The Moduli Space of Genus Six Curves and K-stability: VGIT and the Hassett-Keel Program

Junyan Zhao 851 S Morgan St, 60607, Chicago, Illinois, USA [email protected]

Abstract.

A general curve $C$ of genus six is canonically embedded into the smooth del Pezzo surface $\Sigma\subseteq\mathbb{P}^{1}\times\mathbb{P}^{2}$ of degree $5$ as a divisor in the class $\mathcal{O}_{\Sigma}(2,2)$ . In this article, we study the variation of geometric invariant theory (VGIT) for such pairs $(\Sigma,C)$ , and relate the VGIT moduli spaces to the K-moduli of pairs $(\Sigma,C)$ and the Hassett–Keel program for moduli of genus six curves. We prove that the K-moduli spaces $\overline{M}^{K}(c)$ give the final several steps in the Hassett–Keel program for $\overline{M}_{6}$ .

1. Introduction

This is the second article among three of the author in which we study the moduli space of curves of genus six using moduli spaces of pairs. A general genus six curve $C$ is canonically embedded into the smooth quintic del Pezzo surface $\Sigma$ as a divisor in the class $-2K_{\Sigma}$ . Moreover, this embedding is unique up to $\operatorname{Aut}(\Sigma)$ , which is isomorphic to the symmetric group $\mathfrak{S}_{5}$ . Thus we expect a moduli space parameterizing pairs in which a general member is of the form $(\Sigma,C)$ , where $C\in|-2K_{\Sigma}|$ , to be related to the moduli space $\overline{M}_{6}$ of curves of genus six. A good candidate for such a moduli space is the K-moduli space $\overline{M}^{K}(c)$ , $0<c<1/2$ , parameterizing K-polystable pairs $(X,cD)$ which admit a $\mathbb{Q}$ -Gorenstein smoothing to $(\Sigma,cC)$ . This is the main object studied in [Zha22].

For any $C\in|-2K_{\Sigma}|$ , we may view $(\Sigma,C)$ as a complete intersection in $\mathbb{P}^{1}\times\mathbb{P}^{2}$ of type $(\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{2}}(1,2),\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{2}}(2,2))$ . A natural parameter space for such complete intersections is a projective bundle $\mathbb{P}\mathcal{E}\rightarrow\mathbb{P}^{9}$ , which is equipped with a natural $G=\operatorname{SL}(2)\times\operatorname{SL}(3)$ -action. The Picard rank of $\mathbb{P}\mathcal{E}$ is 2, thus the GIT quotient involves a choice of linearization parameterized by $t\in(0,1/2)$ . We will denote these GIT quotients by $\overline{M}^{\operatorname{GIT}}(t)$ .

The case for curves of genus $4$ is analyzed in [CMJL14], in which the authors proved that the VGIT moduli spaces give the final steps in the Hassett-Keel program for genus 4 curves. Unfortunately, this cannot be expected to hold in our set-up. The main reason is the following: for any $X\in\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{2}}(1,2)$ such that $X\simeq\Sigma$ , the subgroup of $\operatorname{SL}(2)\times\operatorname{SL}(3)$ fixing the point $[X]\in\mathbb{P}^{9}$ is an $\mathfrak{S}_{4}$ , which is not isomorphic to $\operatorname{Aut}(\Sigma)=\mathfrak{S}_{5}$ . The right thing we should expect is a generically $5:1$ map from the VGIT quotient to the moduli spaces which are birational to $\overline{M}_{6}$ . Denote by $\mathbb{P}\mathcal{E}^{K}(c)$ and $\mathbb{P}\mathcal{E}^{\operatorname{GIT}}(t)$ the subsets of $\mathbb{P}\mathcal{E}$ consisting of pairs $(X,D)$ which are c-K-semistable and GIT_t-semistable, respectively. Then we have the following result.

Theorem 1.1.

(Theorem 4.4) For any $c\in(0,1/17)$ , set $t=t(c)=\frac{5c}{4+2c}$ . Then we have $\mathbb{P}\mathcal{E}^{K}(c)=\mathbb{P}\mathcal{E}^{\operatorname{GIT}}(t)$ , and identifications

\overline{M}^{\operatorname{GIT}}(c)\simeq|-2K_{\Sigma}|\mathbin{\mkern-3.0mu/\mkern-6.0mu/\mkern-3.0mu}\mathfrak{S}_{4},\quad\textup{and}\quad\overline{M}^{K}(c)\simeq|-2K_{\Sigma}|\mathbin{\mkern-3.0mu/\mkern-6.0mu/\mkern-3.0mu}\mathfrak{S}_{5}.

Moreover, we have a finite map

\overline{M}^{\operatorname{GIT}}(c)\longrightarrow\overline{M}^{K}(c)

of degree $5$ .

Moreover, by analyzing the GIT stability conditions carefully, we can find out all the GIT walls, i.e. the values $t_{i}$ such that the moduli spaces $\overline{M}^{\operatorname{GIT}}(t)$ change when $t$ varies in $(0,1/2)$ and crosses $t_{i}$ . These walls turn out to be connected with walls for K-moduli spaces.

Theorem 1.2.

(Theorem 4.3) The VGIT walls are

t_{0}=0,\quad t_{1}=\frac{1}{14},\quad t_{2}=\frac{1}{8},\quad t_{3}=\frac{1}{6},\quad t_{4}=\frac{1}{5},\quad t_{5}=\frac{1}{4},\quad t_{6}=\frac{1}{3},\quad t_{7}=\frac{1}{2},

which bijectively correspond via the relation $\displaystyle t(c)=\frac{5c}{4+2c}$ to the K-moduli walls

c_{0}=0,\quad c_{1}=\frac{1}{17},\quad c_{2}=\frac{2}{19},\quad c_{3}=\frac{1}{7},\quad c_{4}=\frac{4}{23},\quad c_{5}=\frac{2}{9},\quad c_{6}=\frac{4}{13},\quad c_{7}=\frac{1}{2}.

Among the walls $t_{i}\in(0,1/2)$ , only the one $t_{1}=\frac{1}{14}$ is a divisorial contraction, and the remaining 5 walls are flips.

Although the GIT quotients are not birational to $\overline{M}_{6}$ , with the help of the projective bundle $\mathbb{P}\mathcal{E}$ , we can identify the K-moduli spaces with some of the log canonical models of $\overline{M}_{6}$ .

Recall that for each $\alpha\in[0,1]$ , we define a log canonical model of $\overline{M}_{6}$ to be

\overline{M}_{6}(\alpha):=\operatorname{Proj}\bigoplus_{n\geq 0}H^{0}\left(\overline{M}_{6},n(K_{\overline{M}_{6}}+\alpha\delta)\right),

where $\delta=\delta_{0}+\delta_{1}+\delta_{2}+\delta_{3}$ is the boundary divisor $\overline{M}_{6}\setminus M_{6}$ . For any genus $g\geq 4$ , when varying $\alpha$ from $1$ to $0$ , the first three walls are at $\alpha_{1}=\frac{9}{11}$ , $\alpha_{2}=\frac{7}{10}$ and $\alpha_{3}=\frac{2}{3}$ (ref. [HH09, HH13, AFSv17]). For $g=6$ , it is also known that the last non-trivial model is given by the GIT quotient $|-2K_{\Sigma}|\mathbin{\mkern-3.0mu/\mkern-6.0mu/\mkern-3.0mu}\operatorname{Aut}(\Sigma)$ (ref. [Mül14, Fed18]). The last main result in this paper is the following.

Theorem 1.3.

(Theorem 5.5) Let $0\leq c\leq\frac{11}{52}$ be a rational number, and $\alpha(c)=\frac{32-19c}{94-68c}\in\left[\frac{16}{47},\frac{97}{276}\right]$ . Then we have an isomorphism

\overline{M}_{6}(\alpha(c))\simeq\overline{M}^{K}(c).

In particular, the last walls of the log canonical models $\overline{M}_{6}(\alpha)$ are

\left\{\frac{16}{47},\frac{35}{102},\frac{29}{55},\frac{41}{118},\frac{22}{63},\frac{47}{134}\right\}.

Acknowledgements The author would like to thank Izzet Coskun, Maksym Fedorchuk, Yuchen Liu and Fei Si for many stimulating discussions, and Ben Gould for comments on the draft.

2. Preliminaries

In this section, we collect some facts about the projective bundle we consider and state some results in VGIT.

2.1. Geometry of projective bundles

Let $\mathbb{P}^{11}=\mathbb{P}H^{0}(\mathbb{P}^{1}\times\mathbb{P}^{2},\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{2}}(1,2))$ be the projective space parameterizing quintic del Pezzo surfaces in $\mathbb{P}^{1}\times\mathbb{P}^{2}$ , and $\mathscr{Q}\subseteq\mathbb{P}^{1}\times\mathbb{P}^{2}\times\mathbb{P}^{11}$ be the universal quintic. Let $\pi_{1}:\mathbb{P}^{1}\times\mathbb{P}^{2}\times\mathbb{P}^{11}\rightarrow\mathbb{P}^{1}\times\mathbb{P}^{2}$ and $\pi_{2}:\mathbb{P}^{1}\times\mathbb{P}^{2}\times\mathbb{P}^{11}\rightarrow\mathbb{P}^{11}$ be the two projections. Consider the locally free sheaf

\mathcal{E}:=\pi_{2*}(\mathcal{O}_{\mathscr{Q}}\otimes\pi_{1}^{*}\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{2}}(2,2))

of rank 16 on $\mathbb{P}^{11}$ and set $\pi:\mathbb{P}\mathcal{E}\rightarrow\mathbb{P}^{11}$ the corresponding projective bundle. Then

\operatorname{Pic}(\mathbb{P}\mathcal{E})\simeq\mathbb{Z}\eta\oplus\mathbb{Z}\xi,

where $\eta=\pi^{*}\mathcal{O}_{\mathbb{P}^{11}}(1)$ and $\xi=\mathcal{O}_{\mathbb{P}\mathcal{E}}(1)$ .

We first compute the Chern characters of $\mathcal{E}$ . Notice that $\mathcal{E}$ fits into the short exact sequence

0\longrightarrow\pi_{2*}(\mathcal{I}_{\mathscr{Q}}\otimes\pi_{1}^{*}\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{2}}(2,2))\longrightarrow\pi_{2*}(\pi_{1}^{*}\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{2}}(2,2))\longrightarrow\mathcal{E}\longrightarrow 0,

and that

\pi_{2*}(\mathcal{I}_{\mathscr{Q}}\otimes\pi_{1}^{*}\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{2}}(2,2))=H^{0}(\mathbb{P}^{1}\times\mathbb{P}^{2},\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{2}}(1,0))\otimes\mathcal{O}_{\mathbb{P}^{11}}(-1),

\pi_{2*}(\pi_{1}^{*}\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{2}}(2,2))=H^{0}(\mathbb{P}^{1}\times\mathbb{P}^{2},\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{2}}(2,2))\otimes\mathcal{O}_{\mathbb{P}^{11}}.

It follows that

\operatorname{ch}(\mathcal{E})=18-2\operatorname{ch}(\mathcal{O}_{\mathbb{P}^{9}}(-1))=18-2\sum_{i=0}^{\infty}\frac{(-1)^{i}\eta^{i}}{i!}.

In particular, we have $c_{1}(\mathcal{E})=2\eta$ and hence the following result.

Proposition 2.1.

The canonical line bundle of $\mathbb{P}\mathcal{E}$ is

K_{\mathbb{P}\mathcal{E}}\sim-14\eta-16\xi.

Proof.

By [Laz04, Section 7.3.A], we have that

K_{\mathbb{P}\mathcal{E}}=\pi^{*}(\omega_{\mathbb{P}^{11}}\otimes\det(\mathcal{E}^{*}))\otimes\mathcal{O}_{\mathbb{P}\mathcal{E}}(-\operatorname{rank}\mathcal{E})=-14\eta-16\xi.

∎

Let $U\subseteq\mathbb{P}\mathcal{E}$ be the Zariski big open subset parameterizing complete intersections $(X,D)$ with $X$ irreducible. Let $i:(\mathscr{X},\mathscr{D})\hookrightarrow\mathbb{P}^{1}\times\mathbb{P}^{2}\times U$ be the tautological family of the pairs, $p_{1},p_{2}$ the projection maps from $(\mathbb{P}^{1}\times\mathbb{P}^{2})\times U$ to the two factors, respectively, and $f:=p_{2}\circ i$ . In summary, we have the following commutative diagram

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Proposition 2.2.

The nef cone of $\mathbb{P}\mathcal{E}$ is generated by $\eta$ and $\eta+\frac{1}{2}\xi$ .

Proof.

It is clear that $\eta$ is an extremal ray. The argument in [Ben14, Theorem 2.7] shows that the other extremal ray is $\eta+\frac{1}{2}\xi$ . ∎

2.2. Variation of GIT

Now our main object is a polarized variety $(X,L)$ together with an action by a reductive group $G$ .

Lemma 2.3.

(ref. [Rad13, Lemma 3.10]) Let $(X,L)$ be a polarized projective variety, and $G$ a reductive group acting on $(X,L)$ . Let $L_{0}$ be a $G$ -linearized line bundle on $X$ . For any rational number $0<\varepsilon\ll 1$ , we define $L_{\pm\varepsilon}:=L\otimes L_{0}^{\otimes(\pm\varepsilon)}$ . Let $X^{ss}(0)$ and $X^{ss}(\pm)$ denote the semistable loci, respectively. Then

(i)

there are open immersions $X^{ss}(\pm)\subseteq X^{ss}(0)$ , and
(ii)

for any $x\in X^{ss}(0)\setminus X^{ss}(\pm)$ , there is a 1-parameter subgroup $\sigma$ of $G$ such that

$\mu^{L}(x,\sigma)=0,\quad\textup{and}\quad\mu^{L_{\pm}}(x,\sigma)<0.$

For simplicity, we denote $X\mathbin{\mkern-3.0mu/\mkern-6.0mu/\mkern-3.0mu}_{L_{\bullet}}G$ by $X\mathbin{\mkern-3.0mu/\mkern-6.0mu/\mkern-3.0mu}_{\bullet}G$ , where $\bullet\in\{+,-,0\}$ . As a consequence, the functorial properties of the GIT quotients give the following commutative diagram:

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where $X^{s}(L_{\bullet})$ are the stable loci, and $X^{s}(L_{\bullet})/G$ are geometric quotients.

Theorem 2.4.

(ref. [Tha96, Theorem 3.3]) With notation as in Lemma 2.3, if both $X\mathbin{\mkern-3.0mu/\mkern-6.0mu/\mkern-3.0mu}_{-}G$ and $X\mathbin{\mkern-3.0mu/\mkern-6.0mu/\mkern-3.0mu}_{+}G$ are non-empty, then $\varphi_{-}$ and $\varphi_{+}$ are proper and birational. If they are both small contractions, then the rational map

g:X\mathbin{\mkern-3.0mu/\mkern-6.0mu/\mkern-3.0mu}_{-}G\dashrightarrow X\mathbin{\mkern-3.0mu/\mkern-6.0mu/\mkern-3.0mu}_{+}G

is a flip with respect to $\mathcal{O}(1)$ on $X\mathbin{\mkern-3.0mu/\mkern-6.0mu/\mkern-3.0mu}_{+}G$ , which is the relative bundle of the projective morphism $g:X\mathbin{\mkern-3.0mu/\mkern-6.0mu/\mkern-3.0mu}_{+}G\rightarrow X\mathbin{\mkern-3.0mu/\mkern-6.0mu/\mkern-3.0mu}_{0}G$ .

3. Stability of points in $\mathbb{P}\mathcal{E}$

In this section, we determine the stability conditions on the projective bundle $\mathbb{P}\mathcal{E}$ using the Hilbert-Mumford (abbv. HM) criterion as the slope $t$ varies.

For any point $(f,[g])\in\mathbb{P}\mathcal{E}$ , we will not distinguish the pair $(f,[g])$ of equations and the corresponding pair $(X,D)$ of schemes. For any polarization $L=a\eta+b\xi$ and any 1-parameter subgroup (abbv. 1-PS) $\lambda$ , we denote by

\mu^{\frac{b}{a}}(X,D;\lambda)=\mu^{a\eta+b\xi}(X,D;\lambda)

the HM-invariant of the point $(X,D)$ . The following result about the HM-invariant will be widely used.

Proposition 3.1.

(ref. [Ben14, Proposition 2.15]) The HM-invariant of a point $(f,[g])=\left(X,D\right)\in\mathbb{P}\mathcal{E}$ is given by

\mu^{a\eta+b\xi}((X,D);\lambda)=a\mu(f;\lambda)+b\mu(g;\lambda),

where $g\in H^{0}(\mathbb{P}^{1}\times\mathbb{P}^{2},\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{2}}(2,2))$ is a representative of $[g]$ with minimal $\lambda$ -weight.

3.1. Stability thresholds for singular surfaces

The main goal in this part is to show that for each type of singular surface $X$ of class $\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{2}}(1,2)$ , the pair $(X,D)$ is not t-semistable for any $D\in|-2K_{X}|$ when $t$ is small. We will see later that the bounds given in this section are indeed the stability thresholds.

Lemma 3.2.

Let $X\subseteq\mathbb{P}^{1}_{u,v}\times\mathbb{P}^{2}_{x,y,z}$ be an irreducible surface of class $\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{2}}(1,2)$ , then $X$ satisfies one of the followings:

(i)

$X$ is smooth;
(ii)

$X$ has a unique $A_{1}$ -singularity;
(iii)

$X$ has exactly two $A_{1}$ -singularities;
(iv)

$X$ has a unique $A_{2}$ -singularity.
(v)

$X$ has a unique $A_{3}$ -singularity.
(vi)

$X$ is singular along a line, which is a fiber of $\mathbb{P}^{1}\times\mathbb{P}^{2}\rightarrow\mathbb{P}^{2}$ .

Proof.

We may assume that

X=\{uf_{2}(x,y,z)+vg_{2}(x,y,z)=0\}

has a singularity at $p=((0:1),(1:0:0))$ . Then $f_{2}$ has no $x^{2}$ term, and $g_{2}$ has no $x^{2},xy,xz$ . After a change of coordinates via a linear action on $(y,z)$ , we may also assume that $f_{2}$ has no $xy$ term. Notice that the coefficient of $xz$ and $y^{2}$ in $f_{2}$ is non-zero: otherwise $X$ will either be singular along the line $((u:v),(1:0:0))$ or have an irreducible component $\{z=0\}$ , a contradiction to the irreducibility of $X$ . Thus if $g_{2}$ has a non-zero term $y^{2}$ , then $X$ has an $A_{1}$ -singularity at $p$ . In this case, we may assume $X$ has another singularity $q=((u_{q}:v_{q}),(x_{q}:y_{q}:z_{q}))$ . Taking partial derivative for $x$ , we see that either $u_{q}=0$ or $z_{q}=0$ .

(a)

If $u_{q}=0$ , then $(y_{q}:z_{q})$ has to be a singular point of $g_{2}$ . If $g_{2}$ is reduced, then $z_{q}=y_{q}=0$ , and hence $q=p$ , which is a contradiction. Thus $g_{2}=(ay+bz)^{2}$ , where $a\neq 0$ . In this case, $X$ has two $A_{1}$ singularities at $p$ and $q$ .
(b)

If $z_{q}=0$ , then taking the $v$ derivative, one gets that either $y_{q}=0$ because $g_{2}$ has a non-zero $y^{2}$ term. In the former case, taking the $x$ derivative, one obtains that $u_{q}=0$ and hence $q=p$ .

Now we assume that $g_{2}$ has no $y^{2}$ term, and thus $g_{2}=z(ay+bz)$ . There are also two cases:

(a)

$a=0$ , and hence $\{g_{2}=0\}$ is a double line tangent to the smooth conic $\{f_{2}=0\}$ at the point $p$ . In this case, $p$ is the only singularity of $X$ , which is of $A_{3}$ -type.
(b)

$a\neq 0$ , and thus $\{g_{2}=0\}=L_{1}\cup L_{2}$ is a union of two lines in $\mathbb{P}^{2}$ , where $L_{1}$ is tangent to the smooth conic $\{f_{2}=0\}$ at the point $p$ , and $L_{2}$ passes through $p$ and intersects the conic properly. In this case, $p$ is the only singularity of $X$ , which is of $A_{2}$ -type.

∎

For any rational number $0<t<1/2$ , we say that $(X,D)$ is t-(semi)stable if $(X,D)$ is GIT-(semi)stable with respect to the polarization $L_{t}:=\eta+t\xi$ . By the HM criterion, this is equivalent to saying that $\mu^{t}(X,D;\lambda)\geq 0$ for any 1-PS $\lambda$ of $(X,D)$ . We say that a 1-PS $\lambda$ induced by the $\mathbb{G}_{m}$ action of weight $(t_{0},t_{1};s_{0},s_{1},s_{2})$ , or for short a 1-PS $\lambda$ of weight $(t_{0},t_{1};s_{0},s_{1},s_{2})$ , if

\lambda\cdot((u:v),(x:y:z))=((\lambda^{t_{0}}u:\lambda^{t_{1}}v),(\lambda^{s_{0}}x:\lambda^{s_{1}}y:\lambda^{s_{2}}z)).

We first show that if $X$ has non-isolated singularities, then $(X,D)$ is never t-semistable for $0<t<1/2$ and $D\in|-2K_{X}|$ .

Proposition 3.3.

If a point $([f],[g])\in\mathbb{P}\mathcal{E}$ is t-semistable for some $0<t<\frac{1}{2}$ , then $X=\mathbb{V}(f)$ is irreducible.

Proof.

Notice that if $X$ is reducible, then it has a component of the class either $\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{2}}(0,1)$ or $\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{2}}(1,0)$ . In the former case, we may assume this component is defined by $\{x=0\}$ . Consider the action $\lambda$ of weight $(0,0;-2,1,1)$ . Then $\mu(f;\lambda)\leq-1$ and $\mu(g;\lambda)\leq 2$ so that $\mu^{t}(X,D;\lambda)<0$ for any $t<\frac{1}{2}$ . In the latter case, assuming this component is defined by $\{u=0\}$ , the action $\lambda$ of weight $(-1,1;0,0,0)$ satisfies $\mu(f;\lambda)=-1$ and $\mu(g;\lambda)\leq 2$ so that $\mu^{t}(X,D;\lambda)<0$ for any $t<\frac{1}{2}$ . ∎

Proposition 3.4.

If $X\subseteq\mathbb{P}^{1}\times\mathbb{P}^{2}$ is singular along a line, which is a fiber over $\mathbb{P}^{2}$ , then $X$ is t-unstable for any $0<t<1/2$ .

Proof.

The defining equation of $X$ is of the form $uf_{2}(y,z)+vf_{2}(y,z)=0$ . Consider the 1-PS $\lambda$ induced by the $\mathbb{G}_{m}$ -action of weight $(0,0;2,-1,-1)$ . Then $\mu(X;\lambda)=-2$ , and hence

\mu^{t}(X,D;\lambda)\leq-2+4t<0

for any $0<t<1/2$ . ∎

Corollary 3.5.

For any $0<t<\frac{1}{2}$ , the locus of t-semistable points $(X,D)\in\mathbb{P}\mathcal{E}$ is contained in the open subset $U$ of $\mathbb{P}\mathcal{E}$ parameterizing complete intersections. Moreover, for any t-semistable point $(X,D)$ , the surface has only isolated singularities of type $A_{1}$ , $A_{2}$ or $A_{3}$ .

Proposition 3.6.

Suppose $X$ is an irreducible singular surface with an $A_{1}$ -singularity. Then $(X,D)$ is t-unstable for any $D\in H^{0}(\mathbb{P}^{1}\times\mathbb{P}^{2},\mathcal{O}_{X}(2,2))$ and $0<t<\frac{1}{14}$ .

Proof.

We may assume $X$ is given by the equation $uyz+vx(x+y+z)=0$ with an $A_{1}$ -singularities $p=((0:0:1),(0:1))$ . Consider the action $\lambda$ of weight $(-3,3;-2,4,-2)$ . Then we have $\mu(X,\lambda)=-1$ , and

\mu^{t}(X,D;\lambda)\leq-1+14t<0

for $0<t<\frac{1}{14}$ . ∎

Proposition 3.7.

Let $X\subseteq\mathbb{P}^{1}\times\mathbb{P}^{2}$ be a quintic del Pezzo surface with two $A_{1}$ -singularities at $p$ and $q$ , and $D\in|-2K_{X}|$ a boundary divisor. Then for $t<\frac{1}{8}$ , the point $(X,D)$ is t-unstable.

Proof.

We may assume $X$ is given by the equation $uyz+vx^{2}=0$ with two $A_{1}$ -singularities at $p=((0:0:1),(0:1))$ and $q=((0:1:0),(0:1))$ . Consider the action $\lambda$ of weight $(-3,3;-2,1,1)$ . Then we have $\mu(f,\lambda)=-1$ , and

\mu^{t}(X,D;\lambda)\leq-1+8t<0

for $0<t<\frac{1}{8}$ . ∎

Proposition 3.8.

Let $X\subseteq\mathbb{P}^{1}\times\mathbb{P}^{2}$ be a quintic del Pezzo surface with an $A_{2}$ -singularity at $p$ , and $D\in|-2K_{X}|$ a boundary divisor. Then for $t<\frac{1}{6}$ , the point $(X,D)$ is t-unstable.

Proof.

We may assume $X$ is given by the equation $uyz+v(xz-y^{2})=0$ with an $A_{2}$ -singularity at $p=((0:0:1),(1:0))$ . Consider the action $\lambda$ of weight $(1,-1;-2,0,2)$ . Then we have $\mu(f,\lambda)=-1$ , and

\mu^{t}(X,D;\lambda)\leq-1+6t<0

for $0<t<\frac{1}{6}$ . ∎

Proposition 3.9.

Let $X\subseteq\mathbb{P}^{1}\times\mathbb{P}^{2}$ be a quintic del Pezzo surface with an $A_{3}$ -singularity at $p$ , and $D\in|-2K_{X}|$ a boundary divisor. Then for $t<\frac{1}{4}$ , the point $(X,D)$ is t-unstable.

Proof.

We may assume $X$ is given by the equation $ux^{2}+v(xz-y^{2})=0$ with an $A_{3}$ -singularity at $p=((0:0:1),(1:0))$ . Consider the action $\lambda$ of weight $(1,-1;-1,0,1)$ . Then we have $\mu(f,\lambda)=-1$ , and

\mu^{t}(X,D;\lambda)\leq-1+4t<0

for $0<t<\frac{1}{4}$ . ∎

3.2. Surfaces with exactly one singularity

From now on, we take the boundary divisor $D$ into consideration. We will determine the value $T$ for which a pair $(X,D)$ is $T$ -semistable but not t-semistable for any $t<T$ . Since the surface $X$ with two $A_{1}$ -singularities is a toric surface, which is more complicated than the others, we first deal with the surfaces with a single singularity.

Lemma 3.10.

Let $X\subseteq\mathbb{P}^{1}\times\mathbb{P}^{2}$ be the quintic del Pezzo surface defined by the equation

uyz+vx(y+z)=0,

which has an $A_{1}$ -singularity at $p=((1:0:0),(1:0))$ . Let $D\in|-2K_{X}|$ be the curve on $X$ which contains a line of multiplicity $4$ , defined by the equation $u^{2}x^{2}=0$ . Then $(X,D)$ is t-semistable if and only if $t=1/14$ .

Proof.

Consider the two one-parameter subgroups given by the $\mathbb{G}_{m}$ -actions of weights $(3,-3;4,-2,-2)$ and $(-3,3;-4,2,2)$ . We see that if $(X,D)$ is t-semistable, then $t=1/14$ . Now we prove the converse: the HM-invariant is non-negative with respect to any 1-PS. Let $\lambda$ be a 1-PS, and we will discuss by cases.

(1)
Suppose the weight of $\lambda$ is $(0,0;a+b,-a,-b)$ . Then we have

$\mu(X;\lambda)=\max\{a,b,-a-b\}.$

We may moreover assume $a\geq b$ . There are three subcases.
1. (i)
  
  $a\geq b\geq-a-b$ : we have $\mu(X;\lambda)=a$ and $\mu(D;\lambda)=2(a+b)\geq a$ . It follows that
  
  $\mu^{t}(X,D;\lambda)\geq\frac{15}{14}a\geq 0$
  
  for $t=1/14$ .
2. (ii)
  
  $-a-b\geq a\geq b$ : we have $\mu(X;\lambda)=-a-b$ and $\mu(D;\lambda)=2(a+b)$ . It follows that
  
  $\mu^{t}(X,D;\lambda)\geq-\frac{6}{7}(a+b)\geq 0$
  
  for $t=1/14$ .
3. (iii)
  
  $a\geq-a-b\geq b$ : we have $\mu(X;\lambda)=a$ and $\mu(D;\lambda)=a\geq 2(a+b)\leq a$ . It follows that
  
  $\mu^{t}(X,D;\lambda)=a+\frac{1}{7}(a+b)=\frac{6}{7}a+\frac{1}{7}(2a+b)\geq 0$
  
  for $t=1/14$ .

(2)

Suppose the weight of $\lambda$ is $(1,-1;a+b,-a,-b)$ . Then we have

\mu(X;\lambda)=\max\{a-1,b-1,1-a-b\}.

Again, we can assume that $a\geq b$ . There are still three subcases to discuss.

(i)

$a\geq b\geq-a-b$ : If $a-1\geq 1-a-b$ , i.e. $b\geq 2-2a$ , then we have $\mu(X;\lambda)=a-1$ and $\mu(D;\lambda)=2+2(a+b)$ . It follows that

\mu^{\frac{1}{14}}(X,D;\lambda)=a-1+\frac{1}{7}(1+a+b)\geq a-1+\frac{1}{7}(3-a)=\frac{2}{7}(3a-2)\geq 0

since we have $a\geq b\geq 2-2a$ . Similarly, if $1-a-b\geq a-1$ , i.e. $b\leq 2-2a$ , then we have $\mu(X;\lambda)=1-a-b$ and $\mu(D;\lambda)=2+2(a+b)$ . It follows that

\begin{split}\mu^{\frac{1}{14}}(X,D;\lambda)&=1-a-b+\frac{1}{7}(1+a+b)\\ &=\frac{2}{7}(4-3a-3b)\\ &=\frac{2}{7}((4-4a-2b)+(a-b))\geq 0.\end{split}

(ii)

$-a-b\geq a\geq b$ : we have $\mu(X;\lambda)=1-a-b$ and $\mu(D;\lambda)=2+2(a+b)$ . It follows that

$\begin{split}\mu^{\frac{1}{14}}(X,D;\lambda)&=1-a-b+\frac{1}{7}(1+a+b)\\ &=\frac{2}{7}(4-3a-3b)\geq 0,\end{split}$

since in this case we always have $a+b\geq 0$ .

(iii)

$a\geq-a-b\geq b$ : If $a-1\geq 1-a-b$ , i.e. $b\geq 2-2a$ , then we have $\mu(X;\lambda)=a-1$ and $\mu(D;\lambda)=2+2(a+b)$ . It follows that

\mu^{\frac{1}{14}}(X,D;\lambda)=a-1+\frac{1}{7}(1+a+b)\geq a-1+\frac{1}{7}(3-a)=\frac{2}{7}(3a-2)\geq 0

since we have $a\geq b\geq 2-2a$ . Similarly, if $1-a-b\geq a-1$ , i.e. $b\leq 2-2a$ , then we have $\mu(X;\lambda)=1-a-b$ and $\mu(D;\lambda)=2+2(a+b)$ . It follows that

\begin{split}\mu^{\frac{1}{14}}(X,D;\lambda)&=1-a-b+\frac{1}{7}(1+a+b)\\ &=\frac{2}{7}(4-3a-3b)\\ &=\frac{2}{7}((4-4a-2b)+(a-b))\geq 0.\end{split}

(3)

The case when the weight of $\lambda$ is $(-1,1;a+b;-a;-b)$ is similar to the case (2), so we omit it here.

∎

Proposition 3.11.

Let $X\subseteq\mathbb{P}^{1}\times\mathbb{P}^{2}$ be a quintic del Pezzo surface with exactly one $A_{1}$ -singularity at $p$ , and $D\in|-2K_{X}|$ a boundary divisor. Then for $t=\frac{1}{14}+\varepsilon$ , the point $(X,D)$ is t-semistable if and only if $p\notin D$ and $D$ contains no curves of multiplicity $4$ .

Proof.

We may assume $X$ is given by the equation $uyz+vx(y+z)=0$ with an $A_{1}$ -singularity at $p=((1:0:0),(1:0))$ . If $D$ passes through $p$ , then any defining equation of $D$ does not contain the term $u^{2}x^{2}$ . Taking the 1-PS $\lambda$ of weight $(3,-3;4,-2,-2)$ , one has that $\mu(X,\lambda)=-1$ and $\mu(D;f)\leq 8$ so that $\mu^{t}(X,D;\lambda)<0$ . In fact, the same argument shows that if $\operatorname{mult}_{p}D\geq 2$ , then $(X,D)$ is t-unstable for any $0<t<1/2$ .

Now we may assume $D$ does not contain $p$ and $D\neq D_{0}:=\{x^{2}u^{2}=0\}$ by Lemma 3.10. Then any defining equation for $D$ has a non-zero $x^{2}u^{2}$ term. The $G_{m}$ -action

\lambda((u:v),(x:y:z))=((\lambda u:\lambda^{-1}v),(\lambda^{2}x:y:z))

induces a 1-PS, where the central fibre is $(X,D_{0})$ and any other fibre is isomorphic to $(X,D)$ . By Lemma 3.10 and openness of GIT-semistability, we have that $(X,D)$ is t-semistable for $t=1/14$ . The wall-crossing structure for VGIT implies that at least one (and hence a general one) of the $D\neq D_{0}$ such that $p\notin D$ satisfies that $(X,D)$ is t-semistable for $t=1/14+\varepsilon$ . Moreover, since for a general such $D$ , the automorphism group of $(X,D)$ is finite, then $(X,D)$ is in fact t-stable for $t=1/14+\varepsilon$ . Notice that we have an exact sequence of group scheme

0\longrightarrow\operatorname{Aut}^{0}(X)=\mathbb{G}_{m}\longrightarrow\operatorname{Stab}(X)\longrightarrow G_{0}\longrightarrow 0,

where $\operatorname{Stab}(X)$ is the stabilizer of $X$ under the $\operatorname{SL}(2)\times\operatorname{SL}(3)$ -action and $G_{0}$ is a finite group. Let $V$ be the affine subspace of $H^{0}(X,\mathcal{O}_{X_{n}}(2,2))$ consisting of polynomials whose coefficient of $x^{2}u^{2}$ is $1$ . Viewing $V$ as a vector space, the $G_{m}$ -action on $V\setminus\{0\}$ is induced by

t((u:v),(x:y:z))=((tu:t^{-1}v),(t^{2}x:y:z)).

Then $(V-\{0\})/\mathbb{G}_{m}$ is a weighted projective space $\mathbb{P}(1^{2},2^{5},3^{5},4^{3})$ with a $G$ -action. Let $E_{+}$ be the exceptional locus of the wall-crossing morphism

\overline{M}^{\operatorname{GIT}}(1/14+\varepsilon)\rightarrow\overline{M}^{\operatorname{GIT}}(1/14).

Observe that there is an injective dominant morphism $E_{+}\hookrightarrow\mathbb{P}(1^{2},2^{5},3^{5},4^{3})/G$ . Since both of them are proper and normal, and the later is irreducible, then this is an isomorphism by Zariski’s Main Theorem (ref. [Har13, Theorem 3.11.4]).

∎

It is seen in the proof to Proposition 3.11 that for the quintic del Pezzo surface $X$ with exactly one $A_{1}$ -singularity, and $D\in|-2K_{X}|$ such that $p\in D$ , we have that $(X,D)$ is t-unstable for any $t<1/2$ if $\operatorname{mult}_{p}D\geq 2$ ; for any $t<1/8$ if $\operatorname{mult}_{p}D=1$ . Now we prove the following result, which is a complement to this statement.

Proposition 3.12.

Let $X\subseteq\mathbb{P}^{1}\times\mathbb{P}^{2}$ be a quintic del Pezzo surface $X$ with exactly one $A_{1}$ -singularity, and $D\in|-2K_{X}|$ such that $\operatorname{mult}_{p}D=1$ . Then $(X,D)$ is t-semistable for $t=1/8$ .

Proof.

The condition $\operatorname{mult}_{p}D=1$ comes down to saying that for any defining equation $g$ of $D$ , $g$ has no $x^{2}u^{2}$ term, but at least one of the coefficients of $u^{2}xy,u^{2}xz,uvx^{2}$ is non-zero. Notice that for any $D$ such that $\operatorname{mult}_{p}D=1$ , there is a $\mathbb{G}_{m}$ -action inducing a degeneration from $(X,D)$ to $(X,D_{0})$ , where $D_{0}$ has only $u^{2}xy,u^{2}xz,uvx^{2}$ . Thus by the openness of GIT-semistability, it suffices to show that $(X,D_{0})$ is $\frac{1}{8}$ -semistable for any $D_{0}$ given by the equation

au^{2}xy+bu^{2}xz+cuvx^{2}=0,

where $(a:b:c)\in\mathbb{P}^{2}$ . Indeed, this is true using the same argument as in the proof of Lemma 3.10. ∎

The next result involves the surfaces with an $A_{2}$ or an $A_{3}$ singularity. The proof is identical to that of Proposition 3.12, hence we wrap up the statements together and omit the details of the proof but only give the 1-parameter subgroups that imposes conditions on the stability thresholds.

Proposition 3.13.

Let $X\subseteq\mathbb{P}^{1}\times\mathbb{P}^{2}$ be a quintic del Pezzo surface with exactly one singularity $p$ , and $D\in|-2K_{X}|$ a boundary divisor.

(1)
Suppose $X=\{uyz+v(xz-y^{2})=0\}$ and $p$ is an $A_{2}$ -singularity.
1. (i)
  
  If $p\notin D$ , then $(X,D)$ is $\frac{1}{6}$ -semistable but not $t$ -semistable for $t<1/4$ ;
2. (ii)
  
  if $\operatorname{mult}_{p}D=1$ , then $(X,D)$ is $\frac{1}{4}$ -semistable but not $t$ -semistable for $t<1/4$ ;
3. (iii)
  
  if $\operatorname{mult}_{p}D\geq 2$ , then $(X,D)$ is t-unstable for any $0<t<1/2$ .
(2)
Suppose $X=\{ux^{2}+v(xz-y^{2})=0\}$ and $p$ is an $A_{3}$ -singularity.
1. (i)
  
  If $p\notin D$ , then $(X,D)$ is $\frac{1}{4}$ -semistable but not $t$ -semistable for $t<1/4$ ;
2. (ii)
  
  if $\operatorname{mult}_{p}D=1$ , then $(X,D)$ is $\frac{1}{3}$ -semistable but not $t$ -semistable for $t<1/3$ ;
3. (iii)
  
  if $\operatorname{mult}_{p}D\geq 2$ , then $(X,D)$ is t-unstable for any $0<t<1/2$ .

Then for $1/4<t<1/3$ , the point $(X,D)$ is t-semistable if and only if $p\notin D$ and $D\neq$

Proof.

Consider the $\mathbb{G}_{m}$ -action of weight $(1,-1:-2,0,2)$ for (1), and weight $(1,-1;-1,0,1)$ for (2). ∎

3.3. Surfaces with two singularities

For the rest of the this section, let us deal with the surface $X$ with exactly two $A_{1}$ -singularities $p,q$ . The complexity stems from the fact that this surface is toric, and hence has a number of $\mathbb{G}_{m}$ -actions.

As before, we may assume the defining equation of $X$ is $uyz+vx^{2}=0$ , and hence $p=((0:0:1),(0:1))$ and $q=((0:1:0),(0:1))$ .

Lemma 3.14.

Let $D\in|-2K_{X}|$ be a divisor. If either $\operatorname{mult}_{p}D\geq 2$ or $\operatorname{mult}_{q}D\geq 2$ , then $(X,D)$ is t-unstable for any $0<t<1/2$ .

Proof.

We may assume that $\operatorname{mult}_{p}D\geq 2$ , i.e. for any defining equation $g$ of $D$ , the coefficients of $v^{2}z^{2},v^{2}xz,v^{2}yz$ and $uvz^{2}$ in $g$ are all zero. Consider the 1-PS $\lambda$ of weight $(-3,3;-2,-2,4)$ . We have that $\mu(X;\lambda)=-1$ and $\mu(D;\lambda)\leq 2$ , thus $\mu^{t}(X,D;\lambda)<0$ for any $0<t<1/2$ . ∎

Observe that $p\notin D$ comes down to saying that any defining equation $g$ of $D$ has non-zero $z^{2}v^{2}$ term, and $\operatorname{mult}_{p}D\leq 1$ is equivalent to saying that at least one of the coefficients of $z^{2}v^{2},v^{2}xz,v^{2}yz,z^{2}uv$ is non-zero. A similar equivalent condition for $q\notin D$ also holds.

Proposition 3.15.

Let $D\in|-2K_{X}|$ be a divisor. Then $(X,D)$ is $\frac{1}{8}$ -semistable if either $p,q\notin D$ or a defining equation of $D$ has a non-zero $v^{2}yz$ term.

Proof.

Notice that any such pair specializes to the pair $(X,D_{0})$ (i.e. one can find a 1-PS, where general fibers are isomorphic to $(X,D)$ , and the central fiber is $(X,D_{0})$ ), where $D_{0}$ is defined by the equation $v^{2}f(y,z)$ for some quadratic homogeneous function in $y$ and $z$ which is not equal to $y^{2}$ or $z^{2}$ . By the openness of GIT-semistability, it suffices prove for the pair $(X,D_{0})$ . This is true by the same argument as in the proof of Lemma 3.10. ∎

To make our notation succinct, we set $a_{x^{2}uv}$ be the coefficient of $x^{2}uv$ in a defining equation of $D$ , and same for the other monomial terms. Now we only need to consider the divisor $D\in|-2K_{X}|$ such that either $a_{v^{2}yz}=a_{v^{2}z^{2}}=0$ or $a_{v^{2}yz}=a_{v^{2}y^{2}}=0$ . We may assume the latter case.

Case 1.

$a_{y^{2}v^{2}}\neq 0$ and $a_{v^{2}xz}\neq 0$ :

Each such pair specializes via the $\mathbb{G}_{m}$ -action of weight $(-3,3;-2,0,2)$ to $(X,D_{1})$ where $D_{0}$ is defined by an equation $v^{2}(y^{2}-axz)$ for some $a\neq 0$ . Consider the same $\mathbb{G}_{m}$ -action on the pair $(X,D_{1})$ , we see that it is t-unstable for any $t<1/6$ . In fact, using the same argument as in Lemma 3.10, we can prove the following.

Lemma 3.16.

The pair $(X,D_{1})$ is $\frac{1}{6}$ -semistable but not t-semistable for any $t<\frac{1}{6}$ .

Case 2.

$a_{y^{2}v^{2}}\neq 0$ , $a_{v^{2}xz}=0$ and $a_{uvz^{2}}\neq 0$ :

Each such pair specializes via the $\mathbb{G}_{m}$ -action of weight $(-6,6;-4,-1,5)$ to $(X,D_{2})$ where $D_{2}$ is defined by an equation $v(vy^{2}+az^{2}u)$ for some $a\neq 0$ . As in the previous case, we have the following result.

Lemma 3.17.

The pair $(X,D_{3})$ is $\frac{1}{5}$ -semistable but not t-semistable for any $t<\frac{1}{5}$ .

For the remaining cases, we can assume moreover $a_{y^{2}v^{2}}=0$ .

Case 3.

$a_{v^{2}xy}\neq 0$ and $a_{v^{2}xz}\neq 0$ :

Each such pair specializes via the $\mathbb{G}_{m}$ -action of weight $(-3,3;-2,1,1)$ to $(X,D_{3})$ where $D_{3}$ is defined by an equation $v^{2}x(y+az)$ for some $a\neq 0$ . Similar as above case, we have the following result.

Lemma 3.18.

The pair $(X,D_{3})$ is $\frac{1}{5}$ -semistable but not t-semistable for any $t<\frac{1}{5}$ .

Now we can assume at least one of $a_{v^{2}xy}$ and $a_{v^{2}xz}$ is zero. Suppose $a_{v^{2}xz}=0$ . There are two remaining cases.

Case 4.

$a_{z^{2}uv}\neq 0$ and $a_{v^{2}xy}\neq 0$ :

Each such pair specializes via the $\mathbb{G}_{m}$ -action of weight $(-3,3;-2,0,2)$ to $(X,D_{4})$ where $D_{4}$ is defined by an equation $v(z^{u}+avxy)$ for some $a\neq 0$ . Similarly, we have the following result.

Lemma 3.19.

The pair $(X,D_{4})$ is $\frac{1}{4}$ -semistable but not t-semistable for any $t<\frac{1}{4}$ .

Case 5.

$a_{z^{2}uv}\neq 0$ , $a_{y^{2}uv}\neq 0$ and $a_{v^{2}xy}=0$ :

In this case, we have $\operatorname{mult}_{p}D,\operatorname{mult}_{q}D\geq 2$ , and thus $(X,D)$ is t-unstable for any $t<1/2$ .

4. Relation to K-moduli spaces

4.1. Computation of CM-line bundles

The main purpose of this section is to compute the CM $\mathbb{Q}$ -line bundle for the family $(\mathscr{X},c\mathscr{D})\rightarrow U\subseteq\mathbb{P}\mathcal{E}$ , where $U$ is the Zariski big open subset of $\mathbb{P}\mathcal{E}$ parameterizing complete intersections $(X,D)$ with $X$ irreducible. Moreover, we will show that the CM $\mathbb{Q}$ -line bundle is proportional to the polarization line bundle for taking VGIT.

Proposition 4.1.

We have that

-f_{*}(-K_{\mathscr{X}/U}-c\mathscr{D})^{3}=3(1-2c)^{2}(4+2c)\left(\eta+\frac{5c}{4+2c}\xi\right).

Proof.

It follows from our construction that

\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{2}\times U}(\mathscr{X})=p_{1}^{*}\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{2}}(1,2)\otimes p_{2}^{*}\pi^{*}\mathcal{O}_{\mathbb{P}^{11}}(1),

\mathcal{O}_{\mathscr{X}}(\mathscr{D})=p_{1}^{*}\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{2}}(2,2)|_{\mathscr{X}}\otimes p_{1}^{*}\mathcal{O}_{U}(1)|_{\mathscr{X}}.

By the adjunction formula, we have that

\begin{split}K_{\mathscr{X}}&=(K_{\mathbb{P}^{1}\times\mathbb{P}^{2}\times U}+\mathscr{X})|_{\mathscr{X}}\\ &=p_{1}^{*}\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{2}}(-1,-1)|_{\mathscr{X}}\otimes p_{2}^{*}(K_{U}+\pi^{*}\mathcal{O}_{\mathbb{P}^{11}}(1))|_{\mathscr{X}},\end{split}

and thus

K_{\mathscr{X}/U}=p_{1}^{*}\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{2}}(-1,-1)|_{\mathscr{X}}\otimes f^{*}\pi^{*}\mathcal{O}_{\mathbb{P}^{11}}(1)|_{\mathscr{X}}.

Adding the class of $\mathscr{D}$ , we obtain that

-K_{\mathscr{X}/U}-c\mathscr{D}=(1-2c)p_{1}^{*}\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{2}}(1,1)|_{\mathscr{X}}\otimes f^{*}(\pi^{*}\mathcal{O}_{\mathbb{P}^{11}}(-1)-c\mathcal{O}_{U}(1)).

For simplicity, we denote $p_{1}^{*}\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{2}}(a,b)|_{\mathscr{X}}$ by $\mathcal{O}_{\mathscr{X}}(a,b)$ . It is easy to see that

(\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{2}}(1,1))^{3}=3,\quad\textup{and}\quad\mathcal{O}_{\mathscr{X}}(1,1)^{2}=5.

Now we compute that

\begin{split}-f_{*}(-K_{\mathscr{X}/U}-c\mathscr{D})^{3}&=-f_{*}((1-2c)^{3}\mathcal{O}_{\mathscr{X}}(1,1)^{3}\\ &\quad\quad\quad-3(1-2c)^{3}\mathcal{O}_{\mathscr{X}}(1,1)^{2}f^{*}(\pi^{*}\mathcal{O}_{\mathbb{P}^{11}}(-1)-c\mathcal{O}_{U}(1))\\ &\quad\quad\quad+3(1-2c)\mathcal{O}_{\mathscr{X}(1,1)}f^{*}(\pi^{*}\mathcal{O}_{\mathbb{P}^{11}}(1)+c\mathcal{O}_{U}(1))^{2}\\ &\quad\quad\quad-f^{*}(\pi^{*}\mathcal{O}_{\mathbb{P}^{11}}(1)+c\mathcal{O}_{U}(1))^{3})\\ &=-3(1-2c)^{3}\pi^{*}\mathcal{O}_{\mathbb{P}^{11}}(1)+3(1-2c)^{2}\cdot 5(\pi^{*}\mathcal{O}_{\mathbb{P}^{11}}(1)+c\mathcal{O}_{U}(1))\\ &=3(1-2c)^{2}\left(5(\eta+c\xi)-(1-2c)\eta\right)\\ &=3(1-2c)^{2}(4+2c)\left(\eta+\frac{5c}{4+2c}\xi\right).\end{split}

∎

Corollary 4.2.

For any rational number $0<c<\frac{11}{52}$ , we set

t=t(c):=\frac{5c}{4+2c}.

Then on the open subset $U$ of $\mathbb{P}\mathcal{E}$ , the CM $\mathbb{Q}$ -line bundle $\Lambda_{\operatorname{CM},f,c}$ and the polarization $\mathcal{L}_{t}:=\eta+t\xi$ are proportional.

4.2. VGIT wall-crossings and K-moduli

Theorem 4.3.

The VGIT walls are

t_{0}=0,\quad t_{1}=\frac{1}{14},\quad t_{2}=\frac{1}{8},\quad t_{3}=\frac{1}{6},\quad t_{4}=\frac{1}{5},\quad t_{5}=\frac{1}{4},\quad t_{6}=\frac{1}{3},\quad t_{7}=\frac{1}{2},

which bijectively correspond via the relation $\displaystyle t(c)=\frac{5c}{4+2c}$ to the K-moduli walls

c_{0}=0,\quad c_{1}=\frac{1}{17},\quad c_{2}=\frac{2}{19},\quad c_{3}=\frac{1}{7},\quad c_{4}=\frac{4}{23},\quad c_{5}=\frac{2}{9},\quad c_{6}=\frac{4}{13},\quad c_{7}=\frac{1}{2}.

Among the walls $t_{i}\in(0,1/2)$ , only the one $t_{1}=\frac{1}{14}$ is a divisorial contraction, and the remaining 5 walls are flips.

Proof.

By the local structure of VGIT, new pairs appear in the GIT moduli stacks $\mathcal{M}^{\operatorname{GIT}}(t)$ for each wall $t$ . By Lemma 3.2, Proposition 4.4 and discussions of stability thresholds in Section 3 about all the pairs that appear in $\mathcal{M}^{\operatorname{GIT}}(t)$ for some $0<t<1/2$ , we conclude that $t_{0},...,t_{7}$ listed above are all the walls for the VGIT quotient $\mathbb{P}\mathcal{E}\mathbin{\mkern-3.0mu/\mkern-6.0mu/\mkern-3.0mu}_{t}\operatorname{SL}(2)\times\operatorname{SL}(3)$ .

Crossing the first wall, we replace $(\Sigma,D_{0})$ by $(X,D)$ , where $D_{0}\in|-2K_{\Sigma}|$ is the curve with a component of multiplicity $4$ , $X$ is a quintic del Pezzo surface with exactly one $A_{1}$ -singularity, and $D\in|2K_{X}|$ does not pass through the singularity of $X$ . Counting dimensions, we see that the wall-crossing map

\overline{M}^{\operatorname{GIT}}\left(\frac{1}{14}+\varepsilon\right)\longrightarrow\overline{M}^{\operatorname{GIT}}\left(\frac{1}{14}\right)

is a divisorial contraction. Similarly, we can prove that the other walls are all flips by computing the dimension of the exceptional locus of $\overline{M}^{\operatorname{GIT}}\left(t_{i}+\varepsilon\right)\rightarrow\overline{M}^{\operatorname{GIT}}\left(t_{i}\right)$ . ∎

Recall that $U\subseteq\mathbb{P}\mathcal{E}$ is the open subset consisting of complete intersections $(X,D)$ . Set $U^{K}(c)$ to be the subset of $U$ parameterizing pairs $(X,D)$ such that $(X,cD)$ is K-semistable, and $U^{\operatorname{GIT}}(c)$ to be the subset of $U$ parameterizing $\operatorname{GIT}_{t(c)}$ -semistable pairs $(X,D)$ .

Theorem 4.4.

For any $c\in(0,1/17)$ , we have $U^{K}(c)=U^{\operatorname{GIT}}(c)$ . Moreover, we have a finite map

\overline{M}^{\operatorname{GIT}}(c)\longrightarrow\overline{M}^{K}(c)

of degree $5$ . In fact, we have that

\overline{M}^{\operatorname{GIT}}(c)\simeq|-2K_{\Sigma}|\mathbin{\mkern-3.0mu/\mkern-6.0mu/\mkern-3.0mu}\mathfrak{S}_{4},\quad\textup{and}\quad\overline{M}^{K}(c)\simeq|-2K_{\Sigma}|\mathbin{\mkern-3.0mu/\mkern-6.0mu/\mkern-3.0mu}\mathfrak{S}_{5}.

Proof.

We may assume that $0<c\ll 1$ . Recall that $U^{K}(c)\simeq\pi^{-1}(V)$ , where $V$ is the open subset of $\mathbb{P}^{11}$ consisting of all smooth surfaces, and $\pi:\mathbb{P}\mathcal{E}\rightarrow\mathbb{P}^{11}$ the projection. From Proposition 3.6, we get that $U^{\operatorname{GIT}}(c)\subseteq U^{K}(c)$ . As a consequence, one obtains a morphism

\overline{M}^{\operatorname{GIT}}(c)\longrightarrow\overline{M}^{K}(c).

As $\overline{M}^{K}(c)$ is irreducible and both of the moduli spaces are proper, then this map is surjective. In particular, the c-K-stability and t(c)-GIT-stability are equivalent on $U$ for $0<c<c_{1}$ .

Notice that $\operatorname{SL}(2)\times\operatorname{SL}(3)$ acts transitively on the open subset of $\mathbb{P}^{11}$ consisting of smooth surfaces, and for any point $[X]$ in this locus, the stabilizer of $[X]$ is isomorphic to $\mathfrak{S}_{4}$ . In fact, viewing $X$ as a pencil of conics $uf_{2}(x,y,z)+vg_{2}(x,y,z)=0$ in $\mathbb{P}^{2}$ , the stabilizer of $X$ consists of $g\in\operatorname{SL}(3)$ fixing the four distinct points

\{(x:y:z)|f_{2}(x,y,z)=g_{2}(x,y,z)=0\}.

It follows that

\overline{M}^{\operatorname{GIT}}(c)\simeq|-2K_{\Sigma}|\mathbin{\mkern-3.0mu/\mkern-6.0mu/\mkern-3.0mu}\mathfrak{S}_{4}.

The last statement is exactly [Zha22, Theorem 1.1]. From this, one can also deduce the morphism $\overline{M}^{\operatorname{GIT}}(c)\rightarrow\overline{M}^{K}(c)$ is finite of degree $5$ . ∎

5. Hassett-Keel program for curves of genus six

Hassett-Keel program aims to understand log minimal models of moduli space $\overline{M}_{g}$ of curves. For any rational number $\alpha\in[0,1]$ such that $K_{\overline{M}_{g}}+\alpha\Delta$ is big, we can define

\overline{M}_{g}(\alpha):=\operatorname{Proj}\bigoplus_{n\geq 0}n\left(K_{\overline{M}_{g}}+\alpha\Delta\right).

A natural question is to look for modular interpretation of $\overline{M}_{g}(\alpha)$ . We refer the reader to [HM06] for results on moduli of curves and the survey [FS13] for the Hassett-Keel program.

In this section, we will prove Theorem 1.3. Let us fix some notation first.

Notation.

Let $U\subseteq\mathbb{P}\mathcal{E}$ be the open subset parameterizing complete intersections $(X,D)$ , where $X$ is irreducible, and $V:=\overline{M}_{6}\setminus(\Delta_{1}\cup\Delta_{2}\cup\Delta_{3})$ be an open subset in $\overline{M}_{6}$ . Let $0<c<\frac{11}{52}$ be a rational number and $\overline{M}_{K}(c)$ be the K-moduli space of K-polystable pairs $(X,cD)$ which admits a $\mathbb{Q}$ -Gorenstein smoothable degeneration to $(\Sigma_{5},cD_{0})$ , where $D_{0}$ is a smooth curve of class $-2K_{\Sigma_{5}}$ . Let $\phi:U\dashrightarrow\overline{M}_{K}(c)$ be the natural rational map, $\psi:U\dashrightarrow V$ be given by $(X,D)\mapsto D$ , and $\varphi:V\rightarrow\overline{M}^{K}(c)$ be $C\mapsto(X,C)$ , where $X$ is the unique quintic del Pezzo surface containing $C$ . Denote by $\overline{\varphi}$ the rational map $\overline{M}_{6}\dashrightarrow\overline{M}^{K}(c)$ , which is an extension of $\varphi$ .

5.1. The first chamber of $\alpha$

Recall that the K-moduli for $0<c\leq 1/17$ is identified with one of the Hassett-Keel models.

Theorem 5.1.

(ref. [Mül14, Proposition 4.3]) Let $0<c\leq\frac{1}{17}$ be a rational number. Then the log canonical model $\overline{M}_{6}(\alpha)$ is isomorphic to $|-2K_{\Sigma_{5}}|/\operatorname{Aut}(\Sigma_{5})$ for $16/47<\alpha\leq 35/102$ , a point for $\alpha=16/47$ , and empty for $\alpha<16/47$ .

Proposition 5.2.

Neither one of $\Delta_{1},\Delta_{2},\Delta_{3}$ is contracted under the natural birational map $\overline{M}_{6}\dashrightarrow|-2K_{\Sigma_{5}}|/\operatorname{Aut}(\Sigma_{5})$ to the point representing the curve $4L_{1}+2L_{2}+2E_{1}+2E_{2}$ .

Proof.

This is an immediate corollary of [Mül14, Proposition 2.1, Proposition 2.2, Proposition 2.3]. ∎

5.2. Wall-crossings in the Hassett-Keel program for $\overline{M}_{6}$

We may assume that $1/17<c<11/52$ .

Proposition 5.3.

Let $\lambda$ be the Hodge line bundle on $V$ , and $\delta_{0}$ the divisor on $V$ consisting of the closure of the locus of irreducible curves. Then we have

\psi^{*}\lambda=6\eta+6\xi,\quad\textup{and}\quad\psi^{*}\delta_{0}=46\eta+47\xi.

Proof.

Let $T\simeq\mathbb{P}^{1}$ be a curve in $\mathbb{P}\mathcal{E}$ given by a general pencil of curves in $|-2K_{X}|$ for a fixed smooth surface $X\in\mathbb{P}^{11}$ . Then we have that

(T.\eta)=0,\quad\textup{and}\quad(T.\xi)=1.

The universal family $\mathscr{C}\rightarrow T$ is the blow-up of $X$ at $(\mathcal{O}_{X}(2,2))^{2}=20$ general points, and hence $\chi_{\operatorname{top}}(\mathscr{C})=27$ . By topological Hurwitz formula, we have that the number of singular fibers is $27-2(-10)=47$ since a general fiber is smooth of genus $6$ , and each singular fiber is a nodal curve with exactly one singularity. On the other hand, the class of $\mathscr{C}$ in $T\times X$ is $\mathcal{O}(1,-2K_{X})$ , and hence by adjunction we have

K_{\mathscr{C}/T}=\mathscr{O}_{\mathscr{C}}(1,-K_{X}).

Thus the degree of the Hodge line bundle $\lambda_{T}$ on $T$ is

\begin{split}\deg\lambda_{T}&=c_{1}({p_{1}}_{*}\omega_{\mathscr{C}/T})\\ &=c_{1}(\mathcal{O}_{\mathbb{P}^{1}}(1)\otimes H^{0}(X,-K_{X}))\\ &=6.\end{split}

Similarly, fix a general surface $S\in|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{2}}(2,2)|$ , and take a general pencil of quintic del Pezzo surfaces in $|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{2}}(1,2)|$ to intersect $S$ , we obtain a one-parameter family of curves $\mathscr{C}^{\prime}\rightarrow T^{\prime}$ over $T^{\prime}\simeq\mathbb{P}^{1}$ . Here we view $T^{\prime}$ as a curve in $\mathbb{P}\mathcal{E}$ , and we have that

(T^{\prime}.\eta)=1,\quad\textup{and}\quad(T^{\prime}.\xi)=0.

The same computation as above yields that

(T^{\prime}.\psi^{*}\delta_{0})=46,\quad\textup{and}\quad(T^{\prime}.\psi^{*}\lambda)=6.

It follows immediately that

\psi^{*}\lambda=6\eta+6\xi,\quad\textup{and}\quad\psi^{*}\delta_{0}=46\eta+47\xi.

∎

Lemma 5.4.

The moduli spaces $\overline{M}_{6}(\alpha)$ and $\overline{M}^{K}(c)$ are all normal.

Proof.

Since the deformation of any curve $[C]\in\overline{M}_{6}(\alpha)$ is unobstructed, there is an étale map

U/\operatorname{Aut}(C)\longrightarrow\overline{M}_{6}(\alpha)

with image an open neighborhood of $[C]$ where $U$ is the first order deformation space of $[C]$ . Since $\operatorname{Aut}(X)$ is finite, then $\overline{M}_{6}(\alpha)$ is normal by a general result of GIT (ref. [MFK94, Chapter 0]). The normality of K-moduli spaces can be proven as in [ADL19, Proposition 4.6(3)]. ∎

Theorem 5.5.

Let $0\leq c\leq\frac{11}{52}$ be a rational number, and $\alpha(c)=\frac{32-19c}{94-68c}\in\left[\frac{16}{47},\frac{97}{276}\right]$ . Then we have an isomorphism

\overline{M}_{6}(\alpha(c))\simeq\overline{M}^{K}(c).

Proof.

Recall that in our setting, there is a commutative diagram

\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 14.13101pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\\\\\crcr}}}\ignorespaces{\hbox{\kern-6.95901pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-10.17084pt\raise-34.2311pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{\phi}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-57.87111pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 27.82446pt\raise 12.21777pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-5.185pt\hbox{$\scriptstyle{\huge{\psi}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 68.13101pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern 38.13101pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 68.13101pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 37.10416pt\raise-39.41858pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.8264pt\hbox{$\scriptstyle{\varphi}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 11.62538pt\raise-57.87111pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-3.0pt\raise-33.41666pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-14.13101pt\raise-68.4622pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\overline{M}^{K}(c)}$}}}}}}}\ignorespaces}}}}\ignorespaces.

We have that

\psi^{*}\varphi^{*}\Lambda_{\operatorname{CM},\overline{M}^{K}_{c},c}=\phi^{*}\Lambda_{\operatorname{CM},\overline{M}^{K}_{c},c}=\Lambda_{\operatorname{CM},U,c}=3(1-2c)^{2}(4+2c)\left(\eta+\frac{5c}{4+2c}\xi\right).

On the other hand, we have that

\psi^{*}(K_{V}+\alpha\delta_{0})=\psi^{*}(13\lambda-(2-\alpha)\delta_{0})=13(6\eta+6\xi)-(2-\alpha)(46\eta+47\xi),

which is proportional, up to a positive constant, to

\eta+\frac{47\alpha-16}{46\alpha-14}\xi=\eta+\frac{5c}{4+2c}\xi.

Thus there is a constant $s>0$ such that $K_{\overline{M}_{6}}+\alpha\delta-\overline{\varphi}^{*}(s\Lambda_{\operatorname{CM},\overline{M}^{K}_{c},c})$ is a $\overline{\varphi}$ -exceptional $\mathbb{Q}$ -divisor.

By Proposition 5.2, we see that the images of the divisors $\Delta_{1},\Delta_{2},\Delta_{3}$ under the rational map

\phi_{0<c\leq 1/17}:\overline{M}_{6}\dashrightarrow|-2K_{\Sigma_{5}}|/\mathfrak{S}_{5}

is not contained in the center of the weighted blow-up

\overline{M}^{K}(1/17+\varepsilon)\longrightarrow\overline{M}^{K}(1/17).

Consider the commutative diagram

\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 6.90001pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\\\\&\crcr}}}\ignorespaces{\hbox{\kern-6.90001pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\overline{M}_{6}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 8.18373pt\raise-42.13443pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.58888pt\hbox{$\scriptstyle{\overline{\varphi}_{0<c\leq 1/17}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 62.05063pt\raise-59.5pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 42.52275pt\raise 9.54245pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.86469pt\hbox{$\scriptstyle{\huge{\overline{\varphi}_{1/17<c\leq 11/52}\quad}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 139.2729pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern 70.08646pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 139.2729pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\overline{M}^{K}(1/17<c\leq 11/52)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 131.28442pt\raise-39.55249pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\sigma}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 91.31058pt\raise-59.5pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-3.0pt\raise-35.04555pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-3.0pt\raise-70.0911pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 30.90001pt\raise-70.0911pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\overline{M}^{K}(0<c\leq 1/17)}$}}}}}}}\ignorespaces}}}}\ignorespaces,

where $\sigma$ is the weighted blow-up followed by some birational modifications in codimension $2$ . Since any non-special curve of genus 6 is contained in a unique ADE quintic del Pezzo surface, then $\overline{\varphi}_{1/17<c\leq 11/52}$ maps the Gieseker-Petri divisor $\overline{\mathcal{GP}}_{6}$ in $\overline{M}_{6}$ birationally to the $\sigma$ -exceptional divisor $E$ . By [Mül14, Proposition 4.3], there is a constant $s^{\prime}>0$ such that $K_{\overline{M}_{6}}+\alpha\delta-s^{\prime}\overline{\varphi}^{*}_{0<c\leq 1/17}\Lambda_{\operatorname{CM},c}$ is $\mathbb{Q}$ -linearly equivalent to

(35/2-51\alpha)\overline{\mathcal{GP}}_{6}+(9-11\alpha)\delta_{1}+(19-29\alpha)\delta_{2}+(34-96\alpha)\delta_{3}.

It follows that

\left(K_{\overline{M}_{6}}+\alpha\delta\right)-s\overline{\varphi}^{*}_{\frac{1}{17}<c\leq\frac{11}{52}}\Lambda_{\operatorname{CM},c}=(9-11\alpha)\delta_{1}+(19-29\alpha)\delta_{2}+(34-96\alpha)\delta_{3},

which is effective when $\alpha\leq\frac{34}{96}$ . Therefore by Lemma 5.4 we deduce

\begin{split}\overline{M}_{6}(\alpha)&=\operatorname{Proj}\bigoplus_{n}H^{0}\left(\overline{M}_{6},n(K_{\overline{M}_{6}}+\alpha\delta)\right)\\ &\simeq\operatorname{Proj}\bigoplus_{n}H^{0}\left(\overline{M}^{K}(c),ns(\Lambda_{\operatorname{CM},c})\right)\\ &=\overline{M}^{K}(c)\end{split}

for $1/17<c\leq 11/52$ . ∎

In [Zha22], we find the walls

\left\{0,\frac{1}{17},\frac{2}{19},\frac{1}{7},\frac{4}{23}\right\}

for K-moduli spaces $\overline{M}^{K}(c)$ . The only wall before $11/52$ that we miss is $1/5$ . This will be displayed in our upcoming work [SZ23]. As a consequence, the last walls of the log canonical models $\overline{M}_{6}(\alpha)$ are

\left\{\frac{16}{47},\frac{35}{102},\frac{29}{55},\frac{41}{118},\frac{22}{63},\frac{47}{134}\right\}.

Remark 5.6.

The value $\alpha(11/52)=329/964$ is not expected to be a wall for the Hassett-Keel. In fact, if it were, then by the local structure of VGIT, there will be divisor appearing in $\overline{M}_{6}(329/964+\varepsilon)$ . However, the Picard group of $\overline{M}_{6}=\overline{M}_{6}(1)$ is of rank four, which is generated by $\Delta_{0},\Delta_{1},\Delta_{2},\Delta_{3}$ and $\overline{\mathcal{GP}}_{6}$ . The divisor $\Delta_{1},\Delta_{2},\Delta_{3}$ should not be the replacement of triple conics in $|-2K_{\Sigma}|$ . In fact, the triple conic is replaced by trigonal curves in the wall-crossing

\overline{M}^{K}(11/52+\varepsilon)\rightarrow\overline{M}^{K}(11/52)\simeq\overline{M}^{K}(11/52-\varepsilon)

for K-moduli, and the locus of trigonal curves in $\overline{M}_{6}$ is of codimension 2. As a result, the K-moduli spaces cannot characterize the Hassett-Keel for all $\alpha\in[0,1]$ .

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The Moduli Space of Genus Six Curves and K-stability: VGIT and the Hassett-Keel Program

Abstract.

1. Introduction

Theorem 1.1.

Theorem 1.2.

Theorem 1.3.

2. Preliminaries

2.1. Geometry of projective bundles

Proposition 2.1.

Proof.

Proposition 2.2.

Proof.

2.2. Variation of GIT

Lemma 2.3.

Theorem 2.4.

3. Stability of points in ℙ​ℰ\mathbb{P}\mathcal{E}

Proposition 3.1.

3.1. Stability thresholds for singular surfaces

Lemma 3.2.

Proof.

Proposition 3.3.

Proof.

Proposition 3.4.

Proof.

Corollary 3.5.

Proposition 3.6.

Proof.

Proposition 3.7.

Proof.

Proposition 3.8.

Proof.

Proposition 3.9.

Proof.

3.2. Surfaces with exactly one singularity

Lemma 3.10.

Proof.

Proposition 3.11.

Proof.

Proposition 3.12.

Proof.

Proposition 3.13.

Proof.

3.3. Surfaces with two singularities

Lemma 3.14.

Proof.

Proposition 3.15.

Proof.

Case 1.

Lemma 3.16.

Case 2.

Lemma 3.17.

Case 3.

Lemma 3.18.

Case 4.

Lemma 3.19.

Case 5.

4. Relation to K-moduli spaces

4.1. Computation of CM-line bundles

Proposition 4.1.

Proof.

Corollary 4.2.

4.2. VGIT wall-crossings and K-moduli

Theorem 4.3.

Proof.

Theorem 4.4.

Proof.

5. Hassett-Keel program for curves of genus six

Notation.

5.1. The first chamber of α\alpha

Theorem 5.1.

Proposition 5.2.

Proof.

5.2. Wall-crossings in the Hassett-Keel program for M¯6\overline{M}_{6}

Proposition 5.3.

Proof.

Lemma 5.4.

Proof.

Theorem 5.5.

Proof.

Remark 5.6.

3. Stability of points in $\mathbb{P}\mathcal{E}$

5.1. The first chamber of $\alpha$

5.2. Wall-crossings in the Hassett-Keel program for $\overline{M}_{6}$