Factorization centers in dimension two and the Grothendieck ring of varieties

Hsueh-Yung Lin, Evgeny Shinder, Susanna Zimmermann Department of Mathematics, National Taiwan University, and National Center for Theoretical Sciences, Taipei, Taiwan. [email protected] School of Mathematics and Statistics, University of Sheffield, Hounsfield Road, S3 7RH, UK, and Hausdorff Center for Mathematics at the University of Bonn, Endenicher Allee 60, 53115. [email protected] Laboratoire angevin de recherche en mathématiques (LAREMA), CNRS, Université d’Angers, 49045 Angers cedex 1, France [email protected]

Abstract.

We initiate the study of factorization centers of birational maps, and complete it for surfaces over a perfect field in this article. We prove that for every birational automorphism $\phi:X\dashrightarrow X$ of a smooth projective surface $X$ over a perfect field $k$ , the blowup centers are isomorphic to the blowdown centers in every weak factorization of $\phi$ . This implies that nontrivial L-equivalences of $0$ -dimensional varieties cannot be constructed based on birational automorphisms of a surface. It also implies that rationality centers are well-defined for every rational surface $X$ , namely there exists a $0$ -dimensional variety intrinsic to $X$ , which is blown up in any rationality construction of $X$ .

Key words and phrases:

Grothendieck ring, Birational automorphism, Algebraic Surface, Factorization center, Sarkisov link

2010 Mathematics Subject Classification:

14E07, 14E30, 14J26 14F20

1. Introduction

One source of motivation in birational geometry comes from studying groups of birational automorphisms of algebraic varieties, in particular the Cremona groups ${\operatorname{Cr}}_{n}(k)={\operatorname{Bir}}(\mathbb{P}^{n}_{k})$ . Each birational automorphism blows up some subschemes and contracts some exceptional divisors. The primary question we study in this paper is:

Question 1.1.

Let $\phi:X\dashrightarrow X$ be a birational automorphism of a smooth projective variety. Do centers blown up by $\phi$ correspond, up to stable birational equivalence, to the exceptional divisors blown down by $\phi$ ?

We note that exceptional divisors are ruled over the corresponding blow up centers, so asking about stable birational equivalence classes is a natural way to compare exceptional divisors with the blow up centers. We give a complete answer to a stronger version of Question 1.1 for surfaces over an arbitrary perfect field, see Theorem 1.3 below. Our proof is an application of the two-dimensional Minimal Model Program [Isk79, Isk96] combined with the Grothendieck ring of varieties, étale cohomology groups and the so-called Gassmann equivalence of Galois sets.

There are three main reasons to study Question 1.1. First of all, it is the structure of birational automorphisms and Cremona groups, in particular their generation by involutions or regularizable elements. In the recent paper [LS22], we explain that the answer to Question 1.1 is negative in various contexts in dimension $n\geq 3$ , and give applications to the structure of the higher Cremona groups. Secondly, Question 1.1 has a tight relationship to the structure of the Grothendieck ring of varieties.

Question 1.2.

(a)

(Larsen-Lunts [LL03], slightly reformulated) If classes of smooth projective varieties $[X]$ and $[Y]$ coincide in the Grothendieck ring, how can we compare the geometry of $X$ and $Y$ ? For instance, are $X$ and $Y$ birational?
(b)

[KS18] What is the geometric meaning of L-equivalence? For instance, if zero-dimensional schemes are L-equivalent, do they have to be isomorphic?
(c)

[(ht] Does the Grothendieck ring of varieties $K_{0}({\mathrm{Var}}/k)$ have torsion elements?

In the direction of (a), the main result of [LL03], which also follows from [Bit04], is that if ${\operatorname{char}}(k)=0$ , equality of classes $[X]$ , $[Y]$ of smooth projective connected varieties implies that $X$ and $Y$ are stably birational. On the other hand, for non-projective smooth connected varieties, the second part of Question (a).(a) is known to have a negative answer (the first such example is [Bor18, proof of Theorem 2.13]). For (b), see [KS18] for conjectural relations to derived equivalence. In §2.2, we explain that L-equivalence of smooth zero-dimensional schemes implies Gassmann equivalence of the corresponding Galois sets, but this does not rule out the possibility of nontrivial L-equivalence between such schemes. Nothing is known about (c). As the Grothendieck ring is a colimit of the truncated groups $K_{0}({\mathrm{Var}}^{\leq n}/k)$ generated by varieties of dimension up to $n$ , we can ask each of the questions in these truncated groups. As a direct consequence of our positive answer to Question 1.1 in dimension two, we are able to answer Question 1.2 completely for $K_{0}({\mathrm{Var}}^{\leq 2}/k)$ ; see Corollary 3.11. Namely, (a) equality of classes of smooth projective varieties in $K_{0}({\mathrm{Var}}^{\leq 2}/k)$ implies birationality, (b) L-equivalence in $K_{0}({\mathrm{Var}}^{\leq 2}/k)$ is trivial and (c) $K_{0}({\mathrm{Var}}^{\leq 2}/k)$ is a free abelian group. We expect that studying Question 1.1 in dimensions $\leq n$ would lead to good control over Question 1.2 for $K_{0}({\mathrm{Var}}^{\leq n}/k)$ .

Finally, answering Question 1.1 positively, or explaining all ways in which it can fail, allows to control all rationality constructions for every rational variety, see the discussion of rationality centers below and §5.2 for more details.

We now explain our answer to Question 1.1 in dimension $2$ over perfect fields. Our main result can be stated in the following way.

Theorem 1.3 (see Theorem 3.5).

Let $k$ be a perfect field. Let $X/k$ be a smooth projective surface and let $\phi\in{\operatorname{Bir}}(X)$ be a birational automorphism. For any factorization of $\phi$ into a sequence of blow ups and blow downs at connected smooth zero-dimensional subschemes, let $Z_{1},\dots,Z_{r}$ (resp. $Z_{1}^{\prime},\dots,Z_{r^{\prime}}^{\prime}$ ) be the centers which get blown up (resp. blown down). Then $r=r^{\prime}$ and there is a reordering under which $Z_{i}\simeq Z_{i}^{\prime}$ over $k$ for all $i=1,\dots,r$ .

Note that Theorem 1.3 is easily seen to hold when $k$ is algebraically closed field, as the Galois actions are trivial and the number of points blown up is equal to the number of exceptional divisors contracted. Over an arbitrary perfect field, Theorem 1.3 is a non-trivial statement. In particular, by this we mean that it is not possible to recover centers of the blow ups simply from the Galois action on the cohomology of the surface; see Remark 2.11 for the technical formulation of this statement in terms of Chow motives, and Example 2.14 for an explicit construction. Neither there seems to exist a straightforward geometric argument: see Example 3.8 for an illustration of the birational geometry involved.

To put Theorem 1.3 into an appropriate context, we introduce a general invariant $c(\phi)$ , keeping track of the factorization centers of the birational map $\phi$ , which is a homomorphism from the groupoid of birational types of surfaces to a free abelian group generated by reduced $k$ -schemes of dimension 0; see Corollary 3.2 for an axiomatic definition of $c(\phi$ ). We then show that $c$ is constant on each ${\operatorname{Bir}}(X,Y)$ and as a consequence $c(\phi)=0$ for every self-map, which implies Theorem 1.3.

To prove that $c$ is constant on ${\operatorname{Bir}}(X,Y)$ , we have to consider each birational type of surfaces that can occur, with geometrically rational (and especially, rational) surfaces being the most interesting ones. For geometrically rational surfaces, by the two-dimensional minimal model program we have to consider birational maps between del Pezzo surfaces and conic bundles. Our proof for uniqueness of factorization centers uses two ingredients: Sarkisov links and Gassmann equivalence. Sarkisov links are certain elementary birational transformations between del Pezzo surfaces and conic bundles which generate the groupoid of birational maps between geometrically rational surfaces. In dimension two, the existence of decomposition into links has been proved and all the links have been classified by Iskovskikh into a finite list [Isk96]. The largest variety of links occurs for what we call models of large degree: these include all minimal geometrically rational surfaces with $K_{X}^{2}\geq 5$ .

We derive Theorem 1.3 for geometrically rational surfaces from a uniform claim we make for all links between minimal geometrically rational surfaces (see Proposition 5.5), which we check for each link in Iskovskikh’s classification. The latter uniform claim is formulated in terms of the (virtual) Néron-Severi Galois set, which is closely related to the Néron-Severi lattice of the surface as a Galois module. These Néron-Severi sets are defined in terms of the Galois action on linear systems of rational curves of low degree (typically pencils of conics and nets of twisted cubics) on del Pezzo surfaces.

On top of the classification of links, to prove the result we use étale cohomology, permutation modules, and the so-called Gassmann triples from group theory: these are triples $(G,H,H^{\prime})$ with $H$ and $H^{\prime}$ subgroups of finite group $G$ such that $\mathbb{C}[G/H]$ and $\mathbb{C}[G/H^{\prime}]$ are isomorphic $G$ -representations (this holds if $H$ , $H^{\prime}$ are conjugate, but the converse if false). Gassmann triples are used to produce arithmetically equivalent fields, see e.g. [Per77, BdS02], isospectral manifolds [Sun85], as well as curves with isomorphic Jacobians [Pra17]. In our dealing with Gassmann equivalence, we follow the approach of [Par13] which generalizes Gassmann triples from a pair of subgroups of $G$ to a pair of $G$ -sets with possibly non-transitive actions. Our Gassmann equivalent sets come from étale cohomology groups and in particular, from Néron-Severi groups of geometrically rational surfaces with their Galois group action. We show that Gassmann equivalence provides a cohomological expression of L-equivalence, see Lemma 2.10 and Remark 2.11. Using the fact that Gassmann triples of small order are trivial allows us to significantly limit the number of Sarkisov links we have to consider in the proof of the main result.

As a by-product of our results on Gassmann equivalence we obtain conditions which forbid non-trivial L-equivalence of zero-dimensional schemes to exist; for example L-equivalent zero-dimensional connected reduced schemes (resp. reduced schemes) of degree $\leq 6$ (resp. $\leq 5$ ) are isomorphic and L-equivalence always implies isomorphism for fields with procyclic Galois groups such as $k=\mathbb{R}$ , $k=\mathbb{F}_{q}$ , see Example 2.12.

Finally for rational surfaces, our main result has the following consequence: if $X$ is a smooth projective rational surface, then there exists a zero-dimensional scheme depending only on $X$ , which will have to be blown up by any birational isomorphism $\phi:\mathbb{P}^{2}\dashrightarrow X$ , see Corollary 5.9; we think of these associated schemes as rationality centers of $X$ . This is in contrast with the higher-dimensional geometry, where the associated rationality centers are not well-defined, even up to stable birational equivalence. For instance, a K3 surface associated to a cubic fourfolds should not be unique up to isomorphism: it should be unique up to derived equivalence [Has16, Remark 27], or possibly up to L-equivalence, as hinted in [KS18, (2.6.1)].

This text is organized as follows. §2 is devoted to Gassmann equivalence of $G$ -sets. We relate it to L-equivalence and provide sufficient conditions for two Gassmann equivalent $G$ -sets to be isomorphic. In §3, we define the invariant $c(\phi)$ which captures the factorization center of a birational map $\phi$ and formulate the main theorem (Theorem 3.5) of the paper, which we prove in §4 and §5. We also study rational curves on del Pezzo surfaces in §4, which will be used to define and study the virtual Néron-Severi sets in §5. At the end of the paper we explain the concept of rationality center for rational surfaces, and illustrate it in the case of del Pezzo surfaces.

Acknowledgements

H.Y.L. was supported by the World Premier International Research centre Initiative (WPI), MEXT, Japan, then by the Ministry of Education Yushan Young Scholar Fellowship (NTU-110VV006) and the National Science and Technology Council (110-2628-M-002-006-). E.S. is supported by EP/T019379/1 “Derived categories and algebraic K-theory of singularities”, and by the ERC Synergy grant “Modern Aspects of Geometry: Categories, Cycles and Cohomology of Hyperkähler Varieties”. S.Z. was supported by the ANR Project FIBALGA ANR-18-CE40-0003-01 and the Project Étoiles montantes of the Région Pays de la Loire. E.S. would like to thank Yujiro Kawamata, Keiji Oguiso, Atsushi Takahashi and Shinnosuke Okawa for supporting his visit to Japan where some of this work has originated. E.S. would also like to thank Ivan Cheltsov, Sergey Galkin, Alexander Kuznetsov, Yuri Prokhorov, Constantin Shramov for discussions and encouragement, and Jean-Louis Colliot-Thélène, Brendan Hassett and Claire Voisin for their comments on this work. H.Y.L. would like to thank the NCTS in Taipei for the hospitality and support during the preparation of this paper. S.Z. would like to thank Jean-Louis Colliot-Thélène for his comments on this work and Jean-Louis Colliot-Thélène, Alexander Merkurjev and Jean-Pierre Serre for indicating references to examples of integral Gassmann triples. The authors would like to thank Keiji Oguiso for the initial discussion he involved on this topic and for making this collaboration possible.

Conventions

We work over a perfect field $k$ unless otherwise specified. All schemes are of finite type over $k$ . By a surface we mean a connected smooth projective (but not necessarily geometrically irreducible) surface $X$ over $k$ .

2. Gassmann equivalence

We explain the basics about the Burnside ring of a profinite group $G$ , the Grothendieck ring of varieties and a homomorphism between them when $G$ is the absolute Galois group. Our presentation is similar to [R1̈1]¹¹1 Note however that [R1̈1] makes an erroneous statement on p. 943 that the homomorphism from the Burnside ring to the representation ring is injective on transitive $G$ -sets (cf. Example 2.7(2)).. This homomorphism allows us to relate L-equivalence of reduced zero-dimensional schemes to Gassmann equivalence of $G$ -sets, and thus to rule out the possibility of nontrivial L-equivalence in small degree.

2.1. Definition and basic properties

Let $G$ be a profinite group and let $\mathbf{Gset}$ be the semi-ring of isomorphism classes of finite $G$ -sets on which $G$ acts continuously for the profinite topology, where finite sets are considered with discrete topology. Continuity of the action is equivalent to the requirement that stabilizer of any point is open (and in particular, a finite index subgroup). Here in $\mathbf{Gset}$ , the addition (resp. multiplication) is defined by disjoint unions (resp. Cartesian products). We define the Burnside ring ${\operatorname{Burn}}(G)$ of $G$ to be the Grothendieck ring associated to $\mathbf{Gset}$ . When $G$ is a finite group, the definition of ${\operatorname{Burn}}(G)$ is the classical one. We sometimes refer to elements of ${\operatorname{Burn}}(G)$ , that is, combinations of isomorphism classes of $G$ -sets with integer coefficients, as virtual $G$ -sets. We note that the number of elements (resp. the number of orbits) in a $G$ -set gives rise to a ring homomorphism (resp. a group homomorphism) ${\operatorname{Burn}}(G)\to\mathbb{Z}$ .

Let $F$ be a field of characteristic zero and let ${\operatorname{Rep}}(G,F)$ be the abelian monoidal category of finite dimensional $G$ -representations over $F$ . Let $K_{0}({\operatorname{Rep}}(G,F))$ be the Grothendieck ring of ${\operatorname{Rep}}(G,F)$ . There is a well-defined ring homomorphism

(2.1)

\mu_{G}:{\operatorname{Burn}}(G)\to K_{0}({\operatorname{Rep}}(G,F))

which sends the class of a continuous finite $G$ -set $A$ to the class of the permutation representation $F[A]$ . We are interested in the kernel of this homomorphism.

Lemma-Definition 2.1.

Let $G$ be a profinite group and let $A$ , $B$ be continuous finite $G$ -sets. Fix a field $F$ of characteristic zero. The following conditions are equivalent:

(1)

$\mu_{G}(A)=\mu_{G}(B)$ in $K_{0}({\operatorname{Rep}}(G,F))$ .
(2)

$F[A]\simeq F[B]$ as $F[G]$ -modules.

We say that two continuous finite $G$ -sets $A$ and $B$ are Gassmann equivalent if they satisfy one of the above equivalent conditions.

Proof.

It is clear that (2) implies (1). Conversely, if $\mu_{G}(A)=\mu_{G}(B)$ , then since $F[A]$ and $F[B]$ are finite dimensional (as $F$ -vector spaces), both $F[A]$ and $F[B]$ admit composition series in ${\operatorname{Rep}}(G,F)$ of finite length [EH18, Lemma 3.9]. The Jordan-Hölder theorem [Wei13, Excercice II.6.3] shows that $F[A]$ and $F[B]$ have the same collection of isomorphism classes of (simple) Jordan-Hölder factors. As both $G$ -modules $F[A]$ and $F[B]$ factor through a finite quotient $G/H$ of $G$ (we can take $H$ to be the intersection of the kernels of the two actions, which is an open subgroup of $G$ , hence has finite index), Maschke’s Theorem implies that $F[A]$ and $F[B]$ are semisimple $G$ -modules. Hence $F[A]\simeq F[B]$ . $\Box$

The following lemma reduces Gassmann equivalence to the case of finite group actions. If $A$ is a $G$ -set, we write $G^{A}$ for the kernel of the action; thus $G^{A}$ is a normal subgroup and $G/G^{A}$ acts on $A$ faithfully.

Lemma 2.2.

Two continuous finite $G$ -sets $A$ and $B$ are Gassmann equivalent if and only if $G^{A}=G^{B}=:H$ and $A$ and $B$ are Gassmann equivalent as $G/H$ -sets.

Proof.

This follows from Lemma 2.1, together with the observation that $G^{A}=\ker{\rho_{A}}$ (and similarly $G^{B}=\ker(\rho_{B})$ ) where $\rho_{A}:G\to{\mathrm{GL}}(V,F)$ is the homomorphism defining the $F[G]$ -module structure of $F[A]$ and $V$ is the underlying $F$ -vector space of $F[A]$ . $\Box$

Lemma 2.3 ([Par13, Proposition 1], cf [Pra17, Definition 1]).

Let $G$ be a finite group and let $A$ and $B$ be finite $G$ -sets. Fix a field $F$ of characteristic zero. The following conditions are equivalent:

(1)

For every $g\in G$ , $g$ fixes the same number of elements in $A$ and in $B$ .
(2)

There exist subgroups $H_{1},\dots,H_{r}$ , $H_{1}^{\prime},\dots,H_{r}^{\prime}$ of $G$ such that

$A\simeq\bigsqcup_{i=1}^{r}G/H_{i},\;\;\;\;B\simeq\bigsqcup_{i=1}^{r}G/H_{i}^{\prime}$

and for each conjugacy class $T\subset G$ these subgroups satisfy

(2.2) $\sum_{i=1}^{r}\frac{|T\cap H_{i}|}{|H_{i}|}=\sum_{i=1}^{r}\frac{|T\cap H_{i}^{\prime}|}{|H_{i}^{\prime}|}.$
(3)

$A$ and $B$ are Gassmann equivalent.

In particular, the kernel of $\mu$ is independent of the choice of $F$ .

Proof.

Equivalence of (1), (2) and (3) for $F=\mathbb{C}$ is proved in [Par13, Proposition 1]. Finally, condition (3) is independent of the choice of $F$ because the functor

{\operatorname{Rep}}(G,\mathbb{Q})\to{\operatorname{Rep}}(G,F)

is injective on isomorphism classes [Ser77, §14.6]. $\Box$

Remark 2.4.

The number of $G$ -orbits in Gassmann equivalent continuous $G$ -sets is the same. This follows from Lemma 2.2 and Lemma 2.3(2), or directly as the number of orbits equals the multiplicity of the trivial representation in the permutation representation.

Proposition 2.5.

Let $G$ be a profinite group and let $A$ and $B$ be Gassmann equivalent continuous finite $G$ -sets. If one of the following conditions is satisfied:

•

$G/G^{A}$ is a cyclic group
•

$A$ is transitive and the stabilizer of a point is normal in $G$
•

$A$ is transitive and $|A|\leq 6$
•

$|A|\leq 5$

then $A$ and $B$ are isomorphic $G$ -sets.

If $\alpha\in{\operatorname{Burn}}(G)$ , then we write $|\alpha|$ for the smallest $\max(|A|,|B|)$ over all possible representations $\alpha=[A]-[B]$ with continuous $G$ -sets $A$ and $B$ . We have the following immediate corollary of Proposition 2.5.

Corollary 2.6.

Let $\alpha\in{\operatorname{Ker}}(\mu_{G})$ . If $|\alpha|\leq 5$ , then $\alpha=0$ .

Proof of Proposition 2.5.

By Lemma 2.2, we can assume that $G$ is a finite group and the $G$ -actions on both $A$ and $B$ are faithful. When $G$ is a finite cyclic group, the result is [Par13, Proposition 4.1]. If $A$ is transitive, then by Remark 2.4, $B$ is also transitive. Let $H$ (resp. $H^{\prime}$ ) be the stabilizer of an element $a\in A$ (resp. $b\in B$ ). Then $G/H$ and $G/H^{\prime}$ are Gassmann equivalent; in this case one refers to $(G,H,H^{\prime})$ as a Gassmann triple. It is well-known and easy to see (e.g. using (2.2) with $r=1$ ) that if $H$ is normal, then $H^{\prime}=H$ . It is a nontrivial computation that if $[G:H]\leq 6$ in Gassmann triple $(G,H,H^{\prime})$ , $H$ and $H^{\prime}$ will be conjugate so that $A$ and $B$ are isomorphic, see [Per77, Proof of Theorem 3] or [BdS02, p.3].

Finally, we need to consider the case when $A$ may not be transitive but $t:=|A|\leq 5$ . We write $A=\{1,\dots,t\}$ and let $S_{t}$ be the permutation group of $t$ elements. Since the $G$ -action on $A$ is assumed to be faithful, it realizes $G$ as a subgroup of $S_{t}$ .

Up to adding trivial $G$ -sets $\{*\}$ , we can assume that $t=5$ . By Remark 2.4, $A$ and $B$ have the same number of $G$ -orbits, and we argue according to the number of orbits $m$ . When $m=5$ , then both $A$ and $B$ are trivial $G$ -sets. When $m=4$ , then both $A$ and $B$ are isomorphic to $\{1,2\}\sqcup\{*\}\sqcup\{*\}\sqcup\{*\}$ and $G\simeq\mathbb{Z}/2\mathbb{Z}$ acts on $\{1,2\}$ by involution. The case $m=1$ is already covered by the third case of Proposition 2.5. So it remains to study the case where $m=2$ or $3$ .

First we show that $G$ uniquely determines the length of the orbits of the $G$ -set $A$ . If $m=3$ , then $A$ is isomorphic to either $A_{3}\sqcup\{*\}\sqcup\{*\}$ or $A_{2}\sqcup A^{\prime}_{2}\sqcup\{*\}$ with $|A_{3}|=3$ and $|A_{2}|=|A^{\prime}_{2}|=2$ . So $G\leq S_{3}$ in the former case and $G\leq\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$ in the latter case. In the former case, since the $G$ -action is transitive on $A_{3}$ , $G$ contains $\mathbb{Z}/3\mathbb{Z}\leq S_{3}$ , so we cannot embed $G$ into $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$ . Therefore $G$ uniquely determines the length of the orbits. If $m=2$ , then $A$ is isomorphic to either $A_{3}\sqcup A_{2}$ or $A_{4}\sqcup\{*\}$ with $|A_{i}|=i$ . So $G\leq S_{3}\times\mathbb{Z}/2\mathbb{Z}$ in the former case and $G\leq S_{4}$ in the latter case. In the former case, since the $G$ -action is transitive on both $A_{3}$ and $A_{2}$ , $G$ contains $\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}\leq S_{3}\times\mathbb{Z}/2\mathbb{Z}$ . Since $S_{4}$ has no elements of order six, we cannot embed $G$ into $S_{4}$ . Hence the lengths of the orbits are determined by $G$ as well.

We still assume that $m=2$ or $3$ . If $A=A^{\prime}\sqcup A^{\prime\prime}$ as $G$ -sets where $A^{\prime}$ is transitive and $A^{\prime\prime}$ is a disjoint union of trivial $G$ -sets, then we also have the same type of decomposition $B=B^{\prime}\sqcup B^{\prime\prime}$ with $|A^{\prime}|=|B^{\prime}|$ . It follows that $\mu_{G}(A^{\prime})=\mu_{G}(B^{\prime})$ , so $A^{\prime}\simeq B^{\prime}$ since both $G$ -sets are transitive. This covers the cases where $A=A_{3}\sqcup\{*\}\sqcup\{*\}$ and $A=A_{4}\sqcup\{*\}$ . If $A=A_{2}\sqcup A^{\prime}_{2}\sqcup\{*\}$ , then either $G=\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$ or $G$ is the diagonal $\mathbb{Z}/2\mathbb{Z}\leq\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$ . In either case we verify that $G$ uniquely determines the $G$ -set structure of $A$ . If $A=A_{3}\sqcup A_{2}$ , then either $G\leq\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$ or $G=S_{3}\times\mathbb{Z}/2\mathbb{Z}$ . Once again in either case, we verify that $G$ uniquely determines the $G$ -set structure of $A$ . $\Box$

The following example shows that the lower bounds on the order of $G$ -sets in Proposition 2.5 are optimal.

Example 2.7.

(1) [Par13, 1.1] Let $G=\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$ and let

H_{1}=\langle(1,0)\rangle,\;\;H_{2}=\langle(0,1)\rangle,\;\;H_{3}=\langle(1,1)\rangle,

and

H_{1}^{\prime}=\{(0,0)\},\;\;H_{2}^{\prime}=H_{3}^{\prime}=G.

Then $H_{1},H_{2},H_{3}$ and $H_{1}^{\prime},H_{2}^{\prime},H_{3}^{\prime}$ satisfy conditions of Lemma 2.3(2) thus give rise to nonisomorphic nontransitive Gassmann equivalent sets of order $6$ with orbit decompositions $2+2+2$ and $4+1+1$ respectively. Indeed for abelian groups the condition (2.2) rewrites as

\sum_{H_{i}\ni g}\frac{1}{|H_{i}|}=\sum_{H^{\prime}_{i}\ni g}\frac{1}{|H^{\prime}_{i}|}\text{ for all $g\in G$}

which is immediately verified.

(2) [Per77, p. 358] Let $G=\mathrm{PSL}_{2}(\mathbb{F}_{7})\simeq\mathrm{PSL}_{3}(\mathbb{F}_{2})$ be the simple group of order $168$ . Let $A$ be the set of $\mathbb{F}_{2}$ -points of the projective plane over $\mathbb{F}_{2}$ , and $B$ be the set of $\mathbb{F}_{2}$ -lines on this plane, that is points of the dual projective plane. Then $A$ , $B$ are transitive $G$ -sets of order $7$ . Using simple linear algebra of the $\mathrm{PSL}_{3}(\mathbb{F}_{2})$ -action on $\mathbb{P}^{2}_{\mathbb{F}_{2}}$ , one shows that $A$ and $B$ satisfy Lemma 2.3(1), so that $A$ and $B$ are Gassmann equivalent, and one can check that they are not isomorphic. See also [BdS02, Theorem 3], which shows that this pair is the only nontrivial Gassmann triple of faithful transitive $G$ -sets of order $7$ .

2.2. L-equivalence and Gassmann equivalence

Let $K_{0}({\mathrm{Var}}/k)$ denote the Grothendieck ring of varieties with generators given by isomorphism classes $[X]$ of schemes of finite type over $k$ and relations generated by cut and paste relations

(2.3)

[X]=[Z]+[X\setminus Z]

for every closed $Z\subset X$ . The ring structure on $K_{0}({\mathrm{Var}}/k)$ is induced by products of schemes. We write $\mathbb{L}=[\mathbb{A}^{1}]$ .

It is known that $\mathbb{L}$ is a zero-divisor [Bor18] and that the annihilator of $\mathbb{L}^{k}$ , $k\geq 1$ encodes deep geometric information. Following [KS18] we call two smooth projective connected varieties $X$ , $Y$ L-equivalent if for some $k\geq 0$ ,

\mathbb{L}^{k}\cdot([X]-[Y])=0.

We sometimes refer to the $[X]=[Y]$ case as trivial L-equivalence. It is currently unknown if zero-dimensional varieties can be nontrivially L-equivalent. The smallest-dimensional example of nontrivial L-equivalence is that of genus one curves over non-closed fields [SZ20]. See [KS18, SZ20] for some details about conjectural relationship between L-equivalence and derived equivalence, and the references therein for the currently known examples. Note that the classes in the Grothendieck ring are insensitive to nonreduced structure, hence when studying L-equivalence we can always assume schemes to be reduced.

Remark 2.8.

For fields of positive characteristic, there exist two alternative definitions of the Grothendieck ring of varieties, hence alternative definitions of L-equivalence.

(1) First of all, one can define the Grothendieck ring $K_{0}^{\operatorname{bl}}({\mathrm{Var}}/k)$ generated by classes of smooth projective varieties with blow up relations as in [Bit04, Theorem 3.1 (bl)]. We have an obvious homomorphism

K_{0}^{\operatorname{bl}}({\mathrm{Var}}/k)\to K_{0}({\mathrm{Var}}/k)

which is known to be an isomorphism if $k$ is a field of characteristic zero [Bit04, Theorem 3.1].

(2) Furthermore, in positive characteristic one can define a modified Grothendieck ring [NS09] by imposing an additional relation of identifying varieties related by universal homeomorphisms (originating from totally inseparable coverings in positive characteristic). It is not known if this additional relation in fact gives rise to a non-isomorphic ring as all the standard invariants which are used to distinguish elements in the Grothendieck ring factor through the modified ring as well [NS09].

Let $k$ be a field and let $G_{k}={\operatorname{Gal}}(\bar{k}/k)$ where $\bar{k}$ denotes the separable closure of $k$ . For a variety $X/k$ (not necessarily smooth or projective) we consider its $\ell$ -adic cohomology $H^{i}_{{\text{\'{e}t}},c}(X_{\overline{k}},\mathbb{Q}_{\ell})$ (with $\ell\neq{\operatorname{char}}(k)$ ) with compact supports as a $G_{k}$ -module. These groups are finite-dimensional $\mathbb{Q}_{\ell}$ -vector spaces [FK88, Remark I.12.16], they vanish outside the range $0\leq i\leq 2\dim(X)$ and give rise to a group homomorphism

(2.4)

\mu_{{\text{\'{e}t}}}:K_{0}({\mathrm{Var}}/k)\to K_{0}({\operatorname{Rep}}(G_{k},\mathbb{Q}_{\ell}))

defined by

\mu_{{\text{\'{e}t}}}([X])=\sum_{i=0}^{2\dim(X)}(-1)^{i}[H^{i}_{{\text{\'{e}t}},c}(X_{\overline{k}},\mathbb{Q}_{\ell})].

We have

(2.5)

\mu_{{\text{\'{e}t}}}(\mathbb{L})=\mu_{{\text{\'{e}t}}}([\mathbb{P}^{1}]-1)=[\mathbb{Q}_{\ell}(-1)]

and furthermore, the projective bundle formula for étale cohomology implies that $\mu_{{\text{\'{e}t}}}$ is a $\mathbb{Z}[\mathbb{L}]$ -module homomorphism where $\mathbb{L}$ acts by multiplication by $\mathbb{Q}_{\ell}(-1)$ on the Galois representations. (The map $\mu_{\text{\'{e}t}}$ is even a ring homomorphism by the Künneth formula [Mil80, Corollary VI.8.23] but we do not need this fact.)

The étale realization (2.4) is useful for extracting information from a class in the Grothendieck of varieties. We record the following example to be used later.

Example 2.9.

Let $X$ be a geometrically rational smooth projective surface. Then since all cohomology classes on $X$ are algebraic, $\mu_{{\text{\'{e}t}}}([X])=[\mathbb{Q}_{\ell}]+[\mathbb{Q}_{\ell}(-2)]+[{\operatorname{NS}}(X_{\overline{k}})\otimes\mathbb{Q}_{\ell}(-1)]$ , where $\mathbb{Q}_{\ell}$ is considered as a trivial one-dimensional $G_{k}$ -representation. In particular, if $X$ , $X^{\prime}$ are two such surfaces, and $[X]=[X^{\prime}]$ , then ${\operatorname{NS}}(X_{\overline{k}})$ and ${\operatorname{NS}}(X^{\prime}_{\overline{k}})$ have the same class in $K_{0}({\operatorname{Rep}}(G_{k},\mathbb{Q}_{\ell}))$ .

We explain how Gassmann equivalence relates to L-equivalence of reduced $k$ -schemes of dimension 0. Let $\mathbf{EtSch}_{k}$ be the semi-ring of $k$ -schemes which are étale over ${\operatorname{Spec}}(k)$ . As we assume $k$ to be perfect, $\mathbf{EtSch}_{k}$ is also the semi-ring of reduced $k$ -schemes of dimension 0. Here in $\mathbf{EtSch}_{k}$ , the addition (resp. multiplication) is defined by disjoint unions (resp. products over ${\operatorname{Spec}}(k)$ ). For every $Z\in\mathbf{EtSch}_{k}$ , its base change $Z_{\bar{k}}$ to the separable closure $\bar{k}$ of $k$ is endowed with a $G_{k}$ -action. By Galois descent, The map $\mathbf{EtSch}_{k}\to\mathbf{Gset}$ sending $Z\in\mathbf{EtSch}_{k}$ to the underlying continuous $G_{k}$ -set of $Z_{\bar{k}}$ is an isomorphism of semi-rings. As we assume $k$ to be perfect, this induces a ring isomorphism

(2.6)

\mathbb{Z}[{\mathrm{Var}}^{0}/k]\simeq{\operatorname{Burn}}(G_{k})

where ${\mathrm{Var}}^{0}/k$ denotes the set of (irreducible) $k$ -varieties of dimension 0.

We have a natural ring homomorphism

(2.7)

\mathbb{Z}[{\mathrm{Var}}^{0}/k]\to K_{0}({\mathrm{Var}}/k)

which sends $Z$ to $[Z]$ . It follows from the blow up presentation of the Grothendieck ring [Bit04, Theorem 3.1] that over fields of characteristic zero (2.7) admits a splitting given by $X\mapsto{\operatorname{Spec}}(H^{0}(X,\mathcal{O}_{X}))$ ( $X$ smooth projective), hence for characteristic zero fields (2.7) is injective.

Throughout this text, the same notation $Z_{\bar{k}}$ (and also $Z$ itself, when it does not lead to any confusion) denotes the underlying $G_{k}$ -set of $Z_{\bar{k}}$ for every étale $k$ -scheme $Z$ .

We work with Gassmann equivalence of such schemes considered as sets with Galois group action.

Lemma 2.10.

Let $Z$ and $Z^{\prime}$ be étale $k$ -schemes. If $Z$ and $Z^{\prime}$ are L-equivalent in the Grothendieck ring or in any of the modifications explained in Remark 2.8, then $Z_{\bar{k}}$ and $Z^{\prime}_{\bar{k}}$ are Gassmann equivalent.

Proof.

We use the étale realization (2.4), which is defined by pre-composition on $K_{0}^{\operatorname{bl}}({\mathrm{Var}}/k)$ , and also factors through the modified Grothendieck ring from Remark 2.8 (2) [NS09, Proposition 4.1.(3)].

For any étale $k$ -scheme $Z$ , we have

(2.8)

\mu_{{\text{\'{e}t}}}([Z])=[H^{0}_{\text{\'{e}t}}(Z_{\bar{k}},\mathbb{Q}_{\ell})]=[\mathbb{Q}_{\ell}[Z_{\bar{k}}]]=\mu_{G_{k}}([Z_{\bar{k}}]).

For each $j\geq 0$ we consider the composition

which is equal to $\mu_{G_{k}}$ (2.1) by (2.5) and (2.8). Thus if $Z$ , $Z^{\prime}$ are L-equivalent, then $Z_{\bar{k}}$ , $Z^{\prime}_{\bar{k}}$ are Gassmann equivalent. $\Box$

Remark 2.11.

Properties of classes $[X]$ of smooth projective varieties in the Grothendieck rings often go in parallel with properties of their Chow motives [And04]. In our situation, by [And04, Exemple 4.1.6.1], if $k$ has characteristic zero, then two étale $k$ -schemes $Z$ , $Z^{\prime}$ are Gassmann equivalent if and only if the Chow motives of $Z$ , $Z^{\prime}$ with rational coefficients are isomorphic. Thus in this setting Lemma 2.10 says that L-equivalence implies isomorphism of Chow motives. The same result is expected for all smooth projective varieties, and it follows from the conjectural uniqueness of direct sum decompositions for Chow motives, see e.g. [G0̈1, Conjecture 2.5, Conjecture 2.6].

Example 2.12.

Let $k$ be a field such that all continuous $G_{k}$ -actions on finite sets factor through a finite cyclic quotient of $G_{k}$ , e.g. $k=\mathbb{R}$ or $k=\mathbb{F}_{q}$ . By Proposition 2.5 cyclic groups do not allow nontrivial Gassmann equivalence, hence by Lemma 2.10 L-equivalence of étale $k$ -schemes implies their isomorphism.

Corollary 2.13.

Let $Z$ and $Z^{\prime}$ be étale $k$ -schemes. If $Z$ and $Z^{\prime}$ are L-equivalent in the Grothendieck ring or any of the modifications explained in Remark 2.8, then $Z\simeq Z^{\prime}$ as soon as one of the following conditions is satisfied:

•

$Z$ is $k$ -irreducible and Galois over ${\operatorname{Spec}}(k)$
•

$Z$ is $k$ -irreducible with $\deg_{k}Z\leq 6$
•

$Z$ satisfies $\deg_{k}Z\leq 5$ .

Proof.

By Lemma 2.10, $Z_{\bar{k}}$ and $Z^{\prime}_{\bar{k}}$ are Gassmann equivalent and by Proposition 2.5 they are isomorphic. $\Box$

Example 2.14.

Translating Example 2.7(1) into the language of étale schemes, we obtain the following. Let $k=\mathbb{Q}$ , and define degree $6$ schemes

	$\displaystyle Z$	$\displaystyle={\operatorname{Spec}}(\mathbb{Q}(\sqrt{\alpha}))\sqcup{\operatorname{Spec}}(\mathbb{Q}(\sqrt{\beta}))\sqcup{\operatorname{Spec}}(\mathbb{Q}(\sqrt{\alpha\beta}))$
	$\displaystyle Z^{\prime}$	$\displaystyle={\operatorname{Spec}}(\mathbb{Q}(\sqrt{\alpha},\sqrt{\beta}))\sqcup{\operatorname{Spec}}(\mathbb{Q})\sqcup{\operatorname{Spec}}(\mathbb{Q}),$

where we choose any $\alpha,\beta\in\mathbb{Q}^{\star}$ which are nontrivial and distinct in $\mathbb{Q}^{\star}/(\mathbb{Q}^{\star})^{2}$ . Then $Z$ , $Z^{\prime}$ are Gassmann equivalent, and have isomorphic Chow motives (see Remark 2.11) but we do not know how to check if they are L-equivalent or not.

Embedding $Z$ , $Z^{\prime}$ into $\mathbb{P}^{2}$ so that both images are in general position (for instance, if $\alpha+\beta\neq 1$ , we can send $Z$ onto $[\pm\sqrt{\alpha}:1:0],[0:\pm\sqrt{\beta}:1],[1:0:\pm\sqrt{\alpha\beta}]$ and $Z^{\prime}$ onto $[\pm\sqrt{\alpha}:\pm\sqrt{\beta}:1],[1:0:1],[0:1:1]$ ), we obtain two del Pezzo surfaces of degree $3$ : $X={\operatorname{Bl}}_{Z}(\mathbb{P}^{2})$ , $X^{\prime}={\operatorname{Bl}}_{Z^{\prime}}(\mathbb{P}^{2})$ . These two surfaces are not isomorphic, as the first one has three $k$ -rational lines, and the second one has five.

However, we have an isomorphism

{\operatorname{NS}}(X_{\bar{\mathbb{Q}}})\otimes\mathbb{Q}\simeq{\operatorname{NS}}(X^{\prime}_{\bar{\mathbb{Q}}})\otimes\mathbb{Q}

of permutation Galois representations, which shows that the associated Galois set is not uniquely defined. One can make a similar example integrally, using integral Gassmann triples as in [Sco93], [Pra17]. See Example 5.12 where we explain that using our techniques we can recover $Z$ (resp. $Z^{\prime}$ ) from $X$ (resp. $X^{\prime}$ ), providing another proof that they are not isomorphic.

Finally note that we do not know if $[X]=[X^{\prime}]$ as by the blow up relation in the Grothendieck ring of varieties, $[X]=[\mathbb{P}^{2}]+\mathbb{L}[Z]$ and $[X^{\prime}]=[\mathbb{P}^{2}]+\mathbb{L}[Z^{\prime}]$ so $[X]=[X^{\prime}]$ would imply the L-equivalence of $Z$ and $Z^{\prime}$ , which is unknown.

3. Factorization centers

In this section we introduce the invariant $c(\phi)$ keeping track of the factorization centers in any decomposition of a birational isomorphism of $\phi$ between surfaces into a sequence of blow ups and blow downs, formulate the main theorem and interpret it in terms of the truncated Grothendieck ring of varieties.

3.1. Formulation of the main result

Fix a perfect field $k$ . Let $\phi:X\dashrightarrow Y$ be a birational isomorphism of smooth projective $k$ -surfaces. By the strong factorization theorem [Man86, Corollary of Lemma III.4.4] (or [Sta18, Lemma 54.17.2]) we have a decomposition

(3.1)

with both $\alpha$ and $\beta$ being compositions of blow ups with smooth centers $Z_{1},\dots,Z_{r}$ and $Z_{1}^{\prime},\dots,Z_{s}^{\prime}$ respectively, which are zero-dimensional smooth schemes. The factorization center of $\phi$ is defined as

(3.2)

c(\phi)=\sum_{i=1}^{r}[Z_{i}]-\sum_{i=1}^{s}[Z_{i}^{\prime}]\in\mathbb{Z}[{\mathrm{Var}}^{0}/k]

(see §2.2 for the definition of $\mathbb{Z}[{\mathrm{Var}}^{0}/k]$ ). We remark that $\mathbb{L}\cdot c(\phi)$ , regarded as an element of $K_{0}({\mathrm{Var}}/k)$ , measures the difference between classes $[Y]$ and $[X]$ in the Grothendieck ring of varieties, see Lemma 3.9. Given that $\mathbb{L}$ is a zero-divisor in the Grothendieck ring, $c(\phi)$ potentially contains more information than $\mathbb{L}\cdot c(\phi)$ .

We explain the well-definedness of $c$ and its basic properties. To do that it is most convenient to consider the groupoid ${\operatorname{Bir}}_{2}/k$ of birational types of surfaces, whose objects are smooth projective surfaces and morphisms are birational isomorphisms.

Recall that if $\mathcal{C}$ is a groupoid, and $G$ a group, a homomorphism from $\mathcal{C}$ to $G$ is a functor from $\mathcal{C}$ to $G$ , where $G$ is considered as a groupoid with one object.

Lemma 3.1.

$c(\phi)$ does not depend on the choice of factorization of $\phi$ and defines a homomorphism $c:{\operatorname{Bir}}_{2}/k\to\mathbb{Z}[{\mathrm{Var}}^{0}/k]$ . Explicitly, for any two birational isomorphisms of surfaces $\phi:X\dashrightarrow X^{\prime}$ , $\psi:X^{\prime}\dashrightarrow X^{\prime\prime}$ we have

c(\psi\circ\phi)=c(\psi)+c(\phi).

In particular, for any surface $X$ we have a homomorphism

c:{\operatorname{Bir}}(X)\to\mathbb{Z}[{\mathrm{Var}}^{0}/k].

Proof.

Consider the diagram (3.1). Let $E_{1},\dots,E_{m}\subset X$ (resp. $E_{1}^{\prime},\dots,E_{n}^{\prime}\subset Y$ ) be the irreducible components of the exceptional divisor of $\phi$ (resp. $\phi^{-1}$ ). Let $D_{1},\dots,D_{t}\subset\widetilde{X}$ be the irreducible divisors which are contracted by both $\alpha$ and $\beta$ . This way the centers $Z_{i},\;i=1,\dots,r$ (resp. $Z_{i}^{\prime},\;i=1,\dots,r^{\prime}$ ) are in one-to-one correspondence with the collection of divisors $\{E_{i}\}_{i=1,\dots,m}\cup\{D_{j}\}_{j=1,\dots,t}$ (resp. $\{E^{\prime}_{i}\}_{i=1,\dots,n}\cup\{D_{j}\}_{j=1,\dots,t}$ ).

Note that each prime divisor $D$ contracted by $\alpha$ or $\beta$ is birational to $\mathbb{P}^{1}\times Z$ , where $Z$ is the center of the corresponding blow up, which can be recovered from $D$ as $Z\simeq{\operatorname{Spec}}(H^{0}(D,\mathcal{O}_{D}))$ . Cancelling out the centers corresponding to $D_{1},\dots,D_{t}$ we obtain

\displaystyle c(\phi)=\sum_{i=1}^{r}[Z_{i}]-\sum_{i=1}^{s}[Z_{i}^{\prime}]=\sum_{i=1}^{m}[{\operatorname{Spec}}(H^{0}(E_{i},\mathcal{O}))]-\sum_{i=1}^{n}[{\operatorname{Spec}}(H^{0}(E_{i}^{\prime},\mathcal{O}))]

thus the expression (3.2) only depends on the exceptional divisors of $\phi$ and $\phi^{-1}$ , hence $c(\phi)$ is independent of the choice of strong factorization (3.1).

To show that $c(\psi\circ\phi)=c(\psi)+c(\phi)$ consider the diagram

where $\tau$ , $\tau^{\prime}$ (resp. $\nu$ , $\nu^{\prime}$ ) provide a strong factorization for $\phi$ (resp. $\psi$ ), and $\mu$ , $\mu^{\prime}$ provide a strong factorization for $\nu^{-1}\tau^{\prime}$ . It is clear that $c$ is additive on regular birational isomorphisms, hence

\begin{split}c(\psi\circ\phi)&=c(\nu^{\prime}\circ\mu^{\prime})-c(\tau\circ\mu)\\ &=c(\nu^{\prime}\circ\mu^{\prime})-c(\nu\circ\mu^{\prime})+c(\tau^{\prime}\circ\mu)-c(\tau\circ\mu)\\ &=c(\nu^{\prime})-c(\nu)+c(\tau^{\prime})-c(\tau)=c(\psi)+c(\phi).\end{split}

$\Box$

Corollary 3.2.

There is a unique assignment $\phi\mapsto c(\phi)\in\mathbb{Z}[{\mathrm{Var}}^{0}/k]$ defined for all birational isomorphisms $\phi$ between smooth projective $k$ -surfaces, satisfying the following axioms

(i)

For any (biregular) isomorphism $\phi$ , $c(\phi)=0$
(ii)

If $\widetilde{X}$ is a blow up of $X$ with smooth connected center $Z$ and $\phi:\widetilde{X}\to X$ the contraction map, then

$c(\phi)=-[Z],\;c(\phi^{-1})=[Z]$
(iii)

For composable birational isomorphisms

$c(\psi\circ\phi)=c(\psi)+c(\phi).$

Proof.

Uniqueness follows from the strong factorization (3.1), and existence is the content of Lemma 3.1. $\Box$

Remark 3.3.

The invariant $c(\phi)$ is generalized to higher-dimensional varieties in [LS22], to the equivariant setting in [KT22] and to varieties with a logarithmic form in [CLKT23]. In all these contexts properties of $c(\phi)$ are analogous to those stated in Corollary 3.2.

Proposition 3.4.

(i)

For any field extension $L/k$ we have a commutative diagram

where the left vertical arrow is an extension of scalars for birational map, the middle vertical arrow is defined by extension of scalars, that is it maps an étale $k$ -scheme $Z$ to the sum of the connected components of $Z\times_{k}L$ , and the right vertical arrow is the restriction of the group action defined through the map ${\operatorname{Gal}}(\bar{L}/L)\to{\operatorname{Gal}}(\bar{k}/k)$ induced by any choice of embedding $\bar{k}\hookrightarrow\bar{L}$ compatible with $k\hookrightarrow L$ .

In particular, applying the diagram to an automorphism $\sigma\in{\operatorname{Aut}}(L/k)$ the left square gives $c(\sigma(\phi))=\sigma(c(\phi))$ , that is $c$ commutes with the group action by the automorphisms of the field.

(ii)

For any finite field extension $L/k$ and surfaces $X$ , $Y$ over $L$ , let $X|_{k}$ , $Y|_{k}$ denote the underlying $k$ -surfaces.²²2Here by $X|_{k}$ we mean the $k$ -surface given by the composition $X\to{\operatorname{Spec}}(L)\to{\operatorname{Spec}}(k)$ (this is not the Weil restriction of scalars). If $L/k$ is nontrivial, the surface $X|_{k}$ is not geometrically connected: $X|_{k}\times_{k}L$ is isomorphic to a disjoint union of $[L:k]$ copies of $X$ . We have the following commutative diagram:

where the middle vertical map is restriction of scalars.

Proof.

By Lemma 3.1, in both (i), (ii) it is sufficient to check a single blow up where the statements are clear. $\Box$

The main result of the paper is the following.

Theorem 3.5.

For any two smooth projective $k$ -surfaces $X$ , $Y$ and any two birational isomorphisms $\phi,\psi:X\dashrightarrow Y$ , we have $c(\phi)=c(\psi)$ . In particular, $c:{\operatorname{Bir}}(X)\to\mathbb{Z}[{\mathrm{Var}}^{0}/k]$ is a zero map.

We prove Theorem 3.5 at the end of §5. The result is straightforward when $k$ is algebraically closed, because in this case the invariant takes values in $\mathbb{Z}\cdot[{\operatorname{Spec}}(k)]\simeq\mathbb{Z}$ , and measures the difference of the ranks of two Néron-Severi groups (cf Proposition 4.4). However, Theorem 3.5 becomes a nontrivial statement when $k$ is an arbitrary perfect field, and its proof depends on the two-dimensional Minimal Model Program. This result is also specific for surfaces and fails to be true in higher dimension, even over algebraically closed fields [LS22].

Example 3.6.

Let $k=\mathbb{R}$ . In this case Theorem 3.5 says that birational automorphisms of surfaces blow up the same number of rational points as the number of rational divisors they blow down, and that they blow up the same number of pairs of complex conjugate points as the number of pairs of complex conjugate divisors they blow down. This can be proved directly by considering the Galois action on the Néron-Severi group (see also Example 2.12).

Remark 3.7.

A different proof of Theorem 3.5 in the case when the surfaces $X$ , $Y$ are rational can be deduced from a more recent result by Lamy and Schneider [LS21] who analyze relations between Sarkisov links to prove that ${\operatorname{Cr}}_{2}(k)$ is generated by involutions. This implies that every homomorphism from ${\operatorname{Cr}}_{2}(k)$ to a free abelian group is trivial (for $k=\mathbb{R}$ one can also deduce this from [Zim18, Theorem 1.1]). On the other hand this is not true for other types of del Pezzo surfaces: Blanc, Schneider and Yasinsky have constructed nontrivial homomorphisms ${\operatorname{Bir}}(X)\to\mathbb{Z}$ for Severi-Brauer surfaces $X$ [BSY22].

Example 3.8.

Consider the following composition $\phi$ of type $II$ links (see Definition 4.2 for links) between del Pezzo surfaces $dP_{d}$ (where $d=K^{2}$ stands for the degree)

where each map ${\operatorname{Bl}}_{i}$ is the blow up (resp. blow down) along a Galois orbit $Z_{i}$ (resp. $Z^{\prime}_{i}$ ) of degree $i$ . See [Isk96] for the general results on links, or see the explicit constructions explained below for the existence of the three links above. We have

c(\phi)=[Z_{2}]-[k]+[Z_{5}]-[Z_{2}^{\prime}]+[k]-[Z_{5}^{\prime}]=[Z_{2}]-[Z_{2}^{\prime}]+[Z_{5}]-[Z_{5}^{\prime}],

so that $c(\phi)=0$ is equivalent to $Z_{2}\simeq Z_{2}^{\prime}\text{ and }Z_{5}\simeq Z_{5}^{\prime}$ .

One way to understand this is to read the diagram as a composition of two different well-known rationality constructions for del Pezzo surace $dP_{5}$ : one blows up a point $P$ and contracts a Galois orbit of five $(-1)$ -curves which are proper preimages of conics through $P$ (connecting $dP_{5}$ with $\mathbb{P}^{2}$ moving right) or one blows up a Galois orbit of two points $Q_{1}$ , $Q_{2}$ and contracts five $(-1)$ -curves obtained as proper preimages of cubics passing through $Q_{1}$ , $Q_{2}$ onto a rational quadric $dP_{8}$ and then transforms it to $\mathbb{P}^{2}$ (connecting $dP_{5}$ with $\mathbb{P}^{2}$ moving left). This way $dP_{5}$ has two potentially different rationality centers that can be blown up by $\mathbb{P}^{2}\dashrightarrow dP_{5}$ , namely $Z_{5}$ and $Z_{5}^{\prime}$ and the result is that these centers are in fact isomorphic (see Corollary 5.9 and Example 5.10).

3.2. Interpretation in terms of the Grothendieck ring of varieties

From the perspective of the Grothendieck ring of varieties, we have the following interpretation of $c(\phi)$ .

Lemma 3.9.

For any birational isomorphism $\phi:X\dashrightarrow Y$ between smooth projective surfaces we have the following identity in $K_{0}({\mathrm{Var}}/k)$

(3.3)

[Y]=[X]+\mathbb{L}\cdot\overline{c}(\phi),

where $\overline{c}$ is the composition of $c$ with the natural homomorphism (2.7) $\mathbb{Z}[{\mathrm{Var}}^{0}/k]\to K_{0}({\mathrm{Var}}/k)$ .

Proof.

Using Lemma 3.1 we see that (3.3) is preserved under compositions of birational isomorphisms. Since birational isomorphisms are decomposed into blow ups and blow downs along $k$ -étale subschemes, it suffices to check Lemma 3.9 for a single blow up, where the result is clear by Corollary 3.2(2) and the blow up formula in the Grothendieck ring. $\Box$

Remark 3.10.

We see that $\mathbb{L}\cdot\overline{c}(\phi)\in K_{0}({\mathrm{Var}}/k)$ only depends on $[X]$ and $[Y]$ , however it is known that $\mathbb{L}$ is a zero-divisor [Bor18] so a priori we cannot divide $\mathbb{L}\cdot\overline{c}(\phi)$ by $\mathbb{L}$ and deduce that $\overline{c}(\phi)$ (or $c(\phi)$ ) only depends on $X$ and $Y$ but not on $\phi$ .

We can informally reformulate Theorem 3.5 by the statement that $0$ -dimensional $k$ -varieties cannot be L-equivalent via surfaces. Let us explain this. For each $n\geq 0$ consider $K_{0}({\mathrm{Var}}^{\leq n}/k)$ , the abelian group generated by isomorphism classes of varieties of dimension $\leq n$ , modulo cut and paste relations (2.3). For each $n\leq m$ there is a (generally non-injective) group homomorphism

K_{0}({\mathrm{Var}}^{\leq n}/k)\to K_{0}({\mathrm{Var}}^{\leq m}/k)

and $K_{0}({\mathrm{Var}}/k)$ , as an abelian group, is the colimit of this system. For every $n$ there is a surjective homomorphism

K_{0}({\mathrm{Var}}^{\leq n}/k)\to\mathbb{Z}[{\operatorname{Bir}}_{n}/k],

where ${\operatorname{Bir}}_{n}/k$ is the set of $k$ -birational classes of dimension $n$ . The kernel of this homomorphism is spanned by varieties of dimension $\leq(n-1)$ , but it may not be isomorphic to $K_{0}({\mathrm{Var}}^{\leq n-1}/k)$ ; see [Zak17] for interpretation of this in terms of an algebraic ${\mathrm{K}}$ -theory spectral sequence.

For $n=0$ , $K_{0}({\mathrm{Var}}^{\leq 0}/k)$ is canonically isomorphic to $\mathbb{Z}[{\mathrm{Var}}^{0}/k]$ , and $K_{0}({\mathrm{Var}}^{\leq 1}/k)$ fits into a split exact sequence

(3.4)

0\to\mathbb{Z}[{\mathrm{Var}}^{0}/k]\to K_{0}({\mathrm{Var}}^{\leq 1}/k)\to\mathbb{Z}[{\operatorname{Bir}}_{1}/k]\to 0.

Our main results can be reformulated as results about $K_{0}({\mathrm{Var}}^{\leq 2}/k)$ :

Corollary 3.11 (of Theorem 3.5).

(i) We have a short exact sequence

(3.5)

0\to\mathbb{Z}[{\mathrm{Var}}^{0}/k]\oplus\mathbb{Z}[{\operatorname{Bir}}_{1}/k]\to K_{0}({\mathrm{Var}}^{\leq 2}/k)\to\mathbb{Z}[{\operatorname{Bir}}_{2}/k]\to 0,

where the last map sends a combination of varieties to the birational classes of its $2$ -dimensional components, and the first map sends a $k$ -variety $Z$ of dimension $0$ to $[Z]$ and a birational class of a curve to the class of its unique smooth projective model.

(ii) We have a (noncanonical) splitting

K_{0}({\mathrm{Var}}^{\leq 2}/k)\simeq\mathbb{Z}[{\mathrm{Var}}^{0}/k]\oplus\mathbb{Z}[{\operatorname{Bir}}_{1}/k]\oplus\mathbb{Z}[{\operatorname{Bir}}_{2}/k],

and in particular $K_{0}({\mathrm{Var}}^{\leq 2}/k)$ is a (torsion-)free abelian group.

(iii) If $Z$ and $Z^{\prime}$ are étale $k$ -schemes, and

\mathbb{L}^{i}\cdot([Z]-[Z^{\prime}])=0

in $K_{0}({\mathrm{Var}}^{\leq 2}/k)$ with $i\in\{0,1,2\}$ (where $\mathbb{L}^{i}\cdot([Z]-[Z^{\prime}])$ is represented by $[\mathbb{A}^{i}\times_{k}Z]-[\mathbb{A}^{i}\times_{k}Z^{\prime}]$ ), then $Z$ and $Z^{\prime}$ are isomorphic.

Proof.

(i) Because resolution of singularities and weak factorization (even strong factorization) are known for surfaces over arbitrary perfect fields, the proof of [Bit04] goes through to show that we have an isomorphism

K_{0}^{\operatorname{bl}}({\mathrm{Var}}^{\leq 2}/k)\to K_{0}({\mathrm{Var}}^{\leq 2}/k),

where the first group is defined by smooth projective varieties of dimension $\leq 2$ and Bittner’s blow up relations. These relations can be equivalently presented as

(3.6)

[Y]=[X]+\mathbb{L}\cdot\bar{c}(\phi)

for all birational isomorphisms $\phi:X\dashrightarrow Y$ between smooth projective surfaces. Indeed, (3.6) include the blow up relations if $\phi:Y\to X$ is a smooth blow up as a particular case and conversely, (3.6) follow as soon as we impose the blow up relations as in the proof of Lemma 3.9. Since curves admit unique smooth projective models, we have the canonical splitting of (3.4) giving

K_{0}({\mathrm{Var}}^{\leq 1}/k)=K_{0}^{\operatorname{bl}}({\mathrm{Var}}^{\leq 1}/k)\simeq\mathbb{Z}[{\mathrm{Var}}^{0}/k]\oplus\mathbb{Z}[{\operatorname{Bir}}_{1}/k].

Furthermore we have an obvious short exact sequence

K_{0}^{\operatorname{bl}}({\mathrm{Var}}^{\leq 1}/k)\to K_{0}^{\operatorname{bl}}({\mathrm{Var}}^{\leq 2}/k)\to\mathbb{Z}[{\operatorname{Bir}}_{2}/k]\to 0

and to prove (3.5) it suffices to show that the first map in the sequence is split-injective.

The splitting is based on Theorem 3.5 and is not canonical. First of all, we choose a smooth projective representative for each birational class of surfaces. If $L$ is a two-dimensional function field, we write $M_{L}$ for the chosen model. We define the splitting

\epsilon:K_{0}^{\operatorname{bl}}({\mathrm{Var}}^{\leq 2}/k)\to K_{0}^{\operatorname{bl}}({\mathrm{Var}}^{\leq 1}/k)

by identity on classes of smooth projective curves and zero-dimensional schemes and if $X$ is a smooth projective surface we define

\epsilon([X])=\mathbb{L}\cdot\overline{c}(M_{k(X)}\overset{\psi}{\dashrightarrow}X)

for any choice of a birational isomorphism $\psi$ between $X$ and its model $M_{k(X)}$ . The fact that this is independent of $\psi$ is the content of Theorem 3.5, and the fact that $\epsilon$ is well-defined, that is preserves the relations (3.6), follows immediately from the property that $c$ is additive on compositions (Lemma 3.1) applied to a composition of birational isomorphisms $M\overset{\psi}{\dashrightarrow}X\overset{\phi}{\dashrightarrow}Y$ (where $M:=M_{k(X)}=M_{k(Y)}$ ).

(ii) follows from (3.5); explicit splittings are constructed in the proof of (i).

(iii) The case $i=2$ is clear as if $\mathbb{A}^{i}\times_{k}Z$ and $\mathbb{A}^{i}\times_{k}Z^{\prime}$ are $k$ -birational, then $Z$ and $Z^{\prime}$ are isomorphic. For $i=1$ , the element $\mathbb{L}\cdot([Z]-[Z^{\prime}])$ is the image of $(-[Z]+[Z^{\prime}],[\mathbb{P}^{1}\times_{k}Z]-[\mathbb{P}^{1}\times_{k}Z^{\prime}])$ under the first map in (3.5), and since this map is injective, $[Z]=[Z^{\prime}]$ , that is $Z$ and $Z^{\prime}$ are isomorphic. Finally the case $i=0$ follows again from (3.5). $\Box$

4. Birational geometry of surfaces

In this section we recall the Minimal Model Program and Sarkisov link decomposition for surfaces, which is used in our proof of Theorem 3.5. In Proposition 4.4 we prove Theorem 3.5 for birational types with particularly simple links. In §4.2 we investigate linear systems of rational curves on del Pezzo surfaces, which will be needed to finish the proof of Theorem 3.5 in §5.

4.1. Birational classification of surfaces and links

Let $X/k$ be a geometrically irreducible surface. Recall that $X$ is called minimal if it does not have a Galois orbit of disjoint $(-1)$ -curves, or equivalently, every regular birational map $X\to X^{\prime}$ from $X$ to a smooth surface is an isomorphism. We say that $X$ is rational if it is birational to $\mathbb{P}^{2}$ over $k$ and $X$ is geometrically rational if it is birational to $\mathbb{P}^{2}$ over $\overline{k}$ . A del Pezzo surface is a smooth projective geometrically irreducible surface with ample anticanonical class.

A conic bundle is a fibered surface $\pi:X\to C$ , with $C$ a smooth projective curve, such that the generic fiber of $\pi$ is a smooth rational curve and $X$ has Picard rank two. For a conic bundle $\pi:X\to C$ , $\pi_{*}\omega_{X/C}^{\vee}$ is locally free of rank $3$ and we have an embedding $X\hookrightarrow\mathbb{P}(\pi_{*}\omega_{X/C}^{\vee})$ over $C$ realizing each fiber of $\pi:X\to C$ as a plane conic (see [Has09, Corollary 3.7]). The number $m$ of geometric singular fibers of a conic bundle equals

(4.1)

m=8-K_{X}^{2}

see e.g. [Isk79, Theorem 3], in particular $K_{X}^{2}\leq 8$ .

We write ${\operatorname{NS}}(X)$ for the Néron-Severi group of divisors modulo algebraic equivalence; by the theorem of Néron-Severi it is a finitely generated abelian group. Each contraction of a Galois orbit of $(-1)$ -curves decreases the rank of the Néron-Severi group, hence a sequence of contractions always terminates to produce a minimal surface birationally equivalent to the given one.

We have the following classification result going back to Enriques, Manin and Iskovskikh, see [Kol96, Theorem III.2.2].

Theorem 4.1 (Minimal Model Program in dimension two).

Any geometrically irreducible minimal surface $X/k$ is isomorphic to exactly one of the following:

•
Non-geometrically rational case
1. (1)
  
  Surface with nef $K_{X}$
2. (2)
  
  Conic bundle $\pi:X\to C$ with $g(C)>0$
•
Geometrically rational case
1. (3)
  
  Conic bundle $X\to C$ with $g(C)=0$
2. (4)
  
  Minimal del Pezzo surface of Picard rank one

In cases (2) and (3), ${\operatorname{NS}}(X)$ is of rank two.

Note that if $k$ is algebraically closed, any conic bundle in (2), (3) has no singular fibers (as components of singular fibers are $(-1)$ -curves), thus (2) consists of ruled surfaces and (3) consists of Hirzebruch surfaces $\mathbb{F}_{n}$ .

Definition 4.2 (Sarkisov links, see 2.2 in [Isk96]).

Let $X\to B$ and $X^{\prime}\to B^{\prime}$ be Mori fiber spaces with $\dim X=\dim X^{\prime}=2$ . Here each Mori fiber space is either a del Pezzo surface of Picard rank one or a conic bundle of Picard rank two over a smooth curve. A Sarkisov link between $X$ and $X^{\prime}$ is a birational map $\nu:X\dashrightarrow X^{\prime}$ satisfying one of the following descriptions.

•

(Type I) $\nu^{-1}$ is the blow up of $X$ at a smooth closed point with $B={\operatorname{Spec}}(k)$ and $X^{\prime}\to B^{\prime}$ a conic bundle.
•
(Type II) We have a commutative diagram

(4.2)

and $\nu=\beta\circ\alpha^{-1}$ where both $\alpha$ and $\beta$ are blow ups at a smooth closed point. In this case,
- –
  
  (Type IIC) either both $X\to B$ and $X^{\prime}\to B^{\prime}$ are conic bundles;
- –
  
  (Type IID) or $B=B^{\prime}={\operatorname{Spec}}(k)$ .
•

(Type III) This is the inverse of link (I).
•

(Type IV) $X\rightarrow B$ , $X^{\prime}\to B^{\prime}$ are conic bundles and $\nu$ is an isomorphism not respecting the conic bundle structures.

We rely on decomposing birational isomorphisms of surfaces into Sarkisov links [Isk96]. Some of the links are easier to deal with.

Lemma 4.3.

If $\nu:X\dashrightarrow X^{\prime}$ is a Sarkisov link of type IIC, then $K_{X}^{2}=K_{X^{\prime}}^{2}$ , ${\operatorname{rk}}\;{\operatorname{NS}}(X)={\operatorname{rk}}\;{\operatorname{NS}}(X^{\prime})$ and $c(\nu)=0$ .

Proof.

Let $\phi:X\to B$ be the conic bundle as in (4.2). By [Isk96, Theorem 2.6], $\nu$ is an elementary transformation of $X\to B$ . More precisely, $\alpha$ in (4.2) is the blow up at a smooth closed point $p\in X$ lying in a smooth fiber $C$ of $\phi$ and $\beta$ in (4.2) is the contraction of the proper transformation $\tilde{C}$ of $C$ under $\alpha$ . Since $\tilde{C}\simeq C\simeq\mathbb{P}^{1}\times_{{\operatorname{Spec}}(k)}\{p\}$ , the blow up center of $\beta$ is isomorphic to $p$ . Hence ${\operatorname{rk}}\;{\operatorname{NS}}(X)={\operatorname{rk}}\;{\operatorname{NS}}(X^{\prime})$ and $c(\nu)=0$ . The equality $K_{X}^{2}=K_{X^{\prime}}^{2}$ follows from the fact that $X\to B$ and $X^{\prime}\to B^{\prime}$ have the same number of geometric singular fibers using (4.1). $\Box$

Recall that the degree of a geometrically rational surface $X$ is defined to be $K_{X}^{2}$ . The next result is a step towards Theorem 3.5.

Proposition 4.4.

Let $X$ , $X^{\prime}$ be a pair of birational minimal geometrically irreducible surfaces. Assume that $X$ is either (1) geometrically irrational or (2) geometrically rational and of degree $\leq 4$ , then the same holds for $X^{\prime}$ and for any birational isomorphism $\phi:X\dashrightarrow X^{\prime}$ we have

(4.3)

c(\phi)=({\operatorname{rk}}\;{\operatorname{NS}}(X^{\prime})-{\operatorname{rk}}\;{\operatorname{NS}}(X))\cdot[{\operatorname{Spec}}(k)].

In particular $c(\phi)=0$ for birational automorphisms of such surfaces.

Proof.

(1) It is clear that $X^{\prime}$ is geometrically irrational and moreover, $X$ and $X^{\prime}$ have the same type in Theorem 4.1. If $X$ , $X^{\prime}$ have nef canonical class, then every birational isomorphism is a biregular isomorphism by [IS96, Corollary 1 in II.7.3] so that $X\simeq X^{\prime}$ and $c(\phi)=0$ as required.

If $X$ , $X^{\prime}$ are conic bundles over a curve of positive genus, then by [Sch22, Corollary 3.2], birational maps between them can be decomposed into Sarkisov links of type IIC, in which case ${\operatorname{rk}}\;{\operatorname{NS}}(X)={\operatorname{rk}}\;{\operatorname{NS}}(X^{\prime})$ and $c(\phi)=0$ by Lemma 4.3.

(2) It follows from the classification of links in [Isk96, Theorem 2.6] that elementary links will only connect $X$ to minimal surfaces of degree $\leq 4$ or conic bundles of degree $3$ and Picard rank $2$ , and that birational isomorphisms between such surfaces will be decomposed into

•

Biregular isomorphisms
•

Type IIC links
•

Bertini and Geiser involutions
•

Blow ups of a rational point (between degree $4$ and degree $3$ surfaces).

For each of these types the claim of Proposition 4.4 is true, namely in the first three cases ${\operatorname{rk}}\;{\operatorname{NS}}(X)={\operatorname{rk}}\;{\operatorname{NS}}(X^{\prime})$ and $c(\phi)=0$ (using Lemma 4.3 for the second case), while in the last case the ranks differ by one and the result is true by definition of $c$ . Equality (4.3) follows by additivity of $c$ under compositions (Lemma 3.1). $\Box$

Remark 4.5.

From the proof of Proposition 4.4, we note that in (4.3), we have actually $c(\phi)=0$ or $c(\phi)=\pm[{\operatorname{Spec}}(k)]$ . Moreover, $c(\phi)=\pm[{\operatorname{Spec}}(k)]$ only happens when $\phi$ is a map between a del Pezzo surface of degree $4$ and a conic bundle of degree $3$ .

4.2. Rational curves on del Pezzo surfaces

Linear systems of rational curves on del Pezzo surfaces are closely related to factorization centers: for instance, to create a rational two-dimensional quadric $X$ , one needs to blow up $\mathbb{P}^{2}$ in $Z_{2}$ , where $Z_{2}$ is degree two étale $k$ -scheme, and to contract a line through the center. This way the original scheme $Z_{2}$ can be recovered as the scheme parametrizing rulings on $X$ (cf Definition 5.2).

Definition 4.6.

Let $j\geq 1$ . We call a complete linear system $|L|$ of curves on a del Pezzo surface $X$ a linear system of degree $j$ rational curves if a general member $C\in|L|$ is a smooth rational curve and $(-K_{X})\cdot L=j$ .

For each $j\geq 1$ we consider $\mathcal{H}^{j}(X)$ , the Hilbert scheme of curves on $X$ with general members of each component being smooth rational curves of degree $j$ . By an easy computation (see Lemma 4.7(i)) $H^{1}(X,L)=0$ , hence $\mathcal{H}^{j}(X)$ is smooth [Kol96, Theorem 2.8]. In fact, over the algebraic closure $\overline{k}$ , each $\mathcal{H}^{j}(X_{\overline{k}})$ is a disjoint union of projective spaces parametrizing effective divisors in the corresponding linear systems. We refer to points of $\mathcal{H}^{1}(X)$ as lines on $X$ , points of $\mathcal{H}^{2}(X)$ as conics and so on. When $X$ is a twisted form of $\mathbb{P}^{1}\times\mathbb{P}^{1}$ (a minimal del Pezzo surface of degree $8$ ), then all curves have even degree, and families of conics on $X$ are also called rulings.

We note that by adjunction formula, a linear system $L$ of rational curves of degree $j$ satisfies

(4.4)

L^{2}=j-2,\;\;(-K_{X})\cdot L=j.

However the latter numerical conditions are not sufficient to deduce that a linear system consists of rational curves. Indeed the linear system $3H+E$ on ${\operatorname{Bl}}_{P}(\mathbb{P}^{2})$ , where $H$ is the class of a line on $\mathbb{P}^{2}$ , and $E$ is the exceptional divisor, satisfies (4.4) with $j=10$ but has a fixed component and contains no rational curves.

We investigate how to find all linear systems of rational curves of a given degree following [Man86]. The first step is to solve (4.4). Let $X$ be a del Pezzo surface of degree $d=K_{X}^{2}$ . Assume $X_{\overline{k}}$ is not isomorphic to $\mathbb{P}^{1}\times\mathbb{P}^{1}$ so that $X_{\overline{k}}$ is a blow up of $\mathbb{P}^{2}$ in $r:=9-d$ points by [Man86, Theorem IV.2.5], [Isk79, §3]. Write $D=aH-\sum_{i=1}^{r}b_{i}E_{i}$ , where $H$ is the pullback of the hyperplane class and $E_{i}$ are the irreducible components of the exceptional divisor. Since $-K_{X}=3H-\sum_{i=1}^{r}E_{i}$ , (4.4) translates into:

(4.5)

\left\{\begin{array}[]{l}\sum_{i=1}^{r}b_{i}=3a-j\\ \sum_{i=0}^{r}b_{i}^{2}=a^{2}-j+2\\ \end{array}\right.

The case $j=1$ is that of lines on del Pezzo surfaces [Man86].

Lemma 4.7.

Let $X$ be a del Pezzo surface of degree $d$ over an algebraically closed field $k$ .

(i)

A linear system of rational curves of degree $j$ satisfies $h^{0}(X,L)=j$ , and higher cohomology of $L$ vanish.
(ii)

Any linear system satisfying (4.4) with $j\geq 1$ is non-empty.
(iii)

A linear system satisfying (4.4) with $1\leq j\leq d-1$ is a linear system of rational curves.
(iv)

Let $1\leq j\leq d-1$ . The assignment $|D|\mapsto|-K_{X}-D|$ establishes a bijection between linear systems of rational curves of degrees $j$ and $d-j$ .

Proof.

(i) Let $C\in|L|$ be a smooth rational curve. Taking cohomology for the short exact sequence

0\to\mathcal{O}_{X}\to\mathcal{O}_{X}(C)\to\mathcal{O}_{C}(C)\to 0

we get $H^{1}(X,\mathcal{O}_{X}(C))=H^{2}(X,\mathcal{O}_{X}(C))=0$ , and hence $h^{0}(X,\mathcal{O}_{X}(C))=j$ by Riemann-Roch.

(ii) We have $h^{2}(X,L)=h^{0}(X,L^{\vee}\otimes\omega_{X})=0$ (because $L^{\vee}\otimes\omega_{X}$ has negative anticanonical degree) and the Riemann-Roch theorem implies

h^{0}(X,L)\geq j.

(iii) If $X\simeq\mathbb{P}^{1}\times\mathbb{P}^{1}$ , then it is easy to see that solutions of (4.4) are precisely divisors $(a,1)$ , $a\geq 0$ and $(1,b)$ , $b\geq 0$ , and these are linear systems of rational curves (no upper bound on the degree needed). Now assume that $X$ is a blow up of $\mathbb{P}^{2}$ in $0\leq r\leq 8$ points. Since $D$ satisfies (4.4), it follows that $-K_{X}-D$ satisfies (4.4) with degree $d-j$ . By (ii) both $|D|$ and $|-K_{X}-D|$ are nonempty. Therefore if $D=aH-\sum_{i=1}^{r}b_{i}E_{i}$ , then $0\leq a\leq 3$ . Solving equations (4.5) with $b_{i}\in\mathbb{Z}$ , under the assumption $1\leq j\leq d-1$ gives rise to the following divisors, which we list up to reordering of the exceptional divisors:

•

$a=0$ : $E_{1}$
•

$a=1$ : $H-\sum_{i=1}^{t}E_{i}$
•

$a=2$ : $2H-\sum_{i=1}^{r-t}E_{i}$
•

$a=3$ : $3H-2E_{1}-\sum_{i=2}^{r}E_{i}$

Indeed, the cases $a=0,1$ are straightforward and the case $a=2,3$ follow via the $D\mapsto-K_{X}-D$ substitution. The $a=1,2$ cases are only possible when $r\leq 7$ ( $d\geq 2$ ). We must have $0\leq t\leq 2$ , and for $r=6$ (resp. $r=7$ ), only allow $t=1,2$ (resp. $t=2$ ). Under these conditions each of the divisors in the list is linearly equivalent to a smooth rational curve.

(iv) Follows from (iii). $\Box$

Proposition 4.8.

Let $X/k$ be a del Pezzo surface of degree $d$ .

(i)

For each $j\geq 1$ , $\mathcal{H}^{j}(X)$ is either empty or a smooth Severi-Brauer fibration of relative dimension $j-1$ over a smooth zero-dimensional scheme $\mathcal{M}^{j}(X)$ . For each $1\leq j\leq d-1$ we have a natural isomorphism $\mathcal{M}^{j}(X)\simeq\mathcal{M}^{d-j}(X)$ .
(ii)

Assume $\mathcal{H}^{j}(X)$ is nonempty and let $Z\subset X$ be a zero-dimensional subscheme of degree $j-1$ . Let $\mathcal{H}^{j}(X,Z)$ be the closed subvariety of $\mathcal{H}^{j}(X)$ parametrizing curves containing $Z$ . If $\mathcal{H}^{j}(X,Z)$ is nonempty and zero-dimensional, then it is isomorphic to $\mathcal{M}^{j}(X)$ .
(iii)

Suppose that $X_{\overline{k}}$ is obtained by blowing up $r\leq 5$ points on $\mathbb{P}^{2}$ with exceptional divisors $E_{1},\ldots,E_{r}$ . Let $H\in{\operatorname{NS}}(X_{\overline{k}})$ be the pullback of the hyperplane class. Then the classes of conics (resp. cubics) on $X_{\overline{k}}$ are $H-E_{i}$ and $2H-\sum_{i=1}^{4}E_{i}$ (resp. $H$ and $2H-\sum_{i=1}^{3}E_{i}$ ).
(iv)
Consider the following cases.
1. ( $dP_{8}$ )
  
  Let $X$ be a minimal del Pezzo surface of degree $8$ . Then $\mathcal{M}^{4}(X)\simeq{\operatorname{Spec}}(k)$ and $\mathcal{M}^{2}(X)\simeq\mathcal{M}^{6}(X)\simeq Z_{2}$ , where $Z_{2}$ is an étale $k$ -scheme of degree 2.
2. ( $dP_{6}$ )
  
  Let $X$ be a del Pezzo surface of degree $6$ . Then
  
  $\mathcal{M}^{2}(X)\simeq\mathcal{M}^{4}(X)\simeq Z_{3},\;\mathcal{M}^{3}(X)\simeq Z_{2}$
  
  for degree two (resp. degree three) étale $k$ -schemes $Z_{2},Z_{3}$ .
3. ( $dP_{5}$ )
  
  Let $X$ be a del Pezzo surface of degree $5$ . Then $\mathcal{M}^{2}(X)\simeq\mathcal{M}^{3}(X)\simeq Z_{5}$ for a degree five étale $k$ -scheme $Z_{5}$ .

Proof.

(i) Since $H^{1}(X,\mathcal{O}_{X})=0$ [Kol96, III.3.2.1], deforming an effective divisor $D$ as a subscheme is equivalent to deforming it in its linear system, so $\mathcal{H}^{j}(X)_{\overline{k}}$ is either empty or a disjoint union of complete linear systems (see [Kol96, I.1.14.2]). Since the Hilbert polynomial of a rational curve $C$ on a del Pezzo surface is determined by its degree $C\cdot(-K_{X})$ , and since the Hilbert scheme is projective [Kol96, Theorem I.1.4], in particular of finite type, it follows that $\mathcal{H}^{j}(X)_{\overline{k}}$ is a finite disjoint union of projective spaces. (Finiteness also follows from the fact there are finitely many solutions for $\eqref{eq:num-rat}$ with fixed $j$ .) By Lemma 4.7(i), these projective spaces have dimension $j-1$ .

Thus we have shown that $\mathcal{H}^{j}(X)$ is a smooth scheme of finite type over $k$ isomorphic over $\overline{k}$ to a finite disjoint union of projective spaces. Let $\mathcal{M}^{j}(X)={\operatorname{Spec}}(H^{0}(\mathcal{H}^{j}(X),\mathcal{O}))$ ; it is a smooth zero-dimensional scheme, and $\mathcal{H}^{j}(X)\to\mathcal{M}^{j}(X)$ is a Severi-Brauer fibration. Finally, the isomorphism $\mathcal{M}^{j}(X)\simeq\mathcal{M}^{d-j}(X)$ is given by $|C|\mapsto|-K_{X}-C|$ , using Lemma 4.7 (iv).

(ii) We claim that the projection $\mathcal{H}^{j}(X,Z)\to\mathcal{M}^{j}(X)$ is an isomorphism. By Galois descent it is sufficient to verify this over the algebraic closure. Each fiber of $\mathcal{H}^{j}(X)_{\bar{k}}\to\mathcal{M}^{j}(X)_{\bar{k}}$ is equal to $|L|$ for some line bundle $L$ . It follows from the assumption that the linear system defined by $H^{0}(X_{\bar{k}},L\otimes I_{Z})\subset H^{0}(X_{\bar{k}},L)$ is a point in $|L|$ where $I_{Z}$ be the ideal sheaf of $Z_{\bar{k}}\subset X_{\bar{k}}$ . Hence $\mathcal{H}^{j}(X,Z)_{\bar{k}}\to\mathcal{M}^{j}(X)_{\bar{k}}$ is an isomorphism.

(iii) is proven by solving (4.5).

(iv) Galois descent identifies smooth zero-dimensional scheme with a set with Galois action. Thus it suffices to verify the numbers of conic bundles and nets of cubics over the algebraic closure in each case. For ( $dP_{8}$ ), the linear system of quartic rational curves on $\mathbb{P}^{1}\times\mathbb{P}^{1}$ is defined by $\mathcal{O}(1,1)$ , and the two pencils of conics $\mathcal{O}(1,0)$ and $\mathcal{O}(0,1)$ . The cases ( $dP_{6}$ ) and ( $dP_{5}$ ) follow from (iii). $\Box$

5. Models of large degree

In this section we deal with surfaces admitting the most interesting birational automorphisms. These are geometrically rational surfaces with minimal models of degree $\geq 5$ . At the end of this section we prove Theorem 3.5 and discuss rationality centers for rational surfaces.

5.1. Virtual Néron-Severi sets

We introduce a type of surfaces $X/k$ which we call models of large degree:

( $dP_{9}$ )

$\mathbb{P}^{2}$ or a Severi-Brauer surface
( $dP_{8}$ )

minimal del Pezzo surface of degree $8$ ; in this case $X_{\overline{k}}\simeq\mathbb{P}^{1}\times\mathbb{P}^{1}$
( $C_{8}$ )

smooth conic bundle over a conic; in this case $X_{\overline{k}}\simeq\mathbb{F}_{n}$ , and we assume $n\geq 1$ to avoid the overlap with ( $dP_{8}$ )
( $dP_{6}$ )

del Pezzo surface of degree $6$
( $dP_{5}$ )

del Pezzo surface of degree $5$

Note that we do not make any assumption on the Picard rank of the surfaces in the list above.

Proposition 5.1.

All minimal geometrically rational surfaces $X$ with $K_{X}^{2}\geq 5$ and all conic bundles over a curve of genus $0$ with $K_{X}^{2}\geq 5$ are among models of large degree.

Proof.

By the work of Iskovskikh [Isk79] explained in Theorem 4.1 (3), (4), $X$ is a del Pezzo surface of Picard rank one or a conic bundle of Picard rank two. Minimal del Pezzo surfaces of degree $K_{X}^{2}\geq 5$ are all among ( $dP_{9}$ ), ( $dP_{8}$ ), ( $dP_{6}$ ), ( $dP_{5}$ ) (del Pezzo surface of degree $7$ is not minimal, see e.g. [Isk79, Corollary on p.37]).

Assume that $X$ is a conic bundle over a curve of genus $0$ ; using (4.1) we have $5\leq K_{X}^{2}\leq 8$ . By [Isk79, Theorem 5], $X$ is model of large degree. $\Box$

Definition 5.2.

For each model of large degree we define its virtual Néron-Severi Galois set $A_{X}\in{\operatorname{Burn}}(G_{k})\simeq\mathbb{Z}[{\mathrm{Var}}^{0}/k]$ as follows. In each case, $Z_{i}$ refers to one of the finite étale $k$ -schemes of degree $i$ introduced in Proposition 4.8(iv).

( $dP_{9}$ )

$A_{X}=[{\operatorname{Spec}}(k)]$
( $dP_{8}$ )

$A_{X}=[Z_{2}]$ , where $Z_{2}$ parametrizes the rulings (that is, the conic bundle structures) on $X_{\overline{k}}$
( $C_{8}$ )

$A_{X}=2[{\operatorname{Spec}}(k)]$
( $dP_{6}$ )

$A_{X}=[Z_{2}]+[Z_{3}]-[{\operatorname{Spec}}(k)]$ , where $Z_{3}$ parametrizes three pencils of conics on $X_{\overline{k}}$ and $Z_{2}$ parametrizes two families of cubics
( $dP_{5}$ )

$A_{X}=[Z_{5}]$ , where $Z_{5}$ parametrizes five pencils of conics on $X_{\overline{k}}$

Note that in all cases except for ( $dP_{6}$ ), the virtual Néron-Severi set is realized as a set. The following Lemma explains the name Néron-Severi set. See (2.1) for the definition of $\mu_{G_{k}}$ .

Lemma 5.3.

If $A_{X}\in{\operatorname{Burn}}(G_{k})$ is the virtual Néron-Severi set of $X$ , and $F$ is any field of characteristic zero, then $\mu_{G_{k}}(A_{X})=[{\operatorname{NS}}(X_{\overline{k}})\otimes F]\in K_{0}({\operatorname{Rep}}(G_{k},F))$ .

Proof.

( $dP_{9}$ )

$A_{X}$ is a set consisting of one element and ${\operatorname{NS}}(X_{\overline{k}})$ is the one-dimensional trivial $G_{k}$ -representation.
( $dP_{8}$ )

$A_{X}$ is the set of rulings on $X$ , and ${\operatorname{NS}}(X_{\overline{k}})$ is freely generated by these rulings.
( $C_{8}$ )

$A_{X}$ is a set with two elements and trivial action. Since $K_{X}$ and the class of a fiber form a basis of ${\operatorname{NS}}(X)\otimes F$ and ${\operatorname{rk}}\,{\operatorname{NS}}(X_{\overline{k}})={\operatorname{rk}}\,{\operatorname{NS}}(\mathbb{F}_{n})=2$ , necessarily ${\operatorname{NS}}(X_{\overline{k}})$ is of rank two with trivial Galois action.
( $dP_{6}$ )

By Proposition 4.8 (iii), ${\operatorname{NS}}(X_{\overline{k}})\otimes F$ is generated by three classes of conics and two classes of cubics, modulo a one-dimensional space with a trivial $G_{k}$ -action; this implies the result.
( $dP_{5}$ )

By Proposition 4.8 (iii), ${\operatorname{NS}}(X_{\overline{k}})\otimes F$ is freely generated by classes of conics $A_{X}$ .

$\Box$

It follows from Lemma 5.3 that the rank of ${\operatorname{NS}}(X)$ equals the number of orbits of the virtual Néron-Severi set $A_{X}$ . In particular, $X$ has Picard rank one if and only $A_{X}$ has only one (virtual) orbit. Since $\mu_{G_{k}}$ is not injective, we can not simply define the virtual Galois set of a surface using $\mu_{G_{k}}(A_{X})=[{\operatorname{NS}}(X_{\overline{k}})\otimes F]$ , see Example 2.14 for an illustration; this is why we define them case by case in Definition 5.2.

Remark 5.4.

The Galois sets forming $A_{X}$ naturally appear in the Chow motive of $X$ [Gil15, (9) and the following Remark] and as semiorthogonal components in the derived category of the corresponding surface [BSS11], [AB18, Propositions 9.8, 10.1]. Furthermore, singular versions of those $k$ -algebras show up in the study of the derived categories of the corresponding singular del Pezzo surfaces [Kuz19], [KKS22], [Xie21].

Proposition 5.5.

Let $X$ be a model of large degree, and let $\phi:X\dashrightarrow X^{\prime}$ be a birational isomorphism to another minimal surface; then $X^{\prime}$ is also a model of large degree, and

(5.1)

c(\phi)=A_{X^{\prime}}-A_{X}.

First we prove that if we apply $\mu_{G_{k}}$ to both sides of (5.1), we have equality.

Lemma 5.6.

Let $X$ , $X^{\prime}$ be models of large degree. Let $\phi:X\leftarrow Y\to X^{\prime}$ be a composition of a blow up and a blow down (the centers of the blow ups can be disconnected, or empty). We have

\mu_{G_{k}}(A_{X}^{\prime})-\mu_{G_{k}}(A_{X})=\mu_{G_{k}}(c(\phi)).

Proof.

Let $Z$ , $Z^{\prime}$ be the centers of the two blow ups. By the blow up formula, we have $[X]+\mathbb{L}\cdot{[Z]}=[Y]=[X^{\prime}]+\mathbb{L}\cdot{[Z^{\prime}]}$ in the Grothendieck ring. Applying the étale realization (2.4) to this equality (cf. Example 2.9), and using Lemma 5.3 with $F=\mathbb{Q}_{\ell}$ we get

(5.2)

\mu_{G_{k}}(A_{X})+[\mathbb{Q}_{\ell}[Z_{\bar{k}}]]=[{\operatorname{NS}}(Y_{\overline{k}})\otimes\mathbb{Q}_{\ell}]=\mu_{G_{k}}(A_{X^{\prime}})+[\mathbb{Q}_{\ell}[Z^{\prime}_{\bar{k}}]]\in K_{0}({\operatorname{Rep}}(G_{k},\mathbb{Q}_{\ell})).

Hence $\mu_{G_{k}}(A_{X}^{\prime})-\mu_{G_{k}}(A_{X})=\mu_{G_{k}}(c(\phi))$ . $\Box$

Before proving Proposition 5.5, we establish some particular cases, which rely on Gassmann equivalence being trivial for Galois sets of order $\leq 5$ .

Lemma 5.7.

Let $X$ , $X^{\prime}$ be models of large degree. Let $\phi:X\leftarrow Y\to X^{\prime}$ be a composition of a blow up and a blow down. Assume that either (1) the two blow ups have isomorphic centers or (2) $K_{Y}^{2}\geq 5$ . Then (5.1) holds for $\phi$ .

Proof.

Let $Z$ , $Z^{\prime}$ be the centers of the two blow ups. We need to show that

\alpha\mathrel{:=}(A_{X}+[Z])-(A_{X^{\prime}}+[Z^{\prime}])=0.

By Lemma 5.6 and Corollary 2.6, it is sufficient to show that $|\alpha|\leq 5$ , that is to represent $\alpha$ as a difference of two $G_{k}$ -sets, each having order $\leq 5$ . Recall that the virtual $G_{k}$ -set $A_{X}$ is represented by a $G_{k}$ -set of order $\leq 5$ , except possibly when $K_{X}^{2}=6$ in which case $A_{X}+[{\operatorname{Spec}}(k)]$ is a $G_{k}$ -set, of order $5$ .

(1) Since $[Z]=[Z^{\prime}]$ , we have $\alpha=A_{X}-A_{X^{\prime}}$ . If $K_{X}^{2}=K_{X^{\prime}}^{2}\neq 6$ , then $|\alpha|\leq 5$ , as $|A_{X}|=|A_{X^{\prime}}|\leq 5$ . If $K_{X}^{2}=K_{X^{\prime}}^{2}=6$ , then $\alpha=(A_{X}+[{\operatorname{Spec}}(k)])-(A_{X^{\prime}}+[{\operatorname{Spec}}(k)])$ also shows that $|\alpha|\leq 5$ .

(2) Assume both $K_{X}^{2}$ , $K_{X^{\prime}}^{2}$ are not equal to $6$ . In this case by (5.2) both $A_{X}+[Z]$ and $A_{X^{\prime}}+[Z^{\prime}]$ have order equal to ${\operatorname{rk}}\;{\operatorname{NS}}(X_{\overline{k}})=10-K_{Y}^{2}\leq 5$ , and the original representation of $\alpha$ shows that $|\alpha|\leq 5$ . The cases when $K_{X}^{2}=6$ or $K_{X^{\prime}}^{2}=6$ are analogous and are left to the reader. $\Box$

Proof of Proposition 5.5.

By [Isk96, Theorem 2.5], any birational isomorphism between minimal geometrically rational surfaces is a composition of Sarkisov links explained in Definition 4.2. Since $c$ is a homomorphism and sends isomorphisms to zero, it suffices to prove Proposition 5.5 for every link of type I, II, or III.

For type I links we write $a\leftarrow b$ for a link $\phi:X\leftarrow X^{\prime}$ with $K_{X}^{2}=a$ , $K_{X^{\prime}}^{2}=b$ . We have the following possibilities according to [Isk96, Theorem 2.6(i)]: $9\leftarrow 8$ , $9\leftarrow 5$ , $8\leftarrow 6$ . Here $X^{\prime}\to B^{\prime}$ is a conic bundle of degree $\geq 5$ , hence $X^{\prime}$ is a model of large degree by Proposition 5.1, and (5.1) follows from Lemma 5.7(2) (with one of the centers empty). Exactly the same argument proves the claim for links of type III.

For type IIC links, the result is true by Lemma 4.3 and Proposition 5.1. For a type IID link $X\leftarrow Y\to X^{\prime}$ , by the first statement of Proposition 4.4(2), $K_{X}^{2}\geq 5$ if and only if $K^{2}_{X^{\prime}}\geq 5$ . Hence $X$ is a model of large degree if and only if $X^{\prime}$ is.

It remains to show (5.1) for each link of type IID. We write $a\leftarrow d\to b$ for a type IID link $X\leftarrow Y\to X^{\prime}$ between surfaces of degree $a$ , $d$ , $b$ . Since $c$ takes values in a torsion-free abelian group, it vanishes on involutions and in particular the Bertini and Geiser involutions; these are links with $d=1$ and $d=2$ respectively in the list of links in [Isk96, Theorem 2.6(ii)]. On the other hand, links with $d\geq 5$ are covered by Lemma 5.7(2).

Thus we only have to consider links with $d=3$ or $d=4$ .

Claim 5.8.

Let $X\xleftarrow{\phi}Y\xrightarrow{\phi^{\prime}}X^{\prime}$ be a link of type IID such that $K_{Y}^{2}\geq 3$ . Let $Z\subset X$ be the blowup center of $\phi$ and $D\subset X$ the divisor contracted by $\phi^{\prime}\circ\phi^{-1}$ . Then each irreducible component of $D_{\bar{k}}$ is a smooth rational curve of degree $\delta$ (according to Definition 4.6) containing exactly $\delta-1$ points of $Z_{\bar{k}}$ . For the type IID links with $K_{Y}^{2}\in\{3,4\}$ listed below, $\delta$ has the following description:

•

$9\leftarrow 4\to 5,9\leftarrow 3\rightarrow 9$ : $\delta=6$ (which are conics in $\mathbb{P}^{2}$ in the classical sense).
•

$8\leftarrow 4\rightarrow 8$ : $\delta=4$ .
•

$8\leftarrow 3\to 5$ : $\delta=6$ .
•

$6\leftarrow 4\to 6$ : $\delta=3$ .
•

$6\leftarrow 3\to 6$ : $\delta=4$ .
•

$5\leftarrow 4\to 9$ : $\delta=2$ .
•

$5\leftarrow 3\to 8$ : $\delta=3$ .

Proof.

Since $K_{Y}^{2}\geq 3$ , $Y_{\bar{k}}$ is obtained by blowing up $r\leq 6$ points on $\mathbb{P}^{2}$ with exceptional divisors $E_{1},\ldots,E_{r}$ . Let $H\in{\operatorname{NS}}(Y_{\bar{k}})$ be the pullback of the hyperplane class of $\mathbb{P}^{2}$ . By solving (4.5), the $(-1)$ -classes on $Y_{\bar{k}}$ are one of the following:

•

$E_{i}$ , with $i\in\{1,\ldots,r\}$ ;
•

$H-E_{i}-E_{j}$ , with $i,j\in\{1,\ldots,r\}$ and $i\neq j$ ;
•

(only when $r=6$ ) $-K_{Y}-H+E_{i}$ , with $i\in\{1,\ldots,6\}$ .

From the above description, any pair of $(-1)$ -curves $E$ and $E^{\prime}$ on $Y_{\bar{k}}$ satisfies $E\cdot E^{\prime}\leq 1$ . Since $X_{\bar{k}}$ is a simultaneous contraction of disjoint $(-1)$ -curves on $Y_{\bar{k}}$ , it follows that each irreducible component $C$ of $D_{\bar{k}}$ is smooth. It also follows that if $C$ has degree $\delta$ , then $C$ contains $\delta-1$ points of $Z_{\bar{k}}$ .

The value of $\delta$ follows from the matrix description in [Isk96, Theorem 2.6] of the action of each link in the classification on the Picard-Manin space. For instance for $6\leftarrow 3\to 6$ , the description in [Isk96, Theorem 2.6(ii), $K_{X}^{2}=6$ , (c)] implies that $D=-2K_{X}$ , which shows that $\delta=(-2K_{X}^{2})/3=4$ . The same argument works for other links. $\Box$

We first consider symmetric links where the centers of the blow up and the blow down are isomorphic, so that they are covered by Lemma 5.7(1):

•

$9\leftarrow 3\rightarrow 9$ : the first map blows up a Galois orbit of six points, and the second one contracts the Galois orbit of the proper preimages of six conics (in the classical sense: they have degree 6 according to Definition 4.6) passing through five of the points by Claim 5.8; these two Galois orbits are isomorphic.
•

$8\leftarrow 4\rightarrow 8$ : we blow up a Galois orbit of four general points on $X$ and contract the Galois orbit of the proper preimages of quartic curves passing through three of the four points by Claim 5.8; these two Galois orbits are isomorphic.

Finally we need to deal with the following links (we list them up to inverses):

•

$8\leftarrow 3\rightarrow 5$ : here $X$ is a quadric with $A_{X}=[Z_{2}]$ and $X^{\prime}$ is a degree five del Pezzo surface with $A_{X^{\prime}}=[Z_{5}^{\prime}]$ . We need to show that the first map has center $Z_{5}^{\prime}$ and the second map has center $Z_{2}$ . By Claim 5.8, the center of the second map parametrizes smooth rational curves of degree $6$ on $X$ passing through an orbit of five points. By Proposition 4.8(ii) and (iv) $(dP_{8})$ , the center of the second map is a subscheme isomorphic to $Z_{2}$ . The center of the first map parametrizes cubics passing through an orbit of two points on $X^{\prime}$ , and thus by Proposition 4.8(ii) and (iv) $(dP_{5})$ is a scheme isomorphic to $Z_{5}^{\prime}$ .
•

$9\leftarrow 4\rightarrow 5$ : it suffices to show that if $Z_{5}$ is the center of the first map and $A_{X^{\prime}}=[Z_{5}^{\prime}]$ , then $Z_{5}\simeq Z_{5}^{\prime}$ . One can verify this directly, as in the previous link, or apply Corollary 2.13 as in the proof of Lemma 5.7(2).
•

$6\leftarrow 3\rightarrow 6$ : we have $A_{X}=[Z_{3}]+[Z_{2}]-1$ and $A_{X^{\prime}}=[Z_{3}^{\prime}]+[Z_{2}^{\prime}]-1$ . We claim that $c(\phi)=[Z_{3}^{\prime}]-[Z_{3}]$ and $Z_{2}\simeq Z_{2}^{\prime}$ . Indeed, by Claim 5.8 the second arrow contracts the proper preimages of smooth rational curves of degree $4$ passing through the Galois orbit of the points blown up by the second arrow, and the latter scheme of conics is isomorphic to $Z_{3}$ by Proposition 4.8(ii) and (iv) $(dP_{6})$ . The same argument with roles of $X$ , $X^{\prime}$ reversed implies that the first arrow blows up $Z_{3}^{\prime}$ . Thus $c(\phi)=[Z_{3}^{\prime}]-[Z_{3}]$ . By Lemma 5.6, we have $\mu_{G_{k}}([Z_{2}])=\mu_{G_{k}}([Z^{\prime}_{2}])$ , hence $Z_{2}\simeq Z_{2}^{\prime}$ by Proposition 2.5.
•

$6\leftarrow 4\rightarrow 6$ : we have $Z_{3}\simeq Z_{3}^{\prime}$ , and $c(\phi)=[Z_{2}^{\prime}]-[Z_{2}]$ similarly to the previous case, using Proposition 4.8(ii), (iv) $(dP_{6})$ , and Proposition 2.5 again.

$\Box$

Proof of Theorem 3.5.

We first assume that $X$ and $Y$ are geometrically irreducible. Composing with contractions to minimal models, and using the additivity of $c$ under composition, we may assume that $X$ and $Y$ are minimal, hence belong to one of the classes from Theorem 4.1.

In the nongeometrically rational case and geometrically rational case with $K_{X}^{2}\leq 4$ the result follows from Proposition 4.4. Finally, in the geometrically rational case with $K_{X}^{2}\geq 5$ the result is Proposition 5.5.

In general, that is if the surfaces $X$ and $Y$ are not geometrically irreducible, write $L_{X}$ and $L_{Y}$ for the fields of regular functions of $X$ and $Y$ ; these are finite field extensions of $k$ . Then $\phi$ induces a $k$ -isomorphism $\sigma:L_{X}\to L_{Y}$ , which allows us to consider both $X$ and $Y$ as smooth projective geometrically irreducible surfaces over $L:=L_{X}$ and $\phi$ becomes a $L$ -birational isomorphism. This way $X$ and $Y$ are restrictions of scalars of geometrically irreducible surfaces over $L$ (this can be thought of as the Stein factorizations for $X$ and $Y$ over ${\operatorname{Spec}}(k)$ ), and the result follows from the geometrically irreducible case considered above and Proposition 3.4(ii). $\Box$

5.2. Rationality centers

The following corollary tells us that the rationality center of a rational surface $X$ is well-defined, that is for any sequence of blow ups and blow downs connecting $\mathbb{P}^{2}$ to $X$ , the virtual Galois set of blow up centers minus the blow down centers is independent of the choice of the birational isomorphism between them.

In the higher-dimensional case such rationality centers are not well-defined (however, see Definition 2.3 in [GS14] for a similar class in the localized Grothendieck ring).

Corollary 5.9 (of Theorem 3.5).

There exists a unique map

\{\text{Isomorphism classes of rational smooth projective surfaces}\}\overset{\mathcal{M}}{\longrightarrow}\mathbb{Z}[{\mathrm{Var}}^{0}/k]

with the following properties:

(1)

We have $\mathcal{M}(X)=A_{X}$ as in Definition 5.2 for models of large degree
(2)

For any birational isomorphism $\phi:\mathbb{P}^{2}\dashrightarrow X$ we have $c(\phi)=[\mathcal{M}(X)]-1$

Proof.

For any rational surface we define $\mathcal{M}(X)$ as $c(\phi)+1$ ; by Theorem 3.5 this is independent of the choice of $\phi:\mathbb{P}^{2}\dashrightarrow X$ . As $\mathcal{M}$ is required to satisfy $[\mathcal{M}(X)]=c(\phi)+1$ , this shows the uniqueness of $\mathcal{M}$ . By Proposition 5.5 this is consistent with Definition 5.2. $\Box$

Example 5.10.

If $X$ is a del Pezzo surface of degree $5$ , by Definition 5.2, $A_{X}=[Z_{5}]$ . Assume that $X$ has Picard rank one which by Lemma 5.3 is equivalent to $Z_{5}$ being irreducible. Then for any rationality construction of $X$ (see Example 3.8), that is a sequence of blow ups and blow downs starting with $\mathbb{P}^{2}$ and ending with $X$ , one of the blow ups will have $Z_{5}$ as its center.

Example 5.11.

Consider the rationality question for a del Pezzo surface $X$ of degree $d\leq 4$ . If $X$ is rational, then it is a result of Iskovskikh deduced from his classification of links [Isk96, Theorem 2.6] that such a surface $X$ can not be minimal, thus it admits a Galois orbit of disjoint $(-1)$ -curves which we can contract via some morphism $\phi:X\to X^{\prime}$ to obtain a minimal rational del Pezzo surface of degree $5$ , $6$ , $8$ or $9$ . In each case the rationality center $A_{X}$ is given in the table. As usual we write $Z_{j}$ or $Z_{j}^{\prime}$ for étale schemes of degree $j$ .

$\deg(X^{\prime})$	$c(\phi^{-1})$	$A_{X^{\prime}}$	$A_{X}$
$5$	$[Z_{5-d}]$	$[Z_{5}^{\prime}]$	$[Z_{5-d}]+[Z_{5}^{\prime}]$
$6$	$[Z_{6-d}]$	$[Z_{2}^{\prime}]+[Z_{3}^{\prime}]-1$	$[Z_{6-d}]+[Z_{3}^{\prime}]+[Z_{2}^{\prime}]-1$
$8$	$[Z_{8-d}]$	$[Z_{2}^{\prime}]$	$[Z_{8-d}]+[Z_{2}^{\prime}]$
$9$	$[Z_{9-d}]$	$1$	$[Z_{9-d}]+1$

Example 5.12.

Consider the two rational cubic surfaces $X$ , $X^{\prime}$ introduced in Example 2.14; they have isomorphic rational permutation Néron-Severi representations, and the associated Galois sets can not be read off from them.

However, the construction of $\mathcal{M}_{X}$ does determine these Galois sets from $X$ itself, as by Corollary 5.9 we have $[Z]=\mathcal{M}_{X}-1$ and $[Z^{\prime}]=\mathcal{M}_{X^{\prime}}-1$ .

References

[AB18] Asher Auel and Marcello Bernardara. Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields. Proc. Lond. Math. Soc. (3), 117(1):1–64, 2018.
[And04] Yves André. Une introduction aux motifs (motifs purs, motifs mixtes, périodes), volume 17 of Panoramas et Synthèses [Panoramas and Syntheses]. Société Mathématique de France, Paris, 2004.
[BdS02] Wieb Bosma and Bart de Smit. On arithmetically equivalent number fields of small degree. In Algorithmic number theory (Sydney, 2002), volume 2369 of Lecture Notes in Comput. Sci., pages 67–79. Springer, Berlin, 2002.
[Bit04] Franziska Bittner. The universal Euler characteristic for varieties of characteristic zero. Compos. Math., 140(4):1011–1032, 2004.
[Bor18] Lev A. Borisov. The class of the affine line is a zero divisor in the Grothendieck ring. J. Algebraic Geom., 27(2):203–209, 2018.
[BSS11] M. Blunk, S. J. Sierra, and S. Paul Smith. A derived equivalence for a degree 6 del Pezzo surface over an arbitrary field. J. K-Theory, 8(3):481–492, 2011.
[BSY22] Jérémy Blanc, Julia Schneider, and Egor Yasinsky. Birational maps of severi-brauer surfaces, with applications to cremona groups of higher rank, 2022.
[CLKT23] Antoine Chambert-Loir, Maxim Kontsevich, and Yuri Tschinkel. Burnside rings and volume forms with logarithmic poles, 2023.
[EH18] Karin Erdmann and Thorsten Holm. Algebras and representation theory. Springer Undergraduate Mathematics Series. Springer, Cham, 2018.
[FK88] Eberhard Freitag and Reinhardt Kiehl. Étale cohomology and the Weil conjecture, volume 13 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1988. Translated from the German by Betty S. Waterhouse and William C. Waterhouse, With an historical introduction by J. A. Dieudonné.
[G0̈1] Lothar Göttsche. On the motive of the Hilbert scheme of points on a surface. Math. Res. Lett., 8(5-6):613–627, 2001.
[Gil15] Stefan Gille. Permutation modules and Chow motives of geometrically rational surfaces. J. Algebra, 440:443–463, 2015.
[GS14] S. Galkin and E. Shinder. The Fano variety of lines and rationality problem for a cubic hypersurface. arXiv:math/arXiv:1405.5154, 2014.
[Has09] Brendan Hassett. Rational surfaces over nonclosed fields. In Arithmetic geometry, volume 8 of Clay Math. Proc., pages 155–209. Amer. Math. Soc., Providence, RI, 2009.
[Has16] Brendan Hassett. Cubic fourfolds, K3 surfaces, and rationality questions. In Rationality problems in algebraic geometry, volume 2172 of Lecture Notes in Math., pages 29–66. Springer, Cham, 2016.
[(ht] Dominik (https://mathoverflow.net/users/37059/dominik). Does the Grothendieck ring of varieties contain torsion? MathOverflow. URL:https://mathoverflow.net/q/225599 (version: 2015-12-08).
[IS96] V. A. Iskovskikh and I. R. Shafarevich. Algebraic surfaces [ MR1060325 (91f:14029)]. In Algebraic geometry, II, volume 35 of Encyclopaedia Math. Sci., pages 127–262. Springer, Berlin, 1996.
[Isk79] Vasily A. Iskovskikh. Minimal model of rational surfaces over arbitrary fields. Izv. Akad. Nauk SSSR Ser. Mat., Translation in Math. USSR Izvestija, 43(1):19–43, 1979.
[Isk96] V. A. Iskovskikh. Factorization of birational mappings of rational surfaces from the point of view of Mori theory. Uspekhi Mat. Nauk, 51(4(310)):3–72, 1996.
[KKS22] Joseph Karmazyn, Alexander Kuznetsov, and Evgeny Shinder. Derived categories of singular surfaces. J. Eur. Math. Soc. (JEMS), 24(2):461–526, 2022.
[Kol96] János Kollár. Rational curves on algebraic varieties, volume 32 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 1996.
[KS18] Alexander Kuznetsov and Evgeny Shinder. Grothendieck ring of varieties, D- and L-equivalence, and families of quadrics. Selecta Math. (N.S.), 24(4):3475–3500, 2018.
[KT22] Andrew Kresch and Yuri Tschinkel. Burnside groups and orbifold invariants of birational maps, 2022.
[Kuz19] Alexander Kuznetsov. Derived Categories of Families of Sextic del Pezzo Surfaces. International Mathematics Research Notices, 06 2019. rnz081.
[LL03] Michael Larsen and Valery A. Lunts. Motivic measures and stable birational geometry. Mosc. Math. J., 3(1):85–95, 259, 2003.
[LS21] Stéphane Lamy and Julia Schneider. Generating the plane cremona groups by involutions, 2021.
[LS22] Hsueh-Yung Lin and Evgeny Shinder. Motivic invariants of birational maps, 2022.
[Man86] Yu. I. Manin. Cubic forms, volume 4 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam, second edition, 1986. Algebra, geometry, arithmetic, Translated from the Russian by M. Hazewinkel.
[Mil80] James S. Milne. Étale cohomology, volume 33 of Princeton Mathematical Series. Princeton University Press, Princeton, N.J., 1980.
[NS09] Johannes Nicaise and Julien Sebag. A note on motivic integration in mixed characteristic, 2009.
[Par13] Ori Parzanchevski. On $G$ -sets and isospectrality. Ann. Inst. Fourier (Grenoble), 63(6):2307–2329, 2013.
[Per77] Robert Perlis. On the equation $\zeta_{K}(s)=\zeta_{K^{\prime}}(s)$ . J. Number Theory, 9(3):342–360, 1977.
[Pra17] Dipendra Prasad. A refined notion of arithmetically equivalent number fields, and curves with isomorphic Jacobians. Adv. Math., 312:198–208, 2017.
[R1̈1] Karl Rökaeus. The class of a torus in the Grothendieck ring of varieties. Amer. J. Math., 133(4):939–967, 2011.
[Sch22] Julia Schneider. Relations in the Cremona group over a perfect field. Ann. Inst. Fourier (Grenoble), 72(1):1–42, 2022.
[Sco93] Leonard L. Scott. Integral equivalence of permutation representations. In Group theory (Granville, OH, 1992), pages 262–274. World Sci. Publ., River Edge, NJ, 1993.
[Ser77] Jean-Pierre Serre. Linear representations of finite groups. Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott, Graduate Texts in Mathematics, Vol. 42.
[Sta18] The Stacks Project Authors. Stacks Project. http://stacks.math.columbia.edu, 2018.
[Sun85] Toshikazu Sunada. Riemannian coverings and isospectral manifolds. Ann. of Math. (2), 121(1):169–186, 1985.
[SZ20] Evgeny Shinder and Ziyu Zhang. L-equivalence for degree five elliptic curves, elliptic fibrations and K3 surfaces. Bull. Lond. Math. Soc., 52(2):395–409, 2020.
[Wei13] Charles A. Weibel. The $K$ -book, volume 145 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2013. An introduction to algebraic $K$ -theory.
[Xie21] Fei Xie. Derived categories of quintic del Pezzo fibrations. Selecta Math. (N.S.), 27(1):Paper No. 4, 32, 2021.
[Zak17] Inna Zakharevich. The annihilator of the Lefschetz motive. Duke Math. J., 166(11):1989–2022, 2017.
[Zim18] Susanna Zimmermann. The Abelianization of the real Cremona group. Duke Math. J., 167(2):211–267, 2018.