\mathcal{Q}}}\subseteq\mathbb{C}^{r}$ described by equations and inequations. The equations are determinantal relations imposed by $\mathcal{Q}$ , some of which are identically satisfied after the normalization. The inequations are the determinantal relations not imposed by $\mathcal{Q}$ . The realization space $\mathcal{S}_{\mathcal{Q}}$ is a bundle over $S_{\mathcal{Q}}$ . In particular, $\mathcal{Q}$ is representable if and only if $S_{\mathcal{Q}}$ is nonempty and $\mathcal{S}_{\mathcal{Q}}$ is irreducible if and only if $S_{\mathcal{Q}}$ is.

Candidate quatroid $\mathcal{Q}_{41}$ is a well-known matroid called the MacLane matroid (see, e.g., [20, 32]) which is realized by a Möbius-Kantor configuration [11, Section 2]. As a matroid, it is non-orientable, not representable over $\mathbb{R}$ , and its realization space is reducible. We give a standard proof regarding its representability.

Lemma 3.22.

The candidate quatroid $\mathcal{Q}_{41}=(\{123,145,167,246,258,357,368,478\},\emptyset)$ is representable over $\mathbb{C}$ but not $\mathbb{R}$ . Its realization space is the union of two distinct orbits of $\mathrm{PGL}_{2}$ .

Proof.

The four points $p_{1},p_{2},p_{4},p_{7}$ are in general linear position, in the sense that no subset of three of them lie on a line. Therefore, we may normalize them with the action of $\mathrm{PGL}_{2}$ to be the four points $[e_{1}],[e_{2}],[e_{3}]$ and $[e_{1}+e_{2}+e_{3}]$ in $\mathbb{P}_{\mathbb{C}}^{2}$ . Using the fact that $123\in I$ , we deduce $p_{3}=[1:z_{1}:0]$ for some $z_{1}$ and similarly, we write $p_{5},p_{6},$ and $p_{8}$ using the linear relations $145,246,258\in I$ , respectively, and in this order. We obtain the representation of $\mathbf{p}$ as the matrix

A_{41}=\begin{bmatrix}1&0&1&0&1&0&1&1\\ 0&1&z_{1}&0&0&1&1&z_{4}\\ 0&0&0&1&z_{2}&z_{3}&1&z_{2}\end{bmatrix}

subject to the determinantal equations imposed by $167,357,368,478\in I$ . Conditions $167$ and $478$ imply that $z_{3}=1=z_{4}$ . Condition $368$ implies that $z_{1}=1-z_{2}$ . Finally, $357$ implies that the last remaining parameter, $z_{2}$ , satisfies the univariate quadratic equation $-z_{2}^{2}+z_{2}=1$ , which has two distinct non-real solutions. For either solution, the inequations hold. Hence the set $S_{\mathcal{Q}_{41}}$ consists of two points. Since the two points are inequivalent for the action of $\mathrm{PGL}_{2}$ , the two corresponding $\mathrm{PGL}_{2}$ -orbits give rise to two disjoint irreducible components of $\mathcal{S}_{\mathcal{Q}_{41}}$ . ∎

We now change focus to the non-representability of $\mathcal{Q}_{63}$ . Our proof relies on the following property: for any realization $\mathbf{p}$ of $\mathcal{Q}_{63}$ , the base locus $Z(L_{\mathbf{p}})$ is reduced. Hence, the matroid underlying the base locus is determined by $\mathcal{Q}_{63}$ (see 3.23) and if such a matroid is non-representable, then $\mathcal{Q}_{63}$ is non-representable as well.

A priori the Cayley–Bacharach point of a configuration p of cubics may coincide with one of the eight points in p. In this case the variety underlying the base locus $Z(L_{\mathbf{p}})$ consists of eight points, but as a scheme, it contains a nonreduced component of degree two supported at the Cayley–Bacharach point. See, for example, Figure 4.

Lemma 3.23.

Let $\mathcal{Q}=(\mathcal{I},\mathcal{J})$ be a Bézoutian candidate quatroid and let $\textbf{p}\in\mathcal{S}_{\mathcal{Q}}$ . If $\mathcal{Q}\not\leq\mathcal{Q}^{\prime}$ for every $\mathcal{Q}^{\prime}$ in the orbit of $\mathcal{Q}_{10}$ , then the Cayley–Bacharach point of $L_{\textbf{p}}$ cannot coincide with any point $p_{i}$ involved in a line condition $I\in\mathcal{I}$ or conic condition $J\in\mathcal{J}$ .

Proof.

Without loss of generality, assume $\mathcal{Q}^{\prime}=\mathcal{Q}_{10}$ . Since $\mathcal{Q}$ is Bézoutian, the base locus $Z(L_{\textbf{p}})$ is $0$ -dimensional by 2.1 and 3.16. Let $p_{9}$ be the Cayley–Bacharach point of $\mathbf{p}$ . By 3.3, we have that $\mathcal{Q}_{77}\leq\mathcal{Q}_{10}$ , and so we prove that if $p_{9}=p_{i}$ for some $p_{i}$ involved in a triple or sextuple of $\mathcal{Q}$ , then either $\mathcal{Q}\leq\mathcal{Q}_{77}$ or $\mathcal{Q}\leq\mathcal{Q}_{10}$ .

Suppose $p_{9}$ coincides with $p_{i}$ for some $i$ involved in a conic condition $J\in\mathcal{J}$ . Without loss of generality, $i=1$ and $J=123456$ . Then $C=q_{J}\ell_{78}$ is a reducible cubic containing $\mathbf{p}$ , hence $C\in L_{\mathbf{p}}$ . By Bézout’s Theorem, the conic $q_{J}$ cannot contain the five points $p_{2},\ldots,p_{6}$ and a scheme of length two supported on $p_{1}$ : if this was the case, the intersection of $q_{J}$ with an irreducible cubic in $L_{\mathbf{p}}$ would have degree (at least) $7$ . Hence, $\ell_{78}$ contains $p_{1}$ . Therefore $\mathcal{Q}\leq\mathcal{Q}_{10}$ .

Now, suppose that $p_{9}$ coincides with $p_{i}$ for some $i$ in a line condition $I\in\mathcal{I}$ . Suppose $i\notin J$ for any $J\in\mathcal{J}$ ; if this is not the case, then $\mathcal{Q}\leq\mathcal{Q}_{10}$ as in the previous case. Without loss of generality $i=1$ and $I=123$ . Then $C=\ell_{123}q_{45678}$ is a reducible cubic containing $\mathbf{p}$ , hence $C\in L_{\mathbf{p}}$ . Similarly to the previous case, by Bézout’s Theorem, the line $\ell_{123}$ does not contain the subscheme of $Z(L_{\textbf{p}})$ of length two supported at $p_{1}$ . Hence, the quadric $q_{45678}$ contains $p_{1}$ . Since $\mathcal{Q}$ does not contain a conic condition involving $i=1$ , the quadric $q_{145678}$ must be the union of two lines. Up to reordering the points, those two lines are $\ell_{145}$ and $\ell_{678}$ implying that $\mathcal{Q}\leq\mathcal{Q}_{77}$ . ∎

Corollary 3.24.

If the base locus $p_{1},\ldots,p_{9}$ of $L_{\textbf{p}}$ is a reduced $0$ -dimensional scheme, then for any sextuple $J\subset\{1,2,\ldots,9\}$ , the points $\{p_{j}\}_{j\in J}$ lie on a quadric if and only if $\{p_{i}\}_{i\not\in J}$ lie on a line.

A candidate quatroid $\mathcal{Q}=(\mathcal{I},\mathcal{J})$ is called exhaustive if every $i\in\{1,2,\ldots,8\}$ is involved in some $I\in\mathcal{I}$ or $J\in\mathcal{J}$ . 3.23 has the following strong consequence.

Corollary 3.25.

Let $\mathcal{Q}$ be an exhaustive Bézoutian quatroid such that $\mathcal{Q}\not\leq\mathcal{Q}^{\prime}$ for every $\mathcal{Q}^{\prime}$ in the orbit of $\mathcal{Q}_{10}$ . Then for every $\textbf{p}\in\mathcal{S}_{\mathcal{Q}}$ , the base locus $Z(L_{\textbf{p}})$ is reduced.

3.25 says that there exist quatroids for which every realization gives a reduced base locus. We say that these quatroids, themselves, have reduced base locus. In this case, knowing the matroid underlying the reduced base locus of the pencil of cubics is equivalent to knowing the quatroid underlying any $8$ -subset of the base locus by 3.24.

Lemma 3.26.

The pair $\mathcal{Q}_{63}=(\{123,145,246,356\},\{125678,134678,234578\})$ is not $\mathbb{C}$ -representable.

Proof.

Suppose towards a contradiction that $\mathbf{p}$ represents $\mathcal{Q}_{63}$ . Note that $\mathcal{Q}_{63}$ is Bézoutian, exhaustive, and $\mathcal{Q}_{63}\not\leq\mathcal{Q}^{\prime}$ for every $\mathcal{Q}^{\prime}$ in the orbit of $\mathcal{Q}_{10}$ . By 3.24, $\mathcal{Q}_{63}$ has reduced base locus and the matroid underlying $Z(L_{\textbf{p}})$ , as represented by nonbases, is

M_{Z(L_{\textbf{p}})}=\{123,145,246,356,349,259,169\}.

This is the Fano matroid, which is famously not $\mathbb{C}$ -representable [28, Prop. 6.4.8]. Hence $\mathcal{Q}_{63}$ is not $\mathbb{C}$ -representable. ∎

Inspired by the proof of 3.26, we name $\mathcal{Q}_{63}$ the Fano candidate.

4. Irreducibility of quatroid strata

In this section, we establish which realization spaces of quatroids are irreducible.

Theorem 4.1.

For every quatroid $\mathcal{Q}$ except those in the orbit of $\mathcal{Q}_{41}$ , the realization space $\mathcal{S}_{\mathcal{Q}}$ is irreducible. If $\mathcal{Q}$ is in the orbit of $\mathcal{Q}_{41}$ , then $\mathcal{S}_{\mathcal{Q}}$ is the union of two irreducible components.

The statement for $\mathcal{Q}_{41}$ has been shown in 3.22. The rest of the proof of Theorem 4.1 is built on a series of reductions, inspired by [9, Thm. 4.5]. To state them precisely, we introduce a definition.

Definition 4.2.

Let $\mathcal{Q}=(\mathcal{I},\mathcal{J})$ be a quatroid and let $i\in\{1,2,\ldots,8\}$ . The deletion of $i$ from $\mathcal{Q}$ is

{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathcal{Q}-\{i\}}=(\{I\in\mathcal{I}\mid i\notin I\},\{J\in\mathcal{J}\mid j\notin J\}).

Proposition 4.3.

Let $\mathcal{Q}=(\mathcal{I},\mathcal{J})$ be a quatroid and $i\in\{1,2,\ldots,8\}$ . Suppose one of the following conditions holds:

•

$i$ is in at most two elements $I_{1},I_{2}$ of $\mathcal{I}$ and no elements of $\mathcal{J}$ ,
•

$i$ is in at most one element of $\mathcal{J}$ and no elements of $\mathcal{I}$ .

Then $\mathcal{S}_{\mathcal{Q}}$ is irreducible if and only if $\mathcal{S}_{\mathcal{Q}-\{i\}}$ is irreducible. In particular, if $\mathcal{Q}=(\mathcal{I},\emptyset)$ and $|\mathcal{I}|\leq 6$ then $\mathcal{S}_{\mathcal{Q}}$ is irreducible.

Proof.

First, notice that if $i$ appears in no elements of $\mathcal{I}$ and $\mathcal{J}$ , then $\mathcal{Q}=\mathcal{Q}-\{i\}$ . Hence, we may assume $i$ appears in at least one element of $\mathcal{I}\cup\mathcal{J}$ . Without loss of generality, let $i=8$ .

Suppose first $i$ belongs to exactly one element of $I\in\mathcal{I}$ , say $I=678$ . Let $\pi:(\mathbb{P}_{\mathbb{C}}^{2})^{8}\to(\mathbb{P}_{\mathbb{C}}^{2})^{7}$ be the projection on the first seven factors. Let $X=\pi(\mathcal{S}_{\mathcal{Q}})$ . First, observe $\mathcal{S}_{\mathcal{Q}-\{i\}}$ is a dense subset of $X\times\mathbb{P}_{\mathbb{C}}^{2}$ : indeed, the line and conic conditions of $\mathcal{Q}-\{i\}$ are the line and conic conditions of $\mathcal{Q}$ not involving $i$ , so they are satisfied by $\mathbf{p}\in(\mathbb{P}_{\mathbb{C}}^{2})^{8}$ if and only if $\pi(\mathbf{p})\in X$ . In particular, $X$ is irreducible if and only if $\mathcal{S}_{\mathcal{Q}-\{i\}}$ is irreducible. Moreover, $\mathcal{S}_{\mathcal{Q}}$ is an open dense subset of a $\mathbb{P}_{\mathbb{C}}^{1}$ -bundle over $X$ , embedded in $X\times\mathbb{P}_{\mathbb{C}}^{2}$ : the fiber over $\mathbf{p}^{\prime}\in X$ is an open subset of the line $\ell_{67}$ . In particular, $X$ is irreducible if and only if $\mathcal{S}_{\mathcal{Q}}$ is irreducible.

If $i$ belongs to exactly one element $J\in\mathcal{J}$ , the proof is similar. Suppose $J=345678$ . The proof follows the same argument described above, with the only difference that $\mathcal{S}_{\mathcal{Q}}$ is an open subset of a fiber bundle over $X$ whose fiber at $\mathbf{p}$ is the conic $q_{34567}$ .

If $i$ belongs to two exactly two elements $I_{1},I_{2}\in\mathcal{I}$ , suppose without loss of generality $I_{1}=678$ and $I_{2}=458$ . Again, the proof is similar. In this case $\mathcal{S}_{\mathcal{Q}}$ is birational to $X$ , with a birational map given by $\phi\colon X\to\mathcal{S}_{\mathcal{Q}}$ defined by $\phi(\mathbf{p}^{\prime})=(\mathbf{p}^{\prime},p_{8})$ where $p_{8}=\ell_{45}\cap\ell_{78}$ . ∎

A second reduction is built on the Cayley–Bacharach construction introduced in Section 3. Let $\mathcal{Q}$ be an exhaustive Bézoutian quatroid such that $\mathcal{Q}\not\leq\mathcal{Q}^{\prime}$ for every $\mathcal{Q}^{\prime}$ in the orbit of $\mathcal{Q}_{10}$ , and let $\mathbf{p}$ be any of its realizations. Then $Z(L_{\mathbf{p}})$ is reduced by 3.25. The underlying matroid of $Z(L_{\mathbf{p}})$ is well-defined by 3.24 and so is the set of quatroids obtained by deleting any of the nine points. If $\mathcal{Q}^{\prime}$ is one such quatroid, write $\mathcal{Q}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\leadsto}\mathcal{Q}^{\prime}$ and say that $\mathcal{Q}^{\prime}$ is a modification of $\mathcal{Q}$ .

Lemma 4.4.

Let $\mathcal{Q},\mathcal{Q}^{\prime}$ be quatroids with $\mathcal{Q}\leadsto\mathcal{Q}^{\prime}$ . If $\mathcal{S}_{\mathcal{Q}^{\prime}}$ is irreducible then so is $\mathcal{S}_{\mathcal{Q}}$ .

Proof.

Let $\mathcal{Q}\leadsto\mathcal{Q}^{\prime}$ and write $\mathcal{M}$ for the matroid of the base locus $Z(L_{\textbf{p}})$ for any $\textbf{p}\in\mathcal{S}_{\mathcal{Q}}$ . Consider the realization space $\mathcal{S}_{\mathcal{M}}\subseteq(\mathbb{P}_{\mathbb{C}}^{2})^{9}$ of that matroid. The variety $\mathcal{S}_{\mathcal{Q}}$ is isomorphic to the variety $\mathcal{K}\subseteq\mathcal{S}_{\mathcal{M}}\subseteq(\mathbb{P}_{\mathbb{C}}^{2})^{9}$ of configurations which are complete intersections of type $(3,3)$ . Since $\mathcal{S}_{\mathcal{Q}^{\prime}}$ is assumed to be irreducible, the map from $\mathcal{K}$ to $\mathcal{S}_{\mathcal{Q}^{\prime}}$ given by forgetting one of the nine points shows that $\mathcal{K}$ is birational to $\mathcal{S}_{\mathcal{Q}^{\prime}}$ . ∎

Based on these results, given a quatroid $\mathcal{Q}$ , there are four ways to reduce the problem of determining irreducibility of $\mathcal{S}_{\mathcal{Q}}$ :

(i)

$\mathcal{Q}$ reduces to $\mathcal{Q}^{\prime}=\mathcal{Q}-\{i\}$ with $i$ involved in at most two line conditions and no conic condition,
(ii)

$\mathcal{Q}$ reduces to $\mathcal{Q}^{\prime}=\mathcal{Q}-\{i\}$ with $i$ involved in no line condition and one conic condition,
(iii)

$\mathcal{Q}$ has at most six line conditions and no conic condition,
(iv)

$\mathcal{Q}\leadsto\mathcal{Q}^{\prime}$ .

In these cases, if $\mathcal{S}_{\mathcal{Q}^{\prime}}$ is irreducible, so is $\mathcal{S}_{\mathcal{Q}}$ .

Lemma 4.5.

All quatroids in orbits other than the $24$ orbits represented by

	$\displaystyle\mathcal{Q}_{4},\mathcal{Q}_{5},\mathcal{Q}_{13},\mathcal{Q}_{19},\mathcal{Q}_{21},\mathcal{Q}_{22},\mathcal{Q}_{25},\mathcal{Q}_{38},\mathcal{Q}_{41},\mathcal{Q}_{46},\mathcal{Q}_{49},\mathcal{Q}_{51},$
	$\displaystyle\mathcal{Q}_{53},\mathcal{Q}_{57},\mathcal{Q}_{58},\mathcal{Q}_{59},\mathcal{Q}_{62},\mathcal{Q}_{66},\mathcal{Q}_{73},\mathcal{Q}_{76},\mathcal{Q}_{81},\mathcal{Q}_{82},\mathcal{Q}_{101},\mathcal{Q}_{123}$

have irreducible realization spaces.

Proof.

The result is obtained by iteratively applying reductions (i)–(iii). The proof is computational and the reductions are performed by the function QuatroidReductions() in the Quatroids.jl package. This function produces the file ReductionProofs.txt. ∎

Lemma 4.6.

The following relationships hold

	$\displaystyle\mathcal{Q}_{4}\leadsto\mathcal{Q}_{17},\quad\,\,\,\,\,\mathcal{Q}_{5}\leadsto\mathcal{Q}_{25},\quad\,\,\,\,\mathcal{Q}_{13}\leadsto\mathcal{Q}_{20},\quad\,\,\mathcal{Q}_{19}\leadsto\mathcal{Q}_{60},\quad\,\,\mathcal{Q}_{22}\leadsto\mathcal{Q}_{30},\quad\,\,\mathcal{Q}_{46}\leadsto\mathcal{Q}_{25},\quad\,\,$
	$\displaystyle\mathcal{Q}_{49}\leadsto\mathcal{Q}_{25},\quad\,\,\mathcal{Q}_{51}\leadsto\mathcal{Q}_{25},\quad\,\,\mathcal{Q}_{53}\leadsto\mathcal{Q}_{40},\quad\,\,\mathcal{Q}_{57}\leadsto\mathcal{Q}_{30},\quad\,\,\mathcal{Q}_{59}\leadsto\mathcal{Q}_{32},\quad\,\,\mathcal{Q}_{62}\leadsto\mathcal{Q}_{32},\quad\,\,$
	$\displaystyle\mathcal{Q}_{66}\leadsto\mathcal{Q}_{36},\quad\,\,\mathcal{Q}_{73}\leadsto\mathcal{Q}_{37},\quad\,\,\mathcal{Q}_{76}\leadsto\mathcal{Q}_{39},\quad\,\,\mathcal{Q}_{81}\leadsto\mathcal{Q}_{45},\quad\,\,\mathcal{Q}_{82}\leadsto\mathcal{Q}_{43}.$

In particular, the realization spaces of the quatroids in the same orbits as

\mathcal{Q}_{4},\mathcal{Q}_{13},\mathcal{Q}_{19},\mathcal{Q}_{22},\mathcal{Q}_{57},\mathcal{Q}_{59},\mathcal{Q}_{62},\mathcal{Q}_{66},\mathcal{Q}_{73},\mathcal{Q}_{81},\mathcal{Q}_{82}

are all irreducible.

Proof.

All quatroids in the statement satisfy the conditions of 3.25: this is verified computationally, see Table 3. The relation $\mathcal{Q}\leadsto\mathcal{Q}^{\prime}$ is also verified computationally via the function Modifications(Q) in the package Quatroids.jl. Note that each quatroid on the right-hand-side of a $\leadsto$ sign, other than $\mathcal{Q}_{25}$ , has already been shown to have irreducible realization space, therefore 4.4 guarantees that the corresponding quatroids on the left-hand-side have irreducible realization spaces. ∎

The following result completes the proof of Theorem 4.1.

Theorem 4.7.

The realization spaces of quatroids

\mathcal{Q}_{21},\mathcal{Q}_{25},\mathcal{Q}_{38},\mathcal{Q}_{58},\mathcal{Q}_{101},\mathcal{Q}_{123}

are irreducible.

Proof.

The auxiliary files include Macaulay2 scripts to reproduce the computations in this proof.

We normalize elements of the realization space in two different ways. Either we normalize a conic conditions to be $x_{0}^{2}-x_{1}x_{2}$ and then three points on such conic to be $[0:1:0],[0:0:1],[1:1:1]$ , or we normalize four points in $\mathbb{P}^{2}_{\mathbb{C}}$ to be $[1:0:0],[0:1:0],[0:0:1],[1:1:1]$ . After this normalization, a point in the realization space of quatroid $\mathcal{Q}_{i}$ is represented by one of the following $3\times 8$ matrices:

\begin{array}[]{rlrl}A_{21}&=\left[\begin{array}[]{cccccccc}0&0&0&1&1&z_{3}&z_{4}&z_{5}\\ 1&0&1&1&z_{2}&z_{3}^{2}&z_{4}^{2}&z_{5}^{2}\\ 0&1&z_{1}&1&1&1&1&1\end{array}\right],&A_{25}&=\left[\begin{array}[]{cccccccc}1&0&1&0&1&1&z_{3}&z_{4}\\ 0&1&z_{1}&0&0&1&1&z_{5}\\ 0&0&0&1&z_{2}&1&1&z_{6}\end{array}\right],\\ {}\hfil\\ A_{38}&=\left[\begin{array}[]{cccccccc}1&1&0&0&1&z_{3}&z_{4}&z_{5}\\ 1&z_{2}&1&0&1&z_{3}^{2}&z_{4}^{2}&z_{5}^{2}\\ z_{1}&z_{1}&0&1&1&1&1&1\end{array}\right],&A_{58}&=\left[\begin{array}[]{cccccccc}1&1&0&0&z_{1}&1&z_{2}&z_{3}\\ z_{5}&1&1&0&z_{1}^{2}&1&z_{2}^{2}&z_{3}^{2}\\ z_{4}&z_{4}&0&1&1&1&1&1\end{array}\right],\\ {}\hfil\\ A_{101}&=\left[\begin{array}[]{cccccccc}1&0&1&1&0&1&0&z_{5}\\ 0&1&z_{1}&z_{2}&0&0&1&1\\ 0&0&0&0&1&z_{3}&z_{4}&z_{4}\\ \end{array}\right],&A_{123}&=\left[\begin{array}[]{cccccccc}0&0&0&0&1&z_{3}&z_{4}&z_{5}\\ 1&0&1&1&1&z_{3}^{2}&z_{4}^{2}&z_{5}^{2}\\ 0&1&z_{1}&z_{2}&1&1&1&1\end{array}\right],\end{array}

where the vectors $(z_{1},\ldots,z_{r})$ are subject to the determinantal relations imposed by the line and conic condition of a quatroid.

We verify in the auxiliary files that the additional conditions cut out an irreducible variety. To prove irreducibility, we reduce to a hypersurface case. To prove irreducibility of a hypersurface, we observe that its singular locus has codimension at least three (in the ambient space). ∎

That every quatroid stratum other than $\mathcal{S}_{\mathcal{Q}_{41}}$ is irreducible guarantees that the notion of generic point makes sense on these strata. For $\mathcal{S}_{\mathcal{Q}_{41}}$ , one may perform every computation exhaustively, since up to the $\mathrm{PGL}_{2}$ -action, this realization space is just two complex conjugate points.

5. The generic number of rational cubics through quatroid strata: a lower bound

Recall from 2.2 that for $\textbf{p}\in\mathcal{P}$ the value $d_{\textbf{p}}$ is the number of rational cubics through p, counted with multiplicity. The irreducibility of a realization space $\mathcal{S}_{\mathcal{Q}}$ guarantees that this number is constant on a dense open subset of $\mathcal{S}_{\mathcal{Q}}$ . Denote by ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}d_{\mathcal{Q}}}$ this generic value. Our goal is to compute $d_{\mathcal{Q}}$ for all quatroids.

In the case of $\mathcal{Q}_{41}$ , it is classically known that, for every $\mathbf{p}\in\mathcal{S}_{\mathcal{Q}_{41}}$ , $d_{\mathbf{p}}=0$ . Indeed, the singular cubics on the pencil $L_{\mathbf{p}}$ are four fully reducible cubics $f_{1},\ldots,f_{4}$ each of the form $\ell_{1}\ell_{2}\ell_{3}$ with $\ell_{1},\ell_{2},\ell_{3}$ linearly independent. We refer, for instance, to [2] for details.

Let ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathfrak{B}}$ be the set of Bézoutian quatroids. By 2.1, if $\mathcal{Q}\notin\mathfrak{B}$ , then $d_{\mathcal{Q}}=0$ . For Bézoutian quatroids, we obtain a lower bound on $d_{\mathcal{Q}}$ via a semicontinuity argument.

Lemma 5.1.

The function $\textbf{p}\mapsto d_{\textbf{p}}$ is lower semicontinuous.

Proof.

By 2.4, $L_{\mathbf{p}}\cap\mathcal{D}$ is either infinite, in which case $d_{\mathbf{p}}=0$ , or a $0$ -dimensional scheme of degree $12$ . The number $d_{\textbf{p}}$ counts, with multiplicity, the number of intersection points $L_{\textbf{p}}\cap\mathcal{D}$ not occurring on the variety of reducible cubics. Equivalently $r_{\mathbf{p}}=12-d_{\mathbf{p}}$ is the degree of the subscheme of $L_{\mathbf{p}}\cap\mathcal{D}$ supported on the subvariety of reducible cubics. Since this subvariety is closed, upper semicontinuity of $r_{\mathbf{p}}$ follows and we conclude that $d_{\mathbf{p}}$ is lower semicontinuous. ∎

Corollary 5.2.

Let $\mathcal{Q}$ be a quatroid. The number $\max_{\textbf{p}\in\mathcal{S}_{\mathcal{Q}}}(d_{\textbf{p}})$ is well-defined and equal to $d_{\mathcal{Q}}$ . In particular, for any $\textbf{p}\in\overline{\mathcal{S}_{\mathcal{Q}}}$ we have that $d_{\textbf{p}}\leq d_{\mathcal{Q}}$ . Consequently, $d_{\mathcal{Q}^{\prime}}\leq d_{\mathcal{Q}}$ if $\mathcal{Q}^{\prime}\leq\mathcal{Q}$ .

A consequence of 5.2 is that any point $\mathbf{p}\in\mathcal{S}_{\mathcal{Q}}$ provides a lower bound $d_{\mathbf{p}}\leq d_{\mathcal{Q}}$ . We compute the number $d_{\mathbf{p}}$ for the rational representatives in RationalRepresentatives.txt using a symbolic calculation. We now discuss how to perform that calculation.

Input: A rational configuration

\textbf{p}=(p_{1},\ldots,p_{8})\in\mathcal{P}

representing a Bézoutian quatroid

Output:

d_{\textbf{p}}

1 initialize

I

to be the ideal of the points

p_{1},\ldots,p_{8}\in\mathcal{P}

2 Find a rational basis

\{C_{0},C_{1}\}

for

I

in degree

3

3 Set

F=\textrm{disc}(t_{0}C_{0}+t_{1}C_{1})\in\mathbb{Q}[t_{0},t_{1}]

4 Factor

F

over

\mathbb{Q}

Optionally, determine the multiplicities of each solution from the powers of the factors

5 Let

R

be the

k

linear factors of

F

with roots

\{[t_{0}^{(i)}:t_{1}^{(i)}]\}_{i=1}^{k}

6 For each

i=1,\ldots,k

factor the cubic

t_{0}^{(i)}C_{0}+t_{1}^{(i)}C_{1}

7 Let

r_{\textbf{p}}

be the number of cubics which factor nontrivially, counted with multiplicity

Optionally, compute the type of each reducible (see Table 1)

8 return $12-r_{\textbf{p}}$

Algorithm 2 Symbolic Computation of

d_{\textbf{p}}

Lemma 5.3.

Let $\textbf{p}\in\mathcal{P}$ be a rational representative of a Bézoutian quatroid. Let $C$ be a reducible cubic through p defined by a cubic form $f$ . Then $f\in\mathbb{P}{S}^{3}\mathbb{Q}^{3}$ and it factors over $\mathbb{Q}$ .

Proof.

Since $C$ is a reducible cubic, then $C$ is a union of either (a) three lines or (b) a line and a conic. In case (a), since p represents a Bézoutian quatroid, then each line contains (at least) two elements of $\mathbf{p}$ by the pigeonhole principle. In particular, since $\mathbf{p}$ is rational, each of the three lines has a rational equation. Similarly, in case (b), since p is Bézoutian, the pigeonhole principal implies that there are at least two points on the line and at least five points on the conic. The equation of a conic through five rational points, as well as the equation of a line through two rational points, has rational coefficients. Consequently, in either (a) or (b), the components of $C$ must be realized by rational forms whose product is $f$ , up to scaling. ∎

algorithm 2 uses 5.3 to find the reducible cubics, and their multiplicities, on a pencil.

Proposition 5.4.

algorithm 2 is correct, in the sense that its output is the number $d_{\mathbf{p}}$ . Moreover, the optional steps correctly determine the multiplicities of each cubic through p and the type of each reducible cubic in the sense of Table 1.

Proof.

Consider a rational configuration $\textbf{p}\in\mathcal{S}_{\mathcal{Q}}$ for a Bézoutian quatroid $\mathcal{Q}$ . By 2.1, $L_{\mathbf{p}}=\langle C_{1},C_{2}\rangle$ and since $\mathbf{p}$ is rational the cubics $C_{1},C_{2}$ can be taken to have rational coefficients. Step $3$ parametrizes the pencil $L_{\mathbf{p}}$ and evaluates the discriminant of plane cubics, $\operatorname{disc}$ , on it. By the separability of $\mathbb{Q}$ , no irreducible factor has a multiple root, and so the multiplicities of the linear factors over $\mathbb{C}$ are witnessed by the powers of the irreducible factors over $\mathbb{Q}$ . This shows that the first optional step correctly detects the multiplicities.

By 5.3 the reducible cubics through rational configurations are themselves defined over $\mathbb{Q}$ , hence all of them appear in the $\mathbb{Q}$ -factorization of $F$ as linear factors. Moreover 5.3 also states that the $\mathbb{C}$ -factorization of each such reducible cubic is realized over $\mathbb{Q}$ . Consequently, those which factor nontrivially are precisely the reducible cubics on $L_{\mathbf{p}}$ , counted with the appropriate multiplicity, and so their number is $r_{\textbf{p}}$ . Since $12=d_{\textbf{p}}+r_{\textbf{p}}$ , the number $d_{\textbf{p}}$ is correctly determined. The type of each individual reducible cubic may be determined in several elementary ways. In countReducibleCubics.m2, we determine the type via their singular locus. ∎

Theorem 5.5.

The following numbers are lower bounds for $d_{\mathcal{Q}_{i}}$ , where $\mathcal{Q}_{i}$ is a Bézoutian quatroid:

i\,\,\,

d_{\mathcal{\textbf{p}}_{i}}

d_{\mathcal{\textbf{p}}_{i}}

d_{\mathcal{\textbf{p}}_{i}}

d_{\mathcal{\textbf{p}}_{i}}

Moreover, for each rational representative $\textbf{p}_{i}$ , the irreducible cubics through $\textbf{p}_{i}$ all appear with multiplicity one and the reducible cubics are either conic+secant lines or triangles (see Table 1).

Proof.

For each quatroid $\mathcal{Q}_{i}$ , with $i\neq 41,63$ , we apply algorithm 2 to the $\mathbb{Q}$ -representative $\textbf{p}_{i}\in\mathcal{S}_{\mathcal{Q}_{i}}$ in RationalRepresentatives.txt. 5.2 implies the results are lower bounds for $d_{{\mathcal{Q}_{i}}}$ . The optional steps are taken to determine the types and multiplicities of all cubics. ∎

Example 5.6.

We detail algorithm 2 on the representative

\textbf{p}=\left[\!\begin{array}[]{cccccccc}-2&2&2&1&-2&1&2&-2\\ 2&-2&-2&2&-1&-2&-1&1\\ 0&2&-2&1&-1&1&-1&2\end{array}\!\right]

of $\mathcal{Q}_{43}=(\{123,145,167,246,357\},\{\})$ . A basis for the ideal of $\mathbf{p}$ in degree $3$ is given by the cubics

	$\displaystyle C_{0}$	$\displaystyle=2x_{0}^{2}x_{1}+2x_{0}x_{1}^{2}-2x_{0}^{2}x_{2}-3x_{0}x_{1}x_{2}-2x_{1}^{2}x_{2}+x_{0}x_{2}^{2}+x_{1}x_{2}^{2}+x_{2}^{3}$
	$\displaystyle C_{1}$	$\displaystyle=8x_{0}^{3}+8x_{1}^{3}+8x_{0}^{2}x_{2}-3x_{0}x_{1}x_{2}+8x_{1}^{2}x_{2}-29x_{0}x_{2}^{2}-29x_{1}x_{2}^{2}-19x_{2}^{3}$

Evaluating the cubic discriminant on the pencil $t_{0}C_{0}+t_{1}C_{1}$ and factoring it yields

\left(t_{1}\right)^{3}\left(t_{0}-19\,t_{1}\right)^{2}\left(t_{0}-13\,t_{1}\right)^{2}\left(t_{0}-5\,t_{1}\right)^{2}\left(375\,t_{0}^{3}-6\,047\,t_{0}^{2}t_{1}-25\,539\,t_{0}t_{1}^{2}+232\,083\,t_{1}^{3}\right).

We see that there are at most three reducible cubics which occur with multiplicity $2$ and at most one reducible cubic occurring with multiplicity $3$ . Since the nonlinear factor appears with multiplicity one, its $\mathbb{C}$ factors appear with multiplicity one as well. By factoring $C_{[t_{0}:t_{1}]}=t_{0}C_{0}+t_{1}C_{1}$ for $[t_{0}:t_{1}]\in\{[0:1],[19:1],[13:1],[5:1]\}$ we see that all these cubics are reducible:

	$\displaystyle C_{[0:1]}$	$\displaystyle=\left(x_{1}-x_{2}\right)\left(x_{0}-x_{2}\right)\left(2\,x_{0}+2\,x_{1}+x_{2}\right)$
	$\displaystyle C_{[19:1]}$	$\displaystyle=\left(x_{0}+x_{1}\right)\left(4\,x_{0}^{2}+15\,x_{0}x_{1}+4\,x_{1}^{2}-15\,x_{0}x_{2}-15\,x_{1}x_{2}-5\,x_{2}^{2}\right)$
	$\displaystyle C_{[13:1]}$	$\displaystyle=\left(x_{0}+x_{1}-3\,x_{2}\right)\left(4\,x_{0}^{2}+9\,x_{0}x_{1}+4\,x_{1}^{2}+3\,x_{0}x_{2}+3\,x_{1}x_{2}+x_{2}^{2}\right)$
	$\displaystyle C_{[5:1]}$	$\displaystyle=\left(x_{0}+x_{1}+x_{2}\right)\left(4\,x_{0}^{2}+x_{0}x_{1}+4\,x_{1}^{2}-5\,x_{0}x_{2}-5\,x_{1}x_{2}-7\,x_{2}^{2}\right)$

The first is the union of three lines with no mutual intersection (i.e. a triangle). The remaining three reducible cubics are conic+secant lines. As a result, $d_{\textbf{p}}=12-3-3\cdot 2=3$ . We display the configuration p and these reducible cubics in Figure 5. ∎

6. The generic number of rational cubics through quatroid strata: an upper bound

In this section, we construct upper bounds for each $d_{\mathcal{Q}}$ , and observe that they coincide with the lower bounds listed in Theorem 5.5. Bounding $d_{\mathcal{Q}}$ from above is equivalent to bounding $r_{\textbf{p}}=12-d_{\textbf{p}}$ from below, for every $\textbf{p}\in\mathcal{S}_{\mathcal{Q}}$ . The value $r_{\textbf{p}}$ occurs as a sum of intersection multiplicities, which is bounded by a sum of multiplicities in the sense of (1):

(2)		$\displaystyle r_{\textbf{p}}=\sum_{\textrm{reducible }C\in L_{\textbf{p}}\cap\mathcal{D}}\textrm{imult}_{C}(L_{\textbf{p}},\mathcal{D})$	$\displaystyle\geq\sum_{\textrm{reducible }C\in L_{\textbf{p}}\cap\mathcal{D}}\textrm{min}_{C\in L\in\textrm{Gr}(2,S^{3}\mathbb{C}^{3})}(\textrm{imult}_{C}(L,\mathcal{D}))$
(2)			$\displaystyle=\sum_{\textrm{reducible }C\in L_{\textbf{p}}\cap\mathcal{D}}\textrm{mult}_{\mathcal{D}}(C)=:{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}m_{\textbf{p}}}.$

Hence, we obtain the bound $d_{\textbf{p}}=12-r_{\textbf{p}}\leq 12-m_{\textbf{p}}$ . Define ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}m_{\mathcal{Q}}}=\min_{\textbf{p}\in\mathcal{S}_{\mathcal{Q}}}(m_{\textbf{p}})$ .

Lemma 6.1.

Let $\mathcal{Q}$ be a Bézoutian quatroid. The value $m_{\mathcal{Q}}$ is well-defined and equal to $m_{\textbf{p}}$ for generic $\textbf{p}\in\mathcal{S}_{\mathcal{Q}}$ .

Proof.

The statement follows from the fact that the multiplicity of a point on a variety is an upper semicontinuous function of the point. ∎

With these definitions, we can bound $d_{\mathcal{Q}}$ from above:

(3)

\displaystyle d_{\mathcal{Q}}=\max_{\textbf{p}\in\mathcal{S}_{\mathcal{Q}}}(d_{\textbf{p}})=\max_{\textbf{p}\in\mathcal{S}_{\mathcal{Q}}}(12-r_{\textbf{p}})=12-\min_{\textbf{p}\in\mathcal{S}_{\mathcal{Q}}}(r_{\textbf{p}})\leq 12-\min_{\textbf{p}\in\mathcal{S}_{\mathcal{Q}}}(m_{\textbf{p}})=12-m_{\mathcal{Q}}

The value $12-m_{\mathcal{Q}}$ may be thought of, for now, as an expected number of rational cubics through $\mathcal{Q}$ .

6.1. An upper bound for $d_{\mathcal{Q}}$ through multiplicities of reducible cubics

Equation (3) gives us a way to compute upper bounds for $d_{\mathcal{Q}}$ . Crucially, as shown in 3.2 a Bézoutian quatroid $\mathcal{Q}$ determines the reducible cubics through any of its representations, and hence the range of the summations in (2). In this section, we determine the values $\textrm{mult}_{\mathcal{D}}(C)$ of the summands.

The multiplicity of a reducible cubic on the discriminant $\mathcal{D}$ is invariant under the action of $\mathrm{PGL}_{2}$ on $\mathbb{P}S^{3}\mathbb{C}^{3}$ . Hence, the contribution of a reducible cubic $C$ to the number $m_{\textbf{p}}$ only depends on the orbit of $C$ under this action. The characterization of these orbits is classical; we refer to [22] for a modern reference. For each orbit, we record in Table 1 its name, a representative $C\in\mathbb{Q}[x_{0},x_{1},x_{2}]$ , and the multiplicity $\mathrm{mult}_{\mathcal{D}}(C)$ . We refer to Section 6.2 for a proof of these multiplicities (see 6.8). Each orbit as well as the orbit-closure containments are illustrated in Figure 6.

Name	Representative	Multiplicity
Nodal Cubic	$x_{0}x_{1}^{2}-x_{2}^{2}(x_{2}-x_{0})$	$1$
Cuspidal Cubic	$x_{0}x_{1}^{2}-x_{2}^{3}$	$2$
Conic+Secant Line	$x_{0}(x_{0}^{2}+x_{1}^{2}-x_{2}^{2})$	$2$
Conic+Tangent Line	$x_{0}(x_{0}x_{1}+x_{2}^{2})$	$3$
Triangle	$x_{0}x_{1}x_{2}$	$3$
Asterisk	$x_{0}x_{1}(x_{0}+x_{1})$	$4$
Double Line + Transverse Line	$x_{0}^{2}x_{1}$	$6$
Triple Line	$x_{0}^{3}$	$8$

Table 1. A name, representative, and the multiplicity for each singular cubic orbit.

Figure 6. Orbit closure containments in the discriminant of singular cubics

A consequence of Theorem 5.5 along with the orbit-closure containments of singular cubics, imply that each reducible cubic through a generic configuration of a Bézoutian quatroid stratum is either a triangle or a conic+secant. In light of this fact, we associate to each quatroid stratum $\mathcal{S}_{Q}$ a string of symbols of the form ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\varnothing^{i}\triangle^{j}}$ which indicates that a generic representative of $\mathcal{S}_{\mathcal{Q}}$ is contained in $i$ reducible cubics of the form conic+secant, and $j$ triangles. These are listed, in black font, in the column Reducibles of Table 4 and Table 5.

Theorem 6.2.

Let $\mathcal{Q}=(\mathcal{I},\mathcal{J})$ be a Bézoutian quatroid. Then

	$\displaystyle m_{\mathcal{Q}}=$	$\displaystyle 2\cdot\|\mathcal{I}\|+2\cdot\|\mathcal{J}\|-2\cdot\Bigl{\|}\Bigl{\{}\{I,J\}\in\mathcal{I}\times\mathcal{J}\mid\|I\cap J\|=1\Bigr{\}}\Bigr{\|}$
		$\displaystyle-\Bigl{\|}\Bigl{\{}\{I_{1},I_{2}\}\subset\mathcal{I}\mid I_{1}\cap I_{2}=\emptyset\Bigr{\}}\Bigr{\|}-\Bigl{\|}\Bigl{\{}\{I_{1},I_{2},I_{3}\}\mid I_{1}\cup I_{2}\cup I_{3}=\{1,\ldots,8\}\Bigr{\}}\Bigr{\|}.$

Proof.

The proof is an overcount along with a justification of correction terms. For each triple and each sextuple, we count two under the erroneous assumption that every triple and sextuple gives rise to its own conic+secant pair, which by Table 1 has multiplicity two. This accounts for the first two summands. We have directly overcounted conic+secant pairs by the third summand.

The fourth and fifth summands cover the cases when the triple did not give rise to a conic+secant, but rather, a triangle. In one case, two of the three lines of that triangle were counted by $2|\mathcal{I}|$ , contributing a total of $4$ to $m_{\mathcal{Q}}$ , which should have been a $3$ by Table 1. Hence the fourth summand corrects for this. The other case is when all three lines of the triangle were counted by $2|\mathcal{I}|$ , in which case the fourth summand reduced the contribution from $6$ to $4$ , and the fifth and final summand corrects this contribution to $3$ ; this correction term only counts once. ∎

Theorem 6.3.

For all $\mathcal{Q}\in\mathfrak{B}$ the numbers computed by Theorem 5.5 agree with the numbers $12-m_{\mathcal{Q}}$ except for the following $24$ quatroid orbits

\{\mathcal{Q}_{10},\mathcal{Q}_{12},\mathcal{Q}_{18},\mathcal{Q}_{21},\mathcal{Q}_{26},\mathcal{Q}_{29},\mathcal{Q}_{31},\mathcal{Q}_{38},\mathcal{Q}_{47},\mathcal{Q}_{56},\mathcal{Q}_{58},\mathcal{Q}_{72}\}=\{\mathcal{Q}\in\mathfrak{B}\mid\mathcal{Q}\leq\mathcal{Q}_{10},\mathcal{Q}\not\leq\mathcal{Q}_{77}\}

\{\mathcal{Q}_{33},\mathcal{Q}_{34},\mathcal{Q}_{35},\mathcal{Q}_{42},\mathcal{Q}_{48},\mathcal{Q}_{49},\mathcal{Q}_{67},\mathcal{Q}_{68},\mathcal{Q}_{69},\mathcal{Q}_{72},\mathcal{Q}_{77}\}=\{\mathcal{Q}\in\mathfrak{B}\mid\mathcal{Q}\leq\mathcal{Q}_{77}\}.

In the above cases, the number $12-m_{\mathcal{Q}}$ is precisely one more than the associated number in Theorem 5.5.

Proof.

The number $m_{\mathcal{Q}}$ is easily computed via the formula in Theorem 6.2. ∎

That the bound from Theorem 6.2 only fails to be an equality by at most one is a stroke of luck. Consequently, it is enough to show that for each of these $24$ quatroids, a configuration p represents it only if $\textrm{imult}_{C}(L_{\textbf{p}},\mathcal{D})>\textrm{mult}_{\mathcal{D}}(C)$ for some reducible $C$ through p.

6.2. A tighter bound for $d_{\mathcal{Q}}$ through tangent cones of reducible cubics

The expected count of rational cubics through generic $\textbf{p}\in\mathcal{S}_{\mathcal{Q}}$ fails to be correct when the line $L_{\textbf{p}}$ intersects $\mathcal{D}$ at some reducible $C\in\mathcal{D}$ with intersection multiplicity greater than $\textrm{mult}_{\mathcal{D}}(C)$ . Such lines are characterized by the tangent cone to $\mathcal{D}$ at $C$ .

Definition 6.4.

The tangent space to $\mathcal{D}$ at $C\in\mathcal{D}$ is the union of lines $L\subset\mathbb{P}S^{3}\mathbb{C}^{3}$ for which

\textrm{imult}_{C}(L,\mathcal{D})>1,

and is denoted ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textrm{T}_{C}\mathcal{D}}$ . The tangent cone to $\mathcal{D}$ at $C$ is the union of lines $L\subset\mathbb{P}S^{3}\mathbb{C}^{3}$ for which

\textrm{imult}_{C}(L,\mathcal{D})>\textrm{mult}_{\mathcal{D}}(C),

and is denoted ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textrm{TC}_{C}\mathcal{D}}$ . Such lines are said to be tangent to $\mathcal{D}$ at $C$ .

Remark 6.5.

We refer to [30, Ch. 2] for more standard definitions of tangent spaces and tangent cones. The equivalent 6.4 streamlines the analysis necessary for our results. ∎

For $p\in\mathbb{P}^{2}_{\mathbb{C}}$ we let ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}H_{p}}\subset\mathbb{P}S^{3}\mathbb{C}^{3}$ be the hyperplane of cubics which vanish at $p$ . The description of tangent spaces at smooth points of the discriminant is classical.

Remark 6.6.

A point $C\in\mathcal{D}$ is a smooth point of $\mathcal{D}$ if and only if it is a nodal cubic. If $C$ is smooth on $\mathcal{D}$ and $p$ is the node of $C$ then

\textrm{T}_{C}\mathcal{D}=\textrm{TC}_{C}\mathcal{D}=H_{p}.

For details, see [12, Example 1.2.3]. ∎

Table 2 extends the tangent cone description of 6.6 beyond nodal cubics. Write ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}kH_{p}}$ for the hyperplane $H_{p}$ counted with multiplicity $k$ . Moreover, if $\ell\in\mathbb{P}S^{1}{\mathbb{C}}^{3}$ is a linear form, write ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathcal{D}_{\ell}^{\prime}}$ for the subset of $\mathbb{P}S^{3}\mathbb{C}^{3}$ of cubics whose intersection with the line $\{\ell=0\}$ is singular.

Proposition 6.7.

The information in Table 2 is correct.

Proof.

The Macaulay2 script tangentConesPlaneCubics.m2 proves the result symbolically. ∎

Cubic Orbit	Curves	Points	Tangent Cone
Nodal Cubic	$C$	$p$ : the node of $C$	$H_{p}$
Cuspidal Cubic	$C$	$p$ : the cusp of $C$	$2H_{p}$
Conic+Secant Line	$q$ : conic, $\ell$ : line	$(p_{1},p_{2})=q\cap\ell$	$H_{p_{1}}\cup H_{p_{2}}$
Conic+Tangent Line	$q$ : conic, $\ell$ : line	$p=q\cap\ell$	$3H_{p}$
Triangle	$\ell_{1},\ell_{2},\ell_{3}$ : lines	$p_{k}=\ell_{i}\cap\ell_{j}$	$H_{p_{1}}\cup H_{p_{2}}\cup H_{p_{3}}$
Asterisk	$\ell_{1},\ell_{2},\ell_{3}$ : lines	$p=\ell_{1}\cap\ell_{2}\cap\ell_{3}$	$4H_{p}$
Double Line+Transverse Line	$\ell_{1}$ : double line, $\ell_{2}$ : line	$p=\ell_{1}\cap\ell_{2}$	$2H_{p}\cup\mathcal{D}^{\prime}_{\ell}$
Triple Line	$\ell$ : line		$2\mathcal{D}^{\prime}_{\ell}$

Table 2. Descriptions of tangent cones to the discriminant

\mathcal{D}

Remark 6.8.

By definition, the multiplicity of a point $C$ on the discriminant $\mathcal{D}$ is the same as the multiplicity of $C$ on $\textrm{TC}_{C}\mathcal{D}$ . Since $\textrm{TC}_{C}\mathcal{D}$ is a cone over $C$ , the multiplicity of $C$ in $\textrm{TC}_{C}$ coincides with the degree of $\textrm{TC}_{C}$ . Hence one may take the Tangent Cone column of Table 2 as the proof of the Multiplicity column of Table 1. We remark that $\mathcal{D}^{\prime}_{\ell}$ has degree four since it is a cone over the variety of singular binary cubics. ∎

Corollary 6.9.

Let $\textbf{p}\in\mathcal{P}$ . The pencil $L_{\textbf{p}}$ is tangent to $\mathcal{D}$ at $C\in L_{\textbf{p}}$ if and only if $C$ is singular at one of the points in p.

Proof.

This follows from the characterization of the tangent cones given in Table 2. ∎

Proposition 6.10.

If p represents $\mathcal{Q}$ for $\mathcal{Q}\leq\mathcal{Q}^{\prime}$ where $\mathcal{Q}^{\prime}$ is in the same orbit as $\mathcal{Q}_{10}$ , then $L_{\textbf{p}}$ tangent to the discriminant. Conversely, if p is a generic point on $\mathcal{S}_{\mathcal{Q}}$ for some $\mathcal{Q}\not\leq\mathcal{Q}^{\prime}$ for every $\mathcal{Q}^{\prime}$ in the orbit of $\mathcal{Q}_{10}$ , then $L_{\textbf{p}}$ is not tangent to the discriminant.

Proof.

Without loss of generality, assume $\mathcal{Q}^{\prime}=\mathcal{Q}_{10}$ . Recall that $\mathcal{Q}_{10}=(\{123\},\{145678\})$ and observe that $\ell_{123}\cdot q_{145678}$ is a reducible cubic $C$ through any realization. It is singular at $p_{1}$ and so $L_{\textbf{p}}$ is tangent to $\mathcal{D}$ at $C$ . Conversely, for all other quatroids, note that the $d_{\textbf{p}}$ computed from Theorem 5.5 agrees with $12-m_{\mathcal{Q}}$ as computed in Theorem 6.3. Therefore, $m_{\textbf{p}}=r_{\textbf{p}}$ and the result follows. ∎

Proposition 6.11.

Let $\textbf{p}\in\mathcal{P}$ represent a Bézoutian quatroid. The following are equivalent:

(1)

$L_{\textbf{p}}$ is tangent to $\mathcal{D}$ ,
(2)

There exists a cubic $C\in L_{\textbf{p}}$ which is singular at a point of p,
(3)

The base locus $Z(L_{\textbf{p}})$ is nonreduced, with one component of length exactly two,
(4)

$d_{\textbf{p}}\leq 12-m_{\textbf{p}}-1$ .

Proof.

The equivalence of parts (1) and (2) is 6.9. The equivalence of parts (1) and (4) follows directly from the definitions of multiplicity and tangent cone.

If part (2) is true, then so is part (3) since a singular point on a cubic has multiplicity at least two, and thus intersects any other cubic of $L_{\mathbf{p}}$ in a scheme of length at least $2$ . That scheme has length exactly two, because eight points of the base locus are necessarily distinct. To see that part (3) implies part (2), note that if the base locus is not reduced, then all cubics in $L_{\textbf{p}}$ have the same tangent at some point $p$ in p. Up to the action of $\mathrm{PGL}_{2}$ , assume $p=[1:0:0]$ and write the equations of two cubics $C_{0},C_{1}\in L_{\textbf{p}}$ as $x_{1}x_{0}^{2}+F_{0}$ and $x_{1}x_{0}^{2}+F_{1}$ where $x_{1}=0$ is the tangent line of $C_{0}$ and $C_{1}$ at $p$ and $F_{i}$ involves subquadratic terms in $x_{0}$ . The cubic $C_{0}-C_{1}$ is singular at $p$ . ∎

Corollary 6.12.

Let $\mathcal{Q}$ be a Bézoutian quatroid. The base locus $Z(L_{\textbf{p}})$ is nonreduced for every $\textbf{p}\in\mathcal{S}_{\mathcal{Q}}$ if and only if $\mathcal{Q}\leq\mathcal{Q}^{\prime}$ for some $\mathcal{Q}^{\prime}$ in the orbit of $\mathcal{Q}_{10}$ .

We conclude this section by summarizing the results we proved before and completing the proof of the main results of the paper.

Theorem 6.13.

For every $\mathcal{Q}\in\mathfrak{Q}$ , the numbers in the $d_{\mathcal{Q}}$ columns of Table 4 and Table 5 correctly give the number of rational cubics through a generic point on $\mathcal{S}_{\mathcal{Q}}$ .

Proof.

As this is the main result of the paper, we summarize the steps we took to achieve it.

For each representable quatroid $\mathcal{Q}$ , we found an explicit representative $\textbf{p}\in\mathcal{S}_{\mathcal{Q}}$ and applied algorithm 2 to p to compute the number $d_{\textbf{p}}$ of rational cubics through that representative. This gave the lower bound $d_{\mathbf{p}}\leq d_{\mathcal{Q}}$ , as shown in Theorem 5.5. Along the way, we showed that the reducible cubics through those representatives are either triangles or conic+secant lines, showing that such reducible orbits are generic. This allowed us to obtain the formula of Theorem 6.2 for $m_{\mathcal{Q}}$ . Together, these results imply

d_{\textbf{p}}\leq d_{\mathcal{Q}}\leq 12-m_{\mathcal{Q}}.

Theorem 6.3 characterizes when this is an equality. When it is not, 6.12 implies the generic base locus is not reduced and so by 6.11 the inequality tightens to

d_{\textbf{p}}\leq d_{\mathcal{Q}}\leq 12-m_{\mathcal{Q}}-1

which again by Theorem 6.3 is an equality. ∎

In Table 4 and Table 5, those Bézoutian quatroids satisfying any of the conditions in 6.11 are indicated by one additional symbol in their Reducibles column. This symbol, written in blue, indicates the type of the reducible cubic at which the intersection multiplicity of the discriminant and $L_{\textbf{p}}$ is (one) larger than its multiplicity, for a generic quatroid representative p. Hence, given a string of symbols of the form $\varnothing^{i}\triangle^{j}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\square}$ for $\square\in\{\varnothing,\triangle\}$ , one may calculate the value in the column $d_{\mathcal{Q}}$ by $d_{\mathcal{Q}}=12-2i-3j-\textbf{1}_{\square}$ , where $\textbf{1}_{\square}$ is the indicator function of the presence of a third symbol.

6.3. The poset of Bézoutian quatroids

We conclude our work with some observations on the poset of Bézoutian quatroids given by the order $\leq$ . In order to display these results, instead of working with the $544748$ Bézoutian quatroids, we work modulo the $\mathfrak{S}_{8}$ -symmetry, and use the $76$ orbits instead. This allows us to fully illustrate the induced poset in Figure 7, where $\mathcal{Q}\leq\mathcal{Q}^{\prime}$ if there exists $\mathcal{Q}^{\prime\prime}$ in the orbit of $\mathcal{Q}^{\prime}$ such that $\mathcal{Q}\leq\mathcal{Q}^{\prime\prime}$ .

We prove that the dimension of the realization space defines a grading on this poset of Bézoutian quatroids, in the sense of [31, Sec. 3.1]. Recall that $\mathcal{Q}^{\prime}$ covers $\mathcal{Q}$ when $\mathcal{Q}^{\prime}$ is minimally larger than $\mathcal{Q}$ . We have the following result.

Theorem 6.14.

The set $\mathfrak{B}$ with the order relation $\leq$ on quatroids is represented in Figure 7. It is partitioned into nine layers $\mathfrak{B}_{0},\ldots,\mathfrak{B}_{8}$ : $\mathcal{Q}\in\mathfrak{B}_{j}$ if and only if $\dim\mathcal{S}_{\mathcal{Q}}=8+j$ . Moreover:

(i)

$\mathcal{Q}_{1}$ is the only quatroid in $\mathfrak{B}_{8}$ ,
(ii)

$\mathcal{Q}_{41}$ is the only quatroid in $\mathfrak{B}_{0}$ ,
(iii)

$\mathfrak{B}$ is graded by dimension: if $\mathcal{Q}^{\prime}$ covers $\mathcal{Q}$ then $\dim\mathcal{S}_{\mathcal{Q}^{\prime}}=\dim\mathcal{S}_{\mathcal{Q}}+1$ ,
(iv)

$\mathfrak{B}$ is graded by number of conditions: if $\mathcal{Q}^{\prime}$ covers $\mathcal{Q}$ then $|\mathcal{I}^{\prime}|+|\mathcal{J}^{\prime}|=|\mathcal{I}|+|\mathcal{J}|+1$ .

The proof of Theorem 6.14 relies on the following technical result, related to 3.3: on the subset of Bézoutian quatroids, the second condition listed in 3.3 completely characterizes the order relation.

Lemma 6.15.

Let $\mathcal{Q}=(\mathcal{I},\mathcal{J}),\mathcal{Q}^{\prime}=(\mathcal{I}^{\prime},\mathcal{J}^{\prime})$ be Bézoutian quatroids. The following are equivalent:

(i)

$\mathcal{Q}\leq\mathcal{Q}^{\prime}$ ,
(ii)

$\mathcal{S}_{\mathcal{Q}}\subseteq\overline{\mathcal{S}_{\mathcal{Q}^{\prime}}}$ ,
(iii)

$\mathcal{I}^{\prime}\subseteq\mathcal{I}$ and $\mathcal{J}^{\prime}\subseteq\mathcal{J}\cup\mathcal{I}^{2}$ , where $\mathcal{I}^{2}$ is the set of all subsets of $\{1,\ldots,8\}$ arising as the union of two elements of $\mathcal{I}$ .

Proof.

The equivalence of (i) and (ii) is the definition of the order relation. The fact that (iii) implies (ii) follows from 3.3. It remains to show that (ii) implies (iii). If $\mathcal{S}_{\mathcal{Q}}\subseteq\overline{\mathcal{S}_{\mathcal{Q}^{\prime}}}$ , every $\mathbf{p}\in\mathcal{S}_{\mathcal{Q}}$ satisfies the linear and quadratic relations imposed by $\mathcal{Q}^{\prime}$ . This immediately implies $\mathcal{I}^{\prime}\subseteq\mathcal{I}$ . To conclude, let $J\in\mathcal{J}^{\prime}$ , and without loss of generality assume $J=123456$ . Since $\mathcal{Q}$ is Bézoutian, there is a unique quadric $q_{123456}$ through a generic $\mathbf{p}\in\mathcal{S}_{\mathcal{Q}}$ . If this quadric is a conic, then $123456\in\mathcal{J}$ . Otherwise it is the union of two lines: since $\mathcal{Q}$ is Bézoutian, no four points lie on a line. Hence, up to relabeling, $q_{123456}=\ell_{123}\ell_{456}$ showing $123,456\in\mathcal{I}$ and $123456\in\mathcal{I}^{2}$ as desired. ∎

A technical consequence of 6.15 is that the closure step in step 6 of algorithm 1 does not alter the pair, whenever the quatroid that is being generated is Bézoutian. Implicitly, this fact yields the grading of $\mathfrak{B}$ by number of conditions, as one can see in the proof of Theorem 6.14.

Proof of Theorem 6.14.

We first prove statement (iv). Let $\mathcal{Q}=(\mathcal{I},\mathcal{J}),\mathcal{Q}^{\prime}=(\mathcal{I}^{\prime},\mathcal{J}^{\prime})$ be quatroids and suppose $\mathcal{Q}^{\prime}$ covers $\mathcal{Q}$ . By 6.15, $\mathcal{I}^{\prime}\subseteq\mathcal{I}$ and $\mathcal{J}^{\prime}\subseteq\mathcal{J}\cup\mathcal{I}^{2}$ . We consider several cases:

•
Suppose there is a condition $I\in\mathcal{I}\setminus\mathcal{I}^{\prime}$ .
- –
  
  If for every $J^{\prime}\in\mathcal{J}$ we have $I\not\subseteq J^{\prime}$ then we claim $\mathcal{I}=\mathcal{I}^{\prime}\cup\{I\}$ and $\mathcal{J}=\mathcal{J}^{\prime}$ : this is immediate because $\mathcal{Q}^{\prime\prime}=(\mathcal{I}^{\prime}\cup\{I\},\mathcal{J}^{\prime})$ satisfies Bézout’s criteria and $\mathcal{Q}^{\prime}$ is minimally larger than $\mathcal{Q}$ .
- –
  
  Otherwise, set $J^{\prime}_{*}=\{J^{\prime}\in\mathcal{J}^{\prime}\mid I\subseteq J^{\prime}\}$ and $I^{\prime}_{*}=\{J^{\prime}\setminus I\mid J^{\prime}\in J^{\prime}_{*}\}$ . Consider $\mathcal{Q}^{\prime\prime}=(\mathcal{I}^{\prime}\cup I^{\prime}_{*}\cup\{I\},\mathcal{J}^{\prime}\setminus J^{\prime}_{*})$ . By 6.15 $\mathcal{Q}^{\prime\prime}\leq\mathcal{Q}^{\prime}$ . Moreover, there is exactly one more condition in $\mathcal{Q}^{\prime\prime}$ than there is in $\mathcal{Q}^{\prime}$ . One may verify that $\mathcal{Q}^{\prime\prime}$ is Bézoutian and so, as before, we conclude $\mathcal{Q}^{\prime\prime}=\mathcal{Q}$ .
•

If $\mathcal{I}=\mathcal{I}^{\prime}$ , then $\mathcal{J}^{\prime}\cap\mathcal{I}^{2}=\mathcal{J}^{\prime}\cap{\mathcal{I}^{\prime}}^{2}=\emptyset$ because $\mathcal{Q}^{\prime}$ satisfies Bézout’s weak criteria. This implies $\mathcal{J}^{\prime}\subseteq\mathcal{J}$ , and by minimality $\mathcal{J}=\mathcal{J}^{\prime}\cup\{J\}$ for some $J$ .

This shows statement (iv). Statement (iii) is then a consequence of statement (iv) and the irreducibility of the strata $\mathcal{S}_{\mathcal{Q}}$ . The structure of the poset in Figure 7 and statement (i) and (ii) follow by direct computation, which can be done in a purely combinatorial way using 6.15. ∎

7. Concluding Remarks

7.1. Positive certificates for non-rationality

Theorem 6.13 allows one to design non-rationality certificates for cubic curves through points in special position. More precisely, there are several quatroid strata $\mathcal{S}_{\mathcal{Q}}$ satisfying $d_{\mathcal{Q}}=0$ : in these cases, for any $\textbf{p}\in\mathcal{S}_{\mathcal{Q}}$ , there are no rational cubics through $\mathbf{p}$ . The same holds passing to the closure, proving the following result.

Theorem 7.1.

Let $\textbf{p}\in\mathcal{S}_{\mathcal{Q}}$ for some $\mathcal{Q}\leq\mathcal{Q}^{\prime}$ with

\mathcal{Q}^{\prime}\text{ in the orbit of an element of }\{\mathcal{Q}_{6},\mathcal{Q}_{32},\mathcal{Q}_{41},\mathcal{Q}_{59},\mathcal{Q}_{62},\mathcal{Q}_{121}\}.

Then there are no rational cubics passing through $\mathbf{p}$ .

We point out that if $\mathcal{Q}\leq\mathcal{Q}_{6}$ or $\mathcal{Q}\leq\mathcal{Q}_{121}$ , then $\mathcal{Q}$ is non-Bézoutian: in this case, there are no irreducible cubics at all passing through $\mathbf{p}$ .

We remark that $\mathcal{Q}_{59}$ and $\mathcal{Q}_{62}$ are exhaustive Bézoutian quatroids so the matroid underlying the (reduced) base locus $Z(L_{\textbf{p}})$ of any $\mathbf{p}\in\mathcal{S}_{\mathcal{Q}}$ is well-defined; in these cases, this is the non-Fano matroid $\mathcal{Q}_{32}$ . Configurations of $\mathcal{Q}_{59}$ appear in [13], in the study of the topology of singular cubics. In a sense, configurations $\mathbf{p}$ as in Theorem 7.1 are forbidden configurations on a rational cubic. More precisely, we have the following consequence.

Corollary 7.2.

Let $C\subset\mathbb{P}_{\mathbb{C}}^{2}$ be an irreducible cubic containing a configuration $\textbf{p}\in\mathcal{S}_{\mathcal{Q}}$ where

\mathcal{Q}\leq\mathcal{Q}^{\prime}\quad\text{ for some }\mathcal{Q}^{\prime}\text{ in the orbit of an element of }\quad\{\mathcal{Q}_{6},\mathcal{Q}_{32},\mathcal{Q}_{41},\mathcal{Q}_{59},\mathcal{Q}_{62},\mathcal{Q}_{121}\}.

Then $C$ is not rational. These forbidden quatroids are illustrated in Figure 8.

To the extent of our knowledge, the existence of forbidden configurations guaranteeing non-rationality gives a novel way to prove non-rationality of a variety. In this sense, these are positive certificates of non-rationality. We leave open the problem of studying positive non-rationality certificates for curves of higher degree, and varieties of higher dimension.

7.2. A finer stratification

We remark that the values $d_{\mathcal{Q}}$ are the number of rational cubics through a generic configuration $\mathbf{p}\in\mathcal{S}_{\mathcal{Q}}$ . A natural question is whether this is the number of rational cubics through every configuration $\mathbf{p}\in\mathcal{S}_{\mathcal{Q}}$ . This is not the case, as observed in the following construction.

Consider the configuration $\mathbf{p}_{z}$ , depending on a complex parameter $z$ and described by the matrix

\mathbf{p}_{z}=\left[\begin{array}[]{cccccccc}1&4&9&16&25&36&1&1\\ 1&2&3&4&5&6&1&-1\\ 1&1&1&1&1&1&0&z\end{array}\right].

For generic $z\in\mathbb{C}$ , the configuration $\mathbf{p}_{z}$ represents quatroid $\mathcal{Q}_{2}$ : the points $p_{1},\ldots,p_{6}$ lie on the conic $x_{1}^{2}-x_{0}x_{2}=0$ , and there is no other linear or quadratic relation. For generic $z$ , $L_{\mathbf{p}_{z}}$ intersects $\mathcal{D}$ in the reducible cubic $C=q_{123456}\ell_{78}$ with multiplicity $2=\mathrm{mult}_{\mathcal{D}}(C)$ , and $10$ additional rational cubics. Hence $d_{\mathbf{p}_{z}}=d_{\mathcal{Q}}=10$ .

Let $\mathbf{p}^{*}$ be the configuration for $z=0$ , which represents $\mathcal{Q}_{2}$ as well. In this case, the line $\ell_{78}$ is tangent to the conic $q_{123456}$ . The pencil $L_{\mathbf{p}^{*}}$ intersects $\mathcal{D}$ at $C$ with intersection multiplicity $3=\mathrm{mult}_{\mathcal{D}}(C)$ and at an additional $9$ distinct rational cubics.

This construction shows that there are special configurations $\mathbf{p}^{*}$ on the quatroid stratum $\mathcal{S}_{\mathcal{Q}_{2}}$ satisfying $d_{\mathbf{p}^{*}}<d_{\mathcal{Q}_{2}}$ . A similar example can be constructed on the other maximal non-uniform quatroid $\mathcal{Q}_{8}$ . A slightly different example can be constructed on $\mathcal{Q}_{78}$ : in this case, for a generic choice of $\mathbf{p}\in\mathcal{S}_{\mathcal{Q}_{78}}$ , the only reducible cubic on $L_{\mathbf{p}}$ is a triangle $C=\ell_{1}\ell_{2}\ell_{3}$ and $L_{\mathbf{p}}$ intersects $\mathcal{D}$ in $C$ with multiplicity $3=\mathrm{mult}_{\mathcal{D}}(C)$ and in $9$ rational cubics, so that $d_{\mathbf{p}}=d_{\mathcal{Q}_{78}}=9$ . There is a locus in $\mathcal{S}_{\mathcal{Q}_{78}}$ where the triangle $C$ degenerates to an asterisk: here $\mathrm{mult}_{\mathcal{D}}(C)=4$ and $d_{\mathbf{p}}=8<d_{\mathcal{Q}}$ .

This phenomenon occurs on other strata. Interestingly, there are strata, such as $\mathcal{S}_{\mathcal{Q}_{40}}$ , with the property that $d_{\mathcal{Q}}=1$ and with a locus of configurations $\mathbf{p}$ such that $d_{\mathbf{p}}=0$ . Understanding these loci, with no rational cubics through them, would provide other non-rationality certificates, in the sense of Section 7.1.

We expect the special loci described in this section always arise with either a conic $+$ line pair $C=q\ell$ degenerating to a conic $+$ tangent pair, or with a triangle $C=\ell_{1}\ell_{2}\ell_{3}$ degenerating to an asterisk. The second phenomenon is linear but it cannot be detected simply by the matroid underlying $\mathbf{p}$ ; however, it is detected by the underlying discriminantal arrangement [1]. We plan to further investigate higher order versions of the discriminantal arrangement in future work.

7.3. Toward higher degree and higher genus

Given two integers $d$ and $g$ , one may consider the locus $V_{d,g}\subseteq\mathbb{P}S^{d}\mathbb{C}^{3}$ of plane curves of degree $d$ and genus $g$ . In [19], Harris answered a conjecture of Severi [29], proving that $V_{d,g}$ is irreducible of dimension $3d+g-1$ . This inspired a body of work surrounding the natural enumerative problem:

Given $3d+g-1$ points in $\mathbb{P}_{\mathbb{C}}^{2}$ , how many elements of $V_{d,g}$ interpolate them?

For generic points, this amounts to computing the degree of (the closure of) $V_{d,g}$ : when $g=0$ , the answer is Kontsevich’s formula [21] and a recursive formula was given for any genus in [10]. The present paper dealt with non-generic instances of the problem when $(d,g)=(3,0)$ , that is the case of rational cubics. Throughout our investigation, we enjoyed a number of nice properties:

•

the cubic discriminant is the union of rational cubics and reducible cubics,
•

the discriminant is cut out by a manageable polynomial,
•

the number of rational cubics through eight generic points is of modest size,
•

there is a known classification of orbit closures of rational and reducible cubics.

We propose the study of curves of higher degree and genus, with similar, higher order methods.

For instance, there are $620$ rational quartics through $11=3\cdot 4+0-1$ generic points. Despite all the “good” properties mentioned above failing in this setting, numerical methods can generate experimental data. We now consider the problem of computing all rational quartics through two interesting matroid strata on $11$ points. A Lüroth quartic is any quartic curve which goes through the $10={{5}\choose{2}}$ intersection points of five lines. The set of Lüroth quartics forms an unwieldy hypersurface of degree $54$ in the space $\mathbb{P}S^{4}\mathbb{C}^{3}$ of homogeneous quartics [6, 25, 27]. These quartics inspire the definition of the Lüroth matroid on $11$ points, which we write in our quatroid format:

{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathcal{Q}_{\textrm{L\"{u}roth}}}=(\{1367,159\underline{10},246\underline{10},2578,3489\},\{\}).

Numerical computations suggest that there are $40$ rational quartics through a Lüroth configuration. In Figure 9 we display one example of a Lüroth configuration and the $12$ real rational quartics which interpolate it.

A more restricted matroid is suggested by a regular pentagon configuration [33] which imposes five additional line conditions on the eleventh point. The underlying matroid is

{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathcal{Q}_{\textrm{Pentagon}}}=(\{1367,159\underline{10},246\underline{10},2578,3489,23\underline{11},18\underline{11},69\underline{11},7\underline{10}\,\underline{11},45\underline{11}\},\{\})

Such a configuration can be realized over $\mathbb{R}$ but not $\mathbb{Q}$ , and it is unique up to the action of $\mathrm{PGL}_{2}$ . We invite the reader to draw this configuration. A numerical calculation suggests that there are $10$ rational quartics interpolating the regular pentagon configuration, none of which are real.

Appendix

Name	Description	Relevant Results
Quatroids.jl	The julia package Quatroids.jl. All commands loaded by this package are indicated by blue font
SimpleMatroids38.txt	File extracted from the database https://www-imai.is.s.u-tokyo.ac.jp/~ymatsu/matroid/ listing all simple rank $3$ matroids on eight elements
GenerateAllMatroids()	Parses the file SimpleMatroids38.txt to obtain an exhaustive list of simple $\mathbb{C}$ -representable matroids of rank at most three on eight elements	Theorem 3.19
AllConicExtensions(M)	Runs algorithm 1 on a matroid $M$ represented by nonbases of size three	Theorem 3.19
GenerateAllCandidateQuatroids()	Runs algorithm 1 all $\mathbb{C}$ -representable matroids of rank at most three on eight elements	Theorem 3.19 Table 4 Table 5
OrbitSizes()	Computes the sizes of each orbit of candidate quatroids	Theorem 3.19 3.21 Table 4 Table 5
Bezoutian()	Returns a boolean vector of length $126$ whose $i$ -th entry indicates whether $\mathcal{Q}_{i}$ is Bézoutian	3.15 Table 4 Table 5
RationalRepresentatives.txt	A file, given in julia (.txt) and Macaulay2 (.m2) format, whose $i$ -th line is a $3\times 8$ matrix of integers whose columns represent $\mathcal{Q}_{i}$	Theorem 3.20
TestingRepresentatives.m2	A Macaulay2 script which confirms that each representative in RationalRepresentatives.txt represents the quatroid claimed	Theorem 3.20
ReducedBaseLocus(Q)	Indicates if the quatroid $\mathcal{Q}$ has reduced base locus due to 3.25	3.26
QuatroidReductions()	Iteratively reduces each quatroid based on the four conditions described in Section 4	4.5
Modifications(Q)	Computes all $\mathcal{Q}^{\prime}$ such that $\mathcal{Q}\leadsto\mathcal{Q}^{\prime}$	4.6
IrreducibilityQ21.m2 etc	A Macaulay2 script which establishes the irreducibility of $\mathcal{Q}_{21}$ as described in the proof of Theorem 4.7. Similar files exist for $\mathcal{Q}_{25},\mathcal{Q}_{38},\mathcal{Q}_{58},\mathcal{Q}_{101},$ and $\mathcal{Q}_{123}$	Theorem 4.7
cubicInvs.m2	A Macaulay2 file containing the cubic discriminant	algorithm 2
countReducibleCubics.m2	Applies algorithm 2 to each of the rational representatives of Bézoutian quatroids in RationalRepresentatives.m2	Theorem 5.5 Table 4 Table 5
QuatroidsWeakUpperBounds()	Evaluates the formula for $m_{\mathcal{Q}}$ from Theorem 6.2 for each Bézoutain quatroid $\mathcal{Q}$	Theorem 6.3
ContainedIn10(Q) ContainedIn77(Q)	Checks $\mathcal{Q}\leq\mathcal{Q}_{10}$ and $\mathcal{Q}\leq\mathcal{Q}_{77}$ respectively	Theorem 6.3
tangentConesPlaneCubics.m2	Computes the tangent cones of each singular cubic	6.7 Table 1 Table 2

Table 3. Descriptions of auxiliary files

Num	$\|$ Orb $\|$	Lines	Conics	Reducibles	$d_{Q}$
$1$	$1$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$		$12$
$2$	$28$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{123456\}}}$	$\varnothing^{1}$	$10$
$3$	$210$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{123456\,\,123478\}}}$	$\varnothing^{2}$	$8$
$4$	$420$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{123456\,\,123478\,\,125678\}}}$	$\varnothing^{3}$	$6$
$5$	$105$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{123456\,\,123478\,\,125678\,\,345678\}}}$	$\varnothing^{4}$	$4$
$6$	$8$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{1234567\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$7$	$1$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{12345678\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$8$	$56$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	$\varnothing^{1}$	$10$
$9$	$840$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{124567\}}}$	$\varnothing^{2}$	$8$
$10$	$168$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{145678\}}}$	$\varnothing^{1}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\varnothing}}$	$9$
$11$	$3360$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{124567\,\,134568\}}}$	$\varnothing^{3}$	$6$
$12$	$840$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{124567\,\,345678\}}}$	$\varnothing^{2}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\varnothing}}$	$7$
$13$	$3360$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{124567\,\,134568\,\,234578\}}}$	$\varnothing^{4}$	$4$
$14$	$168$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{1245678\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$15$	$840$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	$\varnothing^{2}$	$8$
$16$	$3360$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{124678\}}}$	$\varnothing^{3}$	$6$
$17$	$2520$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{234567\}}}$	$\varnothing^{3}$	$6$
$18$	$3360$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{234678\}}}$	$\varnothing^{2}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\varnothing}}$	$7$
$19$	$1680$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{124678\,\,135678\}}}$	$\varnothing^{4}$	$4$
$20$	$10080$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{124678\,\,234567\}}}$	$\varnothing^{4}$	$4$
$21$	$6720$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{124678\,\,235678\}}}$	$\varnothing^{3}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\varnothing}}$	$5$
$22$	$5040$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{124678\,\,135678\,\,234567\}}}$	$\varnothing^{5}$	$2$
$23$	$840$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{2345678\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$24$	$840$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,167\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	$\varnothing^{3}$	$6$
$25$	$840$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,167\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{234567\}}}$	$\varnothing^{4}$	$4$
$26$	$5040$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,167\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{234568\}}}$	$\varnothing^{3}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\varnothing}}$	$5$
$27$	$840$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,167\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{2345678\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$28$	$6720$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,167\,\,246\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	$\varnothing^{4}$	$4$
$29$	$20160$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,167\,\,246\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{234578\}}}$	$\varnothing^{4}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\varnothing}}$	$3$
$30$	$5040$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,167\,\,246\,\,257\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	$\varnothing^{5}$	$2$
$31$	$5040$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,167\,\,246\,\,257\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{345678\}}}$	$\varnothing^{5}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\varnothing}}$	$1$
$32$	$1680$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,167\,\,246\,\,257\,\,347\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	$\varnothing^{6}$	$0$
$33$	$5040$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,167\,\,246\,\,257\,\,347\,\,358\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	$\varnothing^{4}\triangle^{1}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\triangle}}$	$0$
$34$	$20160$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,167\,\,246\,\,257\,\,348\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	$\varnothing^{3}\triangle^{1}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\triangle}}$	$2$
$35$	$10080$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,167\,\,246\,\,257\,\,348\,\,568\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	$\varnothing^{2}\triangle^{2}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\triangle}}$	$1$
$36$	$5040$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,167\,\,246\,\,257\,\,478\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	$\varnothing^{4}\triangle^{1}$	$1$
$37$	$20160$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,167\,\,246\,\,258\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	$\varnothing^{3}\triangle^{1}$	$3$
$38$	$20160$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,167\,\,246\,\,258\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{345678\}}}$	$\varnothing^{3}\triangle^{1}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\varnothing}}$	$2$
$39$	$20160$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,167\,\,246\,\,258\,\,357\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	$\varnothing^{2}\triangle^{2}$	$2$
$40$	$6720$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,167\,\,246\,\,258\,\,357\,\,368\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	$\varnothing^{1}\triangle^{3}$	$1$
$41$	$840$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,167\,\,246\,\,258\,\,357\,\,368\,\,478\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	$\triangle^{4}$	$0$
$42$	$20160$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,167\,\,246\,\,258\,\,378\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	$\varnothing^{1}\triangle^{2}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\triangle}}$	$3$
$43$	$3360$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,167\,\,246\,\,357\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	$\varnothing^{3}\triangle^{1}$	$3$
$44$	$20160$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,167\,\,246\,\,358\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	$\varnothing^{2}\triangle^{1}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\triangle}}$	$4$
$45$	$10080$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,167\,\,248\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	$\varnothing^{2}\triangle^{1}$	$5$
$46$	$10080$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,167\,\,248\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{234567\}}}$	$\varnothing^{3}\triangle^{1}$	$3$
$47$	$20160$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,167\,\,248\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{235678\}}}$	$\varnothing^{2}\triangle^{1}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\varnothing}}$	$4$
$48$	$5040$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,167\,\,248\,\,358\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	$\varnothing^{2}\triangle^{1}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\triangle}}$	$4$
$49$	$5040$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,167\,\,248\,\,358\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{234567\}}}$	$\varnothing^{3}\triangle^{1}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\triangle}}$	$2$
$50$	$20160$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,167\,\,248\,\,368\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	$\varnothing^{1}\triangle^{2}$	$4$
$51$	$20160$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,167\,\,248\,\,368\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{234567\}}}$	$\varnothing^{2}\triangle^{2}$	$2$
$52$	$3360$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,167\,\,248\,\,368\,\,578\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	$\triangle^{3}$	$3$
$53$	$3360$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,167\,\,248\,\,368\,\,578\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{234567\}}}$	$\varnothing^{1}\triangle^{3}$	$1$
$54$	$3360$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,246\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	$\varnothing^{3}$	$6$
$55$	$10080$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,246\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{125678\}}}$	$\varnothing^{4}$	$4$
$56$	$10080$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,246\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{135678\}}}$	$\varnothing^{3}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\varnothing}}$	$5$
$57$	$10080$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,246\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{125678\,\,134678\}}}$	$\varnothing^{5}$	$2$
$58$	$10080$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,246\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{125678\,\,345678\}}}$	$\varnothing^{4}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\varnothing}}$	$3$
$59$	$3360$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,246\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{125678\,\,134678\,\,234578\}}}$	$\varnothing^{6}$	$0$
$60$	$840$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,246\,\,356\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	$\varnothing^{4}$	$4$
$61$	$2520$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,246\,\,356\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{125678\}}}$	$\varnothing^{5}$	$2$
$62$	$2520$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,246\,\,356\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{125678\,\,134678\}}}$	$\varnothing^{6}$	$0$
$63$	$840$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,246\,\,356\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{125678\,\,134678\,\,234578\}}}$	NR	NR

Table 4. Data for each candidate quatroid stratum. NR indicates not representable and NB indicates not Bézoutian.

Num	$\|$ Orb $\|$	Lines	Conics	Reducibles	$d_{Q}$
$64$	$10080$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,246\,\,357\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	$\varnothing^{2}\triangle^{1}$	$5$
$65$	$20160$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,246\,\,357\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{125678\}}}$	$\varnothing^{3}\triangle^{1}$	$3$
$66$	$10080$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,246\,\,357\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{125678\,\,134678\}}}$	$\varnothing^{4}\triangle^{1}$	$1$
$67$	$10080$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,246\,\,357\,\,678\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	$\triangle^{2}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\triangle}}$	$5$
$68$	$10080$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,246\,\,378\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	$\varnothing^{1}\triangle^{1}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\triangle}}$	$6$
$69$	$10080$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,246\,\,378\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{125678\}}}$	$\varnothing^{2}\triangle^{1}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\triangle}}$	$4$
$70$	$5040$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,267\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	$\varnothing^{1}\triangle^{1}$	$7$
$71$	$20160$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,267\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{134678\}}}$	$\varnothing^{2}\triangle^{1}$	$5$
$72$	$5040$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,267\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{345678\}}}$	$\varnothing^{1}\triangle^{1}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\varnothing}}$	$6$
$73$	$20160$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,267\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{134678\,\,234568\}}}$	$\varnothing^{3}\triangle^{1}$	$3$
$74$	$5040$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,267\,\,468\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	$\triangle^{2}$	$6$
$75$	$10080$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,267\,\,468\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{135678\}}}$	$\varnothing^{1}\triangle^{2}$	$4$
$76$	$5040$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,267\,\,468\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{135678\,\,234578\}}}$	$\varnothing^{2}\triangle^{2}$	$2$
$77$	$840$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,678\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	$\triangle^{1}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\triangle}}$	$8$
$78$	$2520$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,145\,\,678\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{234567\}}}$	$\varnothing^{1}\triangle^{1}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\triangle}}$	$6$
$79$	$280$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,456\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	$\triangle^{1}$	$9$
$80$	$2520$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,456\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{124578\}}}$	$\varnothing^{1}\triangle^{1}$	$7$
$81$	$5040$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,456\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{124578\,\,134678\}}}$	$\varnothing^{2}\triangle^{1}$	$5$
$82$	$1680$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123\,\,456\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{124578\,\,134678\,\,235678\}}}$	$\varnothing^{3}\triangle^{1}$	$3$
$83$	$8$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234567\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$84$	$168$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123456\,\,178\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$85$	$28$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{123456\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$86$	$280$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{12345\,\,1678\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$87$	$3360$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{12345\,\,167\,\,268\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$88$	$3360$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{12345\,\,167\,\,268\,\,378\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$89$	$840$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{12345\,\,167\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$90$	$56$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{12345\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$91$	$56$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{12345\,\,678\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$92$	$3360$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234\,\,1567\,\,258\,\,368\,\,478\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$93$	$10080$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234\,\,1567\,\,258\,\,368\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$94$	$5040$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234\,\,1567\,\,258\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$95$	$560$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234\,\,1567\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$96$	$840$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234\,\,156\,\,178\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$97$	$2520$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234\,\,156\,\,178\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{235678\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$98$	$10080$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234\,\,156\,\,178\,\,257\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$99$	$10080$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234\,\,156\,\,178\,\,257\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{345678\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$100$	$2520$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234\,\,156\,\,178\,\,257\,\,268\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$101$	$2520$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234\,\,156\,\,178\,\,257\,\,268\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{345678\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$102$	$10080$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234\,\,156\,\,178\,\,257\,\,268\,\,358\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$103$	$5040$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234\,\,156\,\,178\,\,257\,\,268\,\,358\,\,467\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$104$	$20160$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234\,\,156\,\,178\,\,257\,\,358\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$105$	$20160$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234\,\,156\,\,178\,\,257\,\,358\,\,467\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$106$	$10080$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234\,\,156\,\,178\,\,257\,\,368\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$107$	$1680$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234\,\,156\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$108$	$5040$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234\,\,156\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{235678\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$109$	$10080$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234\,\,156\,\,257\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$110$	$10080$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234\,\,156\,\,257\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{345678\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$111$	$6720$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234\,\,156\,\,257\,\,358\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$112$	$20160$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234\,\,156\,\,257\,\,358\,\,467\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$113$	$6720$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234\,\,156\,\,257\,\,358\,\,678\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$114$	$6720$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234\,\,156\,\,257\,\,367\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$115$	$20160$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234\,\,156\,\,257\,\,368\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$116$	$5040$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234\,\,156\,\,257\,\,368\,\,478\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$117$	$10080$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234\,\,156\,\,257\,\,678\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$118$	$2520$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234\,\,156\,\,278\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$119$	$2520$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234\,\,156\,\,278\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{345678\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$120$	$3360$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234\,\,156\,\,578\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$121$	$70$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$122$	$420$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{125678\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$123$	$210$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{125678\,\,345678\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$124$	$280$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234\,\,567\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$125$	$35$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{1234\,\,5678\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$
$126$	$1$	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\{12345678\}}}$	${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\{\}}}$	NB	${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{0}}$

Table 5. Data for each candidate quatroid stratum. NR indicates not representable and NB indicates not Bézoutian.

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