Exceptional collections on $\Sigma_{2}$

Akira Ishii Graduate School of Mathematics, Nagoya University, Furocho, Chikusa-ku, Nagoya, 464-8602, Japan [email protected] , Shinnosuke Okawa Department of Mathematics, Graduate School of Science, Osaka University
1-1, Machikaneyamacho, Toyonaka, Osaka 560-0043, Japan [email protected] and Hokuto Uehara Department of Mathematical Sciences, Graduate School of Science, Tokyo Metropolitan University, 1-1 Minamiohsawa, Hachioji-shi, Tokyo, 192-0397, Japan [email protected]

Abstract.

Structure theorems for exceptional objects and exceptional collections of the bounded derived category of coherent sheaves on del Pezzo surfaces are established by Kuleshov and Orlov in [MR1286839]. In this paper we propose conjectures which generalize these results to weak del Pezzo surfaces. Unlike del Pezzo surfaces, an exceptional object on a weak del Pezzo surface is not necessarily a shift of a sheaf and is not determined by its class in the Grothendieck group. Our conjectures explain how these complications are taken care of by spherical twists, the categorification of $(-2)$ -reflections acting on the derived category.

This paper is devoted to solving the conjectures for the prototypical weak del Pezzo surface $\Sigma_{2}$ , the Hirzebruch surface of degree $2$ . Specifically, we prove the following results: Any exceptional object is sent to the shift of the uniquely determined exceptional vector bundle by a product of spherical twists which acts trivially on the Grothendieck group of the derived category. Any exceptional collection on $\Sigma_{2}$ is part of a full exceptional collection. We moreover prove that the braid group on 4 strands acts transitively on the set of exceptional collections of length $4$ (up to shifts).

1. Introduction

Semiorthogonal decomposition is among the most fundamental notions of triangulated categories. The finest ones, i.e., semiorthogonal decompositions whose components are equivalent to the bounded derived category of a point, are identified with (full) exceptional collections of the triangulated category. If there is an exceptional collection in a triangulated category, then virtually always there are infinitely many of them because of the group action explained below. Hence the classification of exceptional collections is not obvious at all.

Let $X$ be a smooth projective variety and $\mathbf{D}(X)$ be the bounded derived category of coherent sheaves on $X$ . Concerning the classification of the exceptional collections of $\mathbf{D}(X)$ , the prototypical results are given in [Gorodentsev_1987, Section 5] for $X=\mathbb{P}^{2}$ . The similar results for $X=\mathbb{P}^{1}\times\mathbb{P}^{1}$ are given in [MR966985]. Based on these works, Kuleshov and Orlov established in [MR1286839] the similar results for arbitrary del Pezzo surfaces (see also [MR2108443, Corollary 4.3.2, Theorem 4.3.3, Theorem 4.6.1]). They are summarized as follows.

Theorem 1.1 ([MR1286839]).

Let $X$ be a del Pezzo surface over an algebraically closed field $\mathbf{k}$ .

(1)

Any exceptional object $\mathcal{E}\in\mathbf{D}(X)$ is either a vector bundle or a line bundle on a $(-1)$ -curve, up to shifts.
(2)

The isomorphism class of an exceptional object $\mathcal{E}\in\mathbf{D}(X)$ is determined by its class $[\mathcal{E}]\in\operatorname{K_{0}}\left(X\right)$ , up to shifts by $2\mathbb{Z}$ .
(3)

Any exceptional collection of $\mathbf{D}(X)$ can be extended to a full exceptional collection.
(4)

Conjecture 1.2 below is true for $X$ ; i.e., the action $G_{r}\curvearrowright\mathsf{EC}_{r}(X)$ is transitive.

The symbol $\operatorname{K_{0}}\left(X\right)$ denotes the Grothendieck group of the triangulated category $\mathbf{D}(X)$ . We briefly explain (4). Let $\mathsf{EC}_{r}(X)$ denote the set of isomorphism classes of exceptional collections of length $r$ of $\mathbf{D}(X)$ , where $r=\operatorname{rank}\operatorname{K_{0}}\left(X\right)$ . An important fact relevant to the classification is that there is a standard action of the group $G_{r}\coloneqq\mathbb{Z}^{r}\rtimes\operatorname{Br}_{r}$ on $\mathsf{EC}_{r}(X)$ by shifts and mutations, where $\operatorname{Br}_{r}$ is the braid group on $r$ strands (see Section LABEL:sc:Deformation_and_mutation_of_exceptional_collections for details).

Inspired by the earlier works mentioned above, Bondal and Polishchuk gave the following conjecture.

Conjecture 1.2 ( $=$ [MR1230966, Conjecture 2.2]).

Suppose that $X$ is a smooth projective variety such that $\mathbf{D}(X)$ admits a full exceptional collection of length $r$ . Then the action $G_{r}\curvearrowright\mathsf{EC}_{r}(\mathbf{D}(X))$ is transitive.

In dimensions greater than $2$ , the classification of exceptional collections is widely open. Even for $\mathbb{P}^{3}$ , only partial results seem to be known ([1995alg.geom..7014P]) and Conjecture 1.2 is still open.

The aim of this paper is to investigate the generalization of Theorem 1.1 to weak del Pezzo surfaces; i.e., smooth projective surfaces $X$ whose anti-canonical bundle $\omega_{X}^{-1}$ is nef and big. As it turns out, the generalization is not straightforward at all.

A weak del Pezzo surface $X$ which is not a del Pezzo surface admits at least one (and only finitely many) $(-2)$ -curve(s); i.e., a smooth curve $C\subset X$ with $C^{2}=-2$ and $C\simeq\mathbb{P}^{1}$ . Line bundles on $C$ , as objects of $\mathbf{D}(X)$ , are $2$ -spherical objects and hence yield non-trivial autoequivalences of $\mathbf{D}(X)$ called spherical twists due to Seidel and Thomas [Seidel-Thomas]. For a $2$ -spherical object $\alpha\in\mathbf{D}(X)$ , the corresponding spherical twist $T_{\alpha}$ acts as a reflection on $\operatorname{K_{0}}\left(X\right)$ whereas it always satisfies $T_{\alpha}^{2}\not\simeq\operatorname{id}_{\mathbf{D}(X)}$ . For example, the exceptional object $T_{\mathcal{O}_{C}}^{2}(\mathcal{O}_{X})$ has the same class as $\mathcal{O}_{X}$ in $\operatorname{K_{0}}\left(X\right)$ but is not isomorphic to $\mathcal{O}_{X}$ . In fact, by direct computation one can confirm the following.

\displaystyle\mathcal{H}^{i}\left(T_{\mathcal{O}_{C}}^{2}(\mathcal{O}_{X})\right)\simeq\begin{cases}\mathcal{O}_{X}&i=0\\ \mathcal{O}_{C}&i=1,2\\ 0&\text{otherwise}\end{cases}

(1.1)

Moreover, by applying $T_{\mathcal{O}_{C}}^{2}$ repeatedly, we obtain a collection of infinitely many exceptional objects of unbounded cohomological amplitudes which share the same class in $\operatorname{K_{0}}\left(X\right)$ . Hence Theorem 1.1 (1) (2) are not true for $X$ at all.

Conjecture 1.3, which is the main conjecture of this paper, generalizes Theorem 1.1 to weak del Pezzo surfaces while taking into account all the complications mentioned in the previous paragraph. In a word, it asserts that the failure of Theorem 1.1 (1) (2) on weak del Pezzo surfaces is remedied by spherical twists and that Theorem 1.1 (3) (4) should hold for weak del Pezzo surfaces too. The main theorem of this paper is that Conjecture 1.3 is true for $\Sigma_{2}=\mathbb{P}_{\mathbb{P}^{1}}(\mathcal{O}\oplus\mathcal{O}(2))$ ; namely, the Hirzebruch surface of degree $2$ , a paradigm of weak del Pezzo surfaces. In fact we formulated Conjecture 1.3 by generalizing the results we obtained for $\Sigma_{2}$ , Theorem 1.1, and [MR3758518, Theorem 1.2] simultaneously.

Conjecture 1.3.

Let $X$ be a weak del Pezzo surface over an algebraically closed field $\mathbf{k}$ .

(1)

For any exceptional object $\mathcal{E}\in\mathbf{D}(X)$ , there exists a sheaf $\mathcal{F}$ on $X$ which is either an exceptional vector bundle or a line bundle on a $(-1)$ -curve, a sequence of line bundles on $(-2)$ -curves $\mathcal{L}_{1},\dots,\mathcal{L}_{n}$ , and an integer $m\in\mathbb{Z}$ such that

$\displaystyle\mathcal{E}\simeq(T_{\mathcal{L}_{n}}\circ\cdots\circ T_{\mathcal{L}_{1}})(\mathcal{F})[m].$ (1.2)
(2)

For any pair of exceptional objects $\mathcal{E},\mathcal{E}^{\prime}\in\mathbf{D}(X)$ such that $[\mathcal{E}]=[\mathcal{E}^{\prime}]\in\operatorname{K_{0}}\left(X\right)$ , there are a product $b$ of spherical twists and inverse spherical twists which acts trivially on $\operatorname{K_{0}}\left(X\right)$ and $m\in 2\mathbb{Z}$ such that $\mathcal{E}^{\prime}\simeq b(\mathcal{E})[m]$ . Moreover, for each exceptional object $\mathcal{E}\in\mathbf{D}(X)$ , there is a unique exceptional vector bundle $\mathcal{F}$ such that either $[\mathcal{E}]=[\mathcal{F}]$ or $[\mathcal{E}]=-[\mathcal{F}]$ holds.
(3)

Any exceptional collection on $X$ can be extended to a full exceptional collection.
(4)

Conjecture 1.2 is true for $X$ .

As it is more or less visible, (1), (2), (3), and (4) of Conjecture 1.3 generalizes (1), (2), (3), and (4) of Theorem 1.1, respectively.

To avoid repetition, let us simply state the main result of this paper as follows.

Theorem 1.4 (MAIN THEOREM).

Conjecture 1.3 is true for $X=\Sigma_{2}$ .

More specifically, for $X=\Sigma_{2}$

•

Conjecture 1.3 (1) is solved affirmatively in Section LABEL:sc:Twisting_exceptional_objects_down_to_exceptional_vector_bundles as Theorem LABEL:th:exceptional_objects_are_equivalent_to_vector_bundles.
•

Conjecture 1.3 (2) is solved affirmatively in Section LABEL:sc:Exceptional_objects_sharing_the_same_class as Corollary LABEL:cr:exceptional_objects_in_the_same_numerical_class.
•

Conjecture 1.3 (3) is solved affirmatively in Section LABEL:sc:Constructibility_of_exceptional_collections as Corollary LABEL:cr:constructibility.
•

Conjecture 1.3 (4) is solved affirmatively in Section LABEL:sc:Braid_group_acts_transitively_on_the_set_of_full_exceptional_collections as Theorem LABEL:th:transitivity.

Remark 1.5.

We give some comments on the preceding works which are related to Conjecture 1.3 and Theorem 1.4, and one important consequence of Conjecture 1.3 (4) on the fullness of exceptional collections of maximal length.

•

This paper is a continuation of [MR3431636] by the 2nd and the 3rd authors and almost completely supersedes it. Conjecture 1.3 (1) for $\Sigma_{2}$ is stated as [MR3431636, Conjecture 1.3], and is solved for exceptional sheaves in [MR3431636, Theorem 1.4].
•

Conjecture 1.3 (1) for torsion exceptional sheaves is partially solved in [MR3758518, Theorem 1.2]. A weaker version of Conjecture 1.3 (1) is stated as [MR3758518, Conjecture 1.1].
•

Exceptional objects and exceptional collections of vector bundles on weak del Pezzo surfaces is systematically studied in [MR1604186]. In fact we use some results of this work in this paper.
•

It follows from the definition of mutations that the triangulated subcategory generated by an exceptional collection is invariant under mutation. In particular, the fullness of an exceptional collection is preserved by mutations. On the other hand, any $X$ as in Conjecture 1.3 admits a full exceptional collection. Hence Conjecture 1.3 (4) would imply that any exceptional collection of length equal to $\operatorname{rank}\operatorname{K_{0}}\left(X\right)=\operatorname{rank}\operatorname{Pic}(X)+2$ of $\mathbf{D}(X)$ is full, though it would also follow from Conjecture 1.3 (3).

1.1. Summary of each section and structure of the paper

Section 2 is a preliminary section. We recall the rudiments of mutations and the group $B$ of autoequivalences generated by spherical twists from [Ishii-Uehara_ADC]. Among others, we prove in Corollary LABEL:cr:btriv_is_generated_by_squares that $B^{K_{0}-\mathsf{triv}}$ , the subgroup of $B$ acting trivially on $\operatorname{K_{0}}\left(\Sigma_{2}\right)$ , is generated by squares of spherical twists by line bundles on $C$ .

From Section LABEL:sc:Twisting_exceptional_objects_down_to_exceptional_vector_bundles till the end of the paper, we restrict ourselves to the proof of Theorem 1.4 and in particular discuss the case of $\Sigma_{2}$ only.

Section LABEL:sc:Twisting_exceptional_objects_down_to_exceptional_vector_bundles is the main component of the paper and devoted to the proof of Theorem LABEL:th:exceptional_objects_are_equivalent_to_vector_bundles. Namely, in this section, we prove that for each exceptional object $\mathcal{E}\in\mathbf{D}(\Sigma_{2})$ there is a sequence of integers $a_{1},\dots,a_{n}$ such that $\left(T_{a_{n}}\circ\cdots\circ T_{a_{1}}\right)(\mathcal{E})$ is isomorphic to a shift of a vector bundle. In accordance with the steps of the proof, Section LABEL:sc:Twisting_exceptional_objects_down_to_exceptional_vector_bundles is divided into six subsections.

In Section LABEL:sc:First_properties we prove some properties of the cohomology sheaf $\mathcal{H}^{\bullet}(\mathcal{E})=\bigoplus_{i\in\mathbb{Z}}\mathcal{H}^{i}(\mathcal{E})$ . Among others we show that there is the index $i_{0}\in\mathbb{Z}$ such that $\operatorname{Supp}\mathcal{H}^{i_{0}}(\mathcal{E})=\Sigma_{2}$ and for any $i\neq i_{0}$ the cohomology sheaf $\mathcal{H}^{i}(\mathcal{E})$ , if not $0$ , is a pure sheaf whose reduced support is $C$ . Then in Section LABEL:sc:Properties_of_the_schematic_support, we prove that actually the schematic support of $\mathcal{H}^{i}(\mathcal{E})$ for $i\neq i_{0}$ is $C$ . This is the most technical part of the paper, and actually this had been the main obstacle for the whole work. Fortunately one can use the result of this subsection as a black box to read the rest of the paper.

In Section LABEL:sc:More_on_the_structure we prove that there is a decomposition $\mathcal{H}^{i_{0}}(\mathcal{E})\simeq T\oplus E$ , where $E=E(\mathcal{E})$ is an exceptional sheaf and $T$ is a torsion sheaf. We moreover show that if $\operatorname{tors}E\neq 0$ , then $T$ is a direct sum of copies of $\mathcal{O}_{C}(a)$ for some $a\in\mathbb{Z}$ and that $\operatorname{tors}\mathcal{H}^{\bullet}(\mathcal{E})$ is a direct sum of copies of $\mathcal{O}_{C}(a)$ and $\mathcal{O}_{C}(a+1)$ . This integer $a=a(\mathcal{E})$ plays a central role throughout the paper.

In Section LABEL:sc:Derived_dual_of_exceptional_objects we investigate the relationship between the cohomology sheaves of $\mathcal{E}$ and those of $\mathcal{E}^{\vee}$ , the derived dual of $\mathcal{E}$ . We in particular show in Corollary LABEL:cr:E(cE_vee)_has_non-trivial_torsion that if $\mathcal{E}$ is not isomorphic to a shift of a vector bundle and $\operatorname{tors}E(\mathcal{E})=0$ , then $\operatorname{tors}E(\mathcal{E}^{\vee})\neq 0$ . In this paper we mainly discuss the case $\operatorname{tors}E(\mathcal{E})\neq 0$ , and by this result we can settle the case where $\operatorname{tors}E(\mathcal{E})=0$ by passing to $\mathcal{E}^{\vee}$ .

In Section LABEL:sc:Length_of_the_torsion_part we introduce the notion of the length of the “torsion part” of an object at the generic point $\gamma$ of $C$ . Formally speaking, for $\mathcal{E}$ it is defined as $\ell(\mathcal{E})=\sum_{i\in\mathbb{Z}}\operatorname{length}_{\mathcal{O}_{\Sigma_{2},\gamma}}\operatorname{tors}\mathcal{H}^{i}(\mathcal{E})_{\gamma}$ . It follows that $\mathcal{E}$ is isomorphic to a shift of a vector bundle if and only if $\ell(\mathcal{E})=0$ , and hence it suffices to show that $\ell(T_{c}(\mathcal{E}))<\ell(\mathcal{E})$ for some $c=c(\mathcal{E})\in\mathbb{Z}$ if $\ell(\mathcal{E})>0$ . This is exactly what we achieve in Section LABEL:sc:Proof_of_Theorem_3.1. We show in Theorem LABEL:th:decrease_the_length_by_appropriate_twist that $c=a(\mathcal{E})$ works if $\operatorname{tors}E(\mathcal{E})\neq 0$ and otherwise $c=-a(\mathcal{E}^{\vee})-3$ does.

Spherical objects on the minimal resolution of type $A$ singularity is classified in [Ishii-Uehara_ADC]. More specifically, the proof of Theorem LABEL:th:exceptional_objects_are_equivalent_to_vector_bundles is an adaptation of the proof of [Ishii-Uehara_ADC, Proposition 5.1]. It is, however, much more involved than that of [Ishii-Uehara_ADC, Proposition 5.1]. This is due to the fact that the support of an exceptional object on $\Sigma_{2}$ is never concentrated in the $(-2)$ -curve $C$ . On the contrary the reduced support of a spherical object on $\Sigma_{2}$ is concentrated in $C$ , and it immediately implies that the schematic supports of the cohomology sheaves of the spherical object coincide with $C$ .

Section LABEL:sc:Exceptional_objects_sharing_the_same_class is devoted to the proof of Theorem LABEL:th:exceptional_object_is_K0-trivial_equivalent_to_vector_bundle. It is almost immediately obtained by combining Theorem LABEL:th:exceptional_objects_are_equivalent_to_vector_bundles with a small trick on squares of spherical twists (Proposition LABEL:pr:TaTb_as_product_of_O_(m_C_)_and_squares_of_twists) and the fact that an exceptional vector bundle is uniquely determined by its class in $\operatorname{K_{0}}\left(\Sigma_{2}\right)$ (Lemma LABEL:lm:exceptional_vb_is_determined_by_K0).

Section LABEL:sc:Constructibility_of_exceptional_collections is devoted to the proof of Corollary LABEL:cr:constructibility. Take an exceptional collection $\underline{\mathcal{E}}$ . We first show in Theorem LABEL:th:twisting_exceptional_collection_to_vector_bundles that there is a product of spherical twists $b$ such that $b(\underline{\mathcal{E}})$ consists of vector bundles up to shifts. This is achieved in a one-by-one manner. The key is that if $(\mathcal{B},\mathcal{E})$ is an exceptional pair such that $\mathcal{B}$ is a vector bundle and $\ell(\mathcal{E})>0$ , then, surprisingly enough, $T_{c(\mathcal{E})}(\mathcal{B})$ remains to be a vector bundle (though it may not be isomorphic to $\mathcal{B}$ ). Recall that $c(\mathcal{E})\in\mathbb{Z}$ depends only on $\mathcal{E}$ and that $T_{c(\mathcal{E})}$ strictly decreases the length of $\mathcal{E}$ .

Corollary LABEL:cr:constructibility is known for exceptional collections of vector bundles by a slight generalization Theorem LABEL:th:Constructibility_for_bundle_collections of a result by Kuleshov in [MR1604186]. Applying it to $b(\underline{\mathcal{E}})$ , we immediately obtain the proof of Corollary LABEL:cr:constructibility for $\underline{\mathcal{E}}$ .

Section LABEL:sc:Braid_group_acts_transitively_on_the_set_of_full_exceptional_collections is devoted to the proof of Theorem LABEL:th:transitivity. We use the deformation of $\Sigma_{2}$ to $\mathbb{P}^{1}\times\mathbb{P}^{1}$ . The difficulty is that there are infinitely many exceptional objects on $\Sigma_{2}$ which deform to the same exceptional object on $\mathbb{P}^{1}\times\mathbb{P}^{1}$ (non-uniqueness of the specialization, which is translated into the non-separatedness of the moduli space of semiorthogonal decompositions introduced in [2020arXiv200203303B]). Actually, two exceptional objects have the same deformation to $\mathbb{P}^{1}\times\mathbb{P}^{1}$ if and only if they have the same class in $\operatorname{K_{0}}\left(\Sigma_{2}\right)$ (up to shifts by $2\mathbb{Z}$ ).

In Step LABEL:st:reduction_to_numerically_standard_collection of the proof, we use the fact that the corresponding result is already known by Theorem 1.1 (4) for the del Pezzo surface $\mathbb{P}^{1}\times\mathbb{P}^{1}$ . Since deformation of exceptional collections commutes with mutations, the result for $\mathbb{P}^{1}\times\mathbb{P}^{1}$ immediately implies that for any exceptional collection of length $4$ on $\Sigma_{2}$ there is a sequence of mutations which brings it to a collection which is numerically equivalent to the standard collection $\mathcal{E}^{\mathsf{std}}$ (i.e., having the same classes as $\mathcal{E}^{\mathsf{std}}$ in $\operatorname{K_{0}}\left(\Sigma_{2}\right)$ . See (LABEL:eq:standard_collection) for the definition of $\mathcal{E}^{\mathsf{std}}$ ).

It remains to show that an exceptional collection $\underline{\mathcal{E}}$ on $\Sigma_{2}$ which is numerically equivalent to $\mathcal{E}^{\mathsf{std}}$ can be sent to $\mathcal{E}^{\mathsf{std}}$ by mutations. In Step LABEL:st:from_numerically_standard_to_standard, as an intermediate step, we find $b\in B$ such that $\underline{\mathcal{E}}=b(\mathcal{E}^{\mathsf{std}})$ . We construct such $b$ again in one-by-one manner.

In Step LABEL:st:replace_b_with_mutations we prove that $b$ can be replaced by a sequence of mutations. Thanks to the fact that mutations commute with autoequivalences, it is enough to show the assertion only for $b\in\{T_{0},T_{-1}\}$ . Recall that $T_{0},T_{-1}$ generate $B$ . At this point the problem is concrete enough to be settled by hand.

1.2. Some words on future directions

Though Conjecture 1.3 is stated for arbitrary weak del Pezzo surfaces, in this paper we restrict ourselves to the study of the case of $\Sigma_{2}$ . This is partly because the proof is rather involved already in this case. We nevertheless think that the case of $\Sigma_{2}$ should serve as a paradigm for the further investigations of Conjecture 1.3.

The proof of Theorem 1.4 is based on the classification of rigid sheaves on the $(-2)$ -curve; i.e., the fundamental cycle of the minimal resolution of the $A_{1}$ -singularity. So far such classification is achieved only for the minimal resolution of type $A$ singularities by [Ishii-Uehara_ADC]. If one wants to push the strategy of this paper, it seems inevitable to establish the similar classification for the minimal resolution of type $D$ and type $E$ singularities. See [MR3983387] for results in this direction.

A weak del Pezzo surface over an algebraically closed field $\mathbf{k}$ is isomorphic to either $\mathbb{P}^{1}\times\mathbb{P}^{1},\Sigma_{2}$ , or a blowup of $\mathbb{P}^{2}$ in at most eight points in almost general positions (see, say, [MR2964027, Theorem 8.1.15, Corollary 8.1.24]). Hence our strategy based on the deformation to del Pezzo surfaces, in principle, is applicable to all weak del Pezzo surfaces.

Structure theorems for exceptional collections on $\mathbb{P}^{2}$ play an important role in showing the contractibility of (the main component of) the space of Bridgeland stability conditions $\operatorname{Stab}(\mathbb{P}^{2})$ in [MR3703470]. Our results should be similarly useful for studying $\operatorname{Stab}(\Sigma_{2})$ . More specifically, they should be very closely related to the relationship between $\operatorname{Stab}(\Sigma_{2})$ and $\operatorname{Stab}(\mathbb{P}^{1}\times\mathbb{P}^{1})$ ; see Remark LABEL:rm:stability_conditions for details.

1.3. Notation and convention

We work over an algebraically closed field $\mathbf{k}$ , unless otherwise stated. To ease notation, we will write $\operatorname{ext}^{i}=\dim_{\mathbf{k}}\operatorname{Ext}^{i},h^{i}=\dim_{\mathbf{k}}H^{i},e^{p,q}_{2}=\dim_{\mathbf{k}}E_{2}^{p,q},$ and so on. Below is a list of frequently used symbols.

$\Sigma_{d}$	the Hirzebruch surface of degree $d$ (2.1)
$C$	the $(-2)$ -curve of $\Sigma_{2}$
$f$	the divisor class of a fiber of the morphism $\Sigma_{2}\to\mathbb{P}^{1}$ (2.4)
$\ast^{\vee}$	the derived dual of $\ast\in\mathbf{D}(\Sigma_{2})$ (2.6)
$T_{a}\ (\text{resp. }T^{\prime}_{a})\in\operatorname{Auteq}(\Sigma_{2})$	the (inverse) spherical twist by $\mathcal{O}_{C}(a)$ (2.15)
$B<\operatorname{Auteq}(\Sigma_{2})$	the group of autoequivalences generated by spherical twists (Definition LABEL:df:subgroup_of_spherical_twists)
$\mathsf{EC}_{N},\mathsf{ECVB}_{N},\mathsf{FEC},\mathsf{FECVB}$	various sets of exceptional collections (Definition LABEL:df:sets_of_exceptional_collections)
$\operatorname{Br}_{N}$ (resp. $G_{N}$ )	the braid group on $N$ strands (LABEL:eq:braid_group) (resp. the extension of $\operatorname{Br}_{N}$ by $\mathbb{Z}^{N}$ (LABEL:equation:GN))
$\operatorname{\mathsf{gen}}$	the generalization map for exceptional collections from the central fiber to the generic fiber (LABEL:eq:generalization_map)
$\mathsf{numEC}_{N},\mathsf{numFEC}$	various sets of numerical exceptional collections (Definition LABEL:df:numerical_exceptional_collection)
$\mathcal{E}^{\mathsf{std}}\in\mathsf{FECVB}(\Sigma_{2})$	the standard full exceptional collection of $\mathbf{D}(\Sigma_{2})$ (Definition LABEL:df:the_standard_exceptional_collection_on_Sigma2)
$\mathcal{E}^{\mathsf{std}}_{\xi}\in\mathsf{FECVB}(\mathcal{X}_{\operatorname{\mathsf{gen}}})$	the standard full exceptional collection of $\mathbf{D}(\mathcal{X}_{\operatorname{\mathsf{gen}}})$ (LABEL:eq:standard_collection_generic)
$\mathcal{H}^{\bullet}(\mathcal{E})\ (\text{resp. }\mathcal{H}^{i}(\mathcal{E}))$	the total (resp. $i$ -th) cohomology of $\mathcal{E}\in\mathbf{D}(X)$ with respect to the standard t-structure (LABEL:eq:cohomology_object)
$i_{0}=i_{0}(\mathcal{E})$	the unique index such that $\operatorname{Supp}\mathcal{H}^{i_{0}}(\mathcal{E})=\Sigma_{2}$ (Definition LABEL:df:i0)
$(R,\mathfrak{m})$	the complete local ring of the $A_{1}$ -singularity (Notation LABEL:nt:A1_singularity)
$(0:I)_{M}\subseteq M$	the maximal submodule of $M$ annihilated by the ideal $I\subseteq R$ (LABEL:eq:annihilator)
$\mathcal{I}_{C}\subset\mathcal{O}_{\Sigma_{2}}$	the ideal sheaf of $C\subset\Sigma_{2}$ (LABEL:equation:cIC)
$D(\ast)$	the dual $\mathbf{k}$ -vector space of $\ast$ (LABEL:equation:k-dual)
$\mathcal{H}^{i_{0}}(\mathcal{E})\simeq E(\mathcal{E})\oplus T(\mathcal{E})$	the canonical decomposition into an exceptional sheaf and a torsion sheaf (Lemma LABEL:lm:exceptional_sheaf_is_a_direct_summand)
$\mathcal{T}(\mathcal{E}),\ \mathcal{F}(\mathcal{E})$	the torsion (resp. the torsion free) part of $\mathcal{H}^{i_{0}}(\mathcal{E})$ (Definition LABEL:df:E_and_cF)
$a,s,t\in\mathbb{Z}$	integers specified by the irreducible decomposition of $\mathcal{T}$ (LABEL:eq:decomposition_of_cT) (see also Lemma LABEL:lm:exceptional_sheaf_is_a_direct_summand)
$\ell(\ast)$	“length of the torsion part of $\ast$ ” at the generic point $\gamma$ of $C$ (Definition LABEL:df:length)
$b,r,s\in\mathbb{Z}$	integers specified by the irreducible decomposition of an exceptional vector bundle restricted to $C$ (Lemma LABEL:lm:exceptional_vector_bundle_restricted_to_C)

Acknowledgements

During the preparation of this paper, A.I. was partially supported by JSPS Grants-in-Aid for Scientific Research (19K03444). S.O. was partially supported by JSPS Grants-in-Aid for Scientific Research (16H05994, 16H02141, 16H06337, 18H01120, 20H01797, 20H01794). H.U. was partially supported by JSPS Grants-in-Aid for Scientific Research (18K03249).

2. Preliminaries

2.1. The Hirzebruch surfaces

The Hirzebruch surface of degree $d\in\mathbb{Z}_{\geq 0}$ is the ruled surface

\displaystyle p\colon\Sigma_{d}\coloneqq\mathbb{P}_{\mathbb{P}^{1}}\left(\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(d)\right)\to\mathbb{P}^{1}.

(2.1)

The history of Hirzebruch surfaces goes back, at least, to the first paper by Hirzebruch [MR45384]. An explicit isotrivial degeneration of $\Sigma_{d}$ to $\Sigma_{d^{\prime}}$ for $d>d^{\prime}$ and $d-d^{\prime}\in 2\mathbb{Z}$ is constructed in [MR153033, p.86 Example]. As a special case, there exists a smooth projective morphism (defined over $\operatorname{Spec}\mathbb{Z}$ )

\displaystyle\mathcal{X}\to\mathbb{A}^{1}_{t}

(2.2)

such that the fiber over $t=0$ is isomorphic to $\Sigma_{2}$ and the restriction of the family over the open subscheme $\mathbb{G}_{m}\hookrightarrow\mathbb{A}^{1}$ is isomorphic to the trivial family $\left(\mathbb{P}^{1}\times\mathbb{P}^{1}\right)\times\mathbb{G}_{m}\stackrel{{\scriptstyle\mathrm{pr}_{2}}}{{\to}}\mathbb{G}_{m}$ .

In this paper we investigate the bounded derived category of coherent sheaves on $\Sigma_{2}$ , which is the most basic example of weak del Pezzo surfaces. We let $C\subset\Sigma_{2}$ denote the unique negative curve, and $f$ the (linear equivalence class of) the fiber of $p$ . Recall that

\displaystyle\operatorname{Pic}\Sigma_{2}=\mathbb{Z}C\oplus\mathbb{Z}f,

(2.3)

where

\displaystyle C^{2}=-2,f^{2}=0,C.f=1.

(2.4)

The anti-canonical bundle is given by

\displaystyle-K_{\Sigma_{2}}=2C+4f.

(2.5)

2.2. Derived category, spherical twist, and the autoequivalence group of $\Sigma_{2}$

Definition 2.1.

For a quasi-compact scheme $Y$ , we let $\operatorname{Perf}Y$ denote the perfect derived category of $Y$ with the standard structure of a triangulated category. When $Y$ is equipped with a morphism to $\operatorname{Spec}\mathbf{k}$ , we think of $\operatorname{Perf}Y$ as a triangulated $\mathbf{k}$ -linear category. It comes with the natural symmetric monoidal structure given by the tensor product over $\mathcal{O}_{Y}$ , but we do not take it into account unless otherwise stated.

When $Y$ is a smooth and projective variety over a field $\mathbf{k}$ , we identify $\operatorname{Perf}Y$ with the bounded derived category $\mathbf{D}(Y)$ of coherent sheaves on $Y$ .

The following equivalence of tensor triangulated categories

\displaystyle{}^{\vee}\colon\left(\operatorname{Perf}Y\right)^{\mathrm{op}}\xrightarrow{\sim}\operatorname{Perf}Y;\quad\mathcal{E}\mapsto\mathcal{E}^{\vee}\coloneqq\mathbb{R}\mathop{{\mathcal{H}}om}\nolimits_{Y}(\mathcal{E},\mathcal{O}_{Y})

(2.6)

will be called the derived dual.

One can easily verify that there exits a canonical natural isomorphism

\displaystyle\operatorname{id}\stackrel{{\scriptstyle\sim}}{{\Rightarrow}}{}^{\vee\vee}.

(2.7)

Definition 2.2.

For smooth projective varieties $X,Y$ over $\operatorname{Spec}\mathbf{k}$ and an object $K\in\mathbf{D}(X\times_{\mathbf{k}}Y)$ , the integral transform by the kernel $K$ will be denoted and defined as follows.

\displaystyle\Phi_{K}\coloneqq\Phi_{K}^{X\to Y}\colon\mathbf{D}(X)\to\mathbf{D}(Y);\quad E\mapsto\mathbb{R}p_{Y\ast}\left(p_{X}^{\ast}E\otimes_{X\times_{\mathbf{k}}Y}^{\mathbb{L}}K\right)

(2.8)

Let $X$ be a smooth projective variety over a field $\mathbf{k}$ . Recall that an object $\alpha\in\mathbf{D}(X)$ is spherical if $\alpha\otimes_{\mathcal{O}_{X}}\omega_{X}\simeq\alpha$ and $\mathop{\mathbb{R}\mathrm{Hom}}\nolimits_{X}(\alpha,\alpha)\simeq\mathbf{k}\oplus\mathbf{k}[-\dim X]$ . The spherical twist by $\alpha$ is the endofunctor

\displaystyle T_{\alpha}\coloneqq\Phi_{K_{\alpha}}

(2.9)

defined by the kernel $K_{\alpha}\coloneqq\operatorname{cone}\left(\alpha^{\vee}\boxtimes\alpha\xrightarrow{\operatorname{ev}}\mathcal{O}_{\Delta_{X}}\right)$ .

Consider the exchange automorphism

\displaystyle\operatorname{\mathsf{swap}}\colon X\times X\to X\times X;\,(x,y)\mapsto(y,x).

(2.10)

Recall that

\displaystyle\left(\operatorname{\mathsf{swap}}^{*}K_{\alpha}\right)^{\vee}\otimes p_{1}^{\ast}\omega_{X}[\dim X]

(2.11)

is called the right adjoint kernel of $K_{\alpha}$ and it enjoys the following adjoint property (see, say, [MR2244106, Definition 5.7]).

\displaystyle T_{\alpha}=\Phi_{K_{\alpha}}\dashv\Phi_{\left(\operatorname{\mathsf{swap}}^{*}K_{\alpha}\right)^{\vee}\otimes p_{1}^{\ast}\omega_{X}[\dim X]}\eqqcolon T^{\prime}_{\alpha}

(2.12)

It follows that $T_{\alpha}$ is an autoequivalence, so that $T^{\prime}_{\alpha}\simeq T_{\alpha}^{-1}$ . Note that there exists the obvious isomorphism

\displaystyle\operatorname{\mathsf{swap}}^{\ast}K_{\alpha}\simeq K_{\alpha^{\vee}},

(2.13)

so that

\displaystyle T^{\prime}_{\alpha^{\vee}}\simeq\Phi_{K_{\alpha}^{\vee}\otimes p_{1}^{\ast}\omega_{X}[\dim X]}.

(2.14)

A typical example of a spherical object on $\Sigma_{2}$ is $\mathcal{O}_{C}(a)$ for $a\in\mathbb{Z}$ . The corresponding (inverse) spherical twist will be denoted as follows, for short.

\displaystyle T_{a}\coloneqq T_{\mathcal{O}_{C}(a)},\quad T^{\prime}_{a}\coloneqq T_{\mathcal{O}_{C}(a)}^{-1}

(2.15)

By definition, for each spherical object $\alpha$ and any object $E\in\mathbf{D}(X)$ , there are standard triangles as follows. To ease notation, we simply let $\otimes_{\mathcal{O}_{X}}$ denote the derived tensor product. The morphisms $\varepsilon$ and $\eta$ are the evaluation and the coevaluation maps respectively, both of which are obtained from the standard adjoint pair of functors $-\otimes_{\mathcal{O}_{X}}\ast\dashv\mathop{\mathbb{R}\mathrm{Hom}}\nolimits_{X}(\ast,-)$ .

	$\displaystyle\mathop{\mathbb{R}\mathrm{Hom}}\nolimits_{X}\left(\alpha,E\right)\otimes_{\mathbf{k}}\alpha\xrightarrow{\varepsilon}E\to T_{\alpha}(E)\xrightarrow{+1}$		(2.16)
	$\displaystyle T^{\prime}_{\alpha}(E)\to E\xrightarrow{\eta}\mathop{\mathbb{R}\mathrm{Hom}}\nolimits_{X}\left(E,\alpha\right)^{\vee}\otimes_{\mathbf{k}}\alpha\xrightarrow{+1}$		(2.17)

Lemma 2.3 ([Ishii-Uehara_ADC, Lemma 4.14]).

For each $\Phi\in\operatorname{Auteq}(X)$ , there is an isomorphism of autoequivalences as follows.

\displaystyle\Phi\circ T_{\alpha}\circ\Phi^{-1}\simeq T_{\Phi(\alpha)}

(2.18)

In particular, for any $a,m\in\mathbb{Z}$ it holds that

(2.19)

Exceptional collections on Σ2\Sigma_{2}