# Exceptional collections on $\Sigma_{2}$

Akira Ishii  Graduate School of Mathematics, Nagoya University, Furocho,
Chikusa-ku, Nagoya, 464-8602, Japan<EMAIL_ADDRESS>,
Shinnosuke Okawa  Department of Mathematics, Graduate School of Science, Osaka
University
1-1, Machikaneyamacho, Toyonaka, Osaka 560-0043, Japan
<EMAIL_ADDRESS>and  Hokuto Uehara  Department of Mathematical
Sciences, Graduate School of Science, Tokyo Metropolitan University, 1-1
Minamiohsawa, Hachioji-shi, Tokyo, 192-0397, Japan<EMAIL_ADDRESS>
###### 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 [KO94]. 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).

###### Contents

  1. 1 Introduction
    1. 1.1 Summary of each section and structure of the paper
    2. 1.2 Some words on future directions
    3. 1.3 Notation and convention
  2. 2 Preliminaries
    1. 2.1 The Hirzebruch surfaces
    2. 2.2 Derived category, spherical twist, and the autoequivalence group of $\Sigma_{2}$
    3. 2.3 Deformation and mutation of exceptional collections
    4. 2.4 Results obtained via deformation of $\Sigma_{2}$ to $\mathbb{P}^{1}\times\mathbb{P}^{1}$
  3. 3 Twisting exceptional objects down to exceptional vector bundles
    1. 3.1 First properties of $\mathcal{H}^{\bullet}(\mathcal{E})$
    2. 3.2 Properties of the schematic support of $\operatorname{tors}\mathcal{H}^{\bullet}(\mathcal{E})$
    3. 3.3 More on the structure of $\mathcal{H}^{\bullet}(\mathcal{E})$
    4. 3.4 Derived dual of exceptional objects
    5. 3.5 Length of the torsion part
    6. 3.6 Proof of Theorem 3.1
  4. 4 Exceptional objects sharing the same class in $\operatorname{K_{0}}\left(\Sigma_{2}\right)$
  5. 5 Constructibility of exceptional collections
  6. 6 Braid group acts transitively on the set of full exceptional collections

##  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 [GR87, Section 5] for $X=\mathbb{P}^{2}$. The similar results for
$X=\mathbb{P}^{1}\times\mathbb{P}^{1}$ are given in [Rud88]. Based on these
works, Kuleshov and Orlov established in [KO94] the similar results for
arbitrary del Pezzo surfaces (see also [GK04, Corollary 4.3.2, Theorem 4.3.3,
Theorem 4.6.1]). They are summarized as follows.

######  Theorem 1.1 ([KO94]).

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

  1. (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. (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. (3)

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

  4. (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 2.3 for details).

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

######  Conjecture 1.2 ($=$[BP93, 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 ([Pos95]) 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 [ST01]. 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 [CJ18, Theorem 1.2]
simultaneously.

######  Conjecture 1.3.

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

  1. (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. (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. (3)

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

  4. (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 3 as Theorem 3.1.

  * •

Conjecture 1.3 (2) is solved affirmatively in Section 4 as Corollary 4.4.

  * •

Conjecture 1.3 (3) is solved affirmatively in Section 5 as Corollary 5.6.

  * •

Conjecture 1.3 (4) is solved affirmatively in Section 6 as Theorem 6.1.

######  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 [OU15] by the 2nd and the 3rd authors and
almost completely supersedes it. Conjecture 1.3 (1) for $\Sigma_{2}$ is stated
as [OU15, Conjecture 1.3], and is solved for exceptional _sheaves_ in [OU15,
Theorem 1.4].

  * •

Conjecture 1.3 (1) for torsion exceptional sheaves is partially solved in
[CJ18, Theorem 1.2]. A weaker version of Conjecture 1.3 (1) is stated as
[CJ18, Conjecture 1.1].

  * •

Exceptional objects and exceptional collections of vector bundles on weak del
Pezzo surfaces is systematically studied in [Kul97]. 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 [IU05].
Among others, we prove in Corollary 2.11 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 3 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 3 is the main component of the paper and devoted to the proof of
Theorem 3.1. 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 3 is divided into six
subsections.

In Section 3.1 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 3.2, 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 3.3 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 3.4 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 3.27 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 3.5 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 3.6. We show in Theorem 3.32 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 [IU05]. More specifically, the proof of Theorem 3.1 is an
adaptation of the proof of [IU05, Proposition 5.1]. It is, however, much more
involved than that of [IU05, 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 4 is devoted to the proof of Theorem 4.3. It is almost immediately
obtained by combining Theorem 3.1 with a small trick on squares of spherical
twists (Proposition 2.8) and the fact that an exceptional _vector bundle_ is
uniquely determined by its class in
$\operatorname{K_{0}}\left(\Sigma_{2}\right)$ (Lemma 2.33).

Section 5 is devoted to the proof of Corollary 5.6. Take an exceptional
collection $\underline{\mathcal{E}}$. We first show in Theorem 5.4 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 5.6 is known for exceptional collections of vector bundles by a
slight generalization Theorem 5.5 of a result by Kuleshov in [Kul97]. Applying
it to $b(\underline{\mathcal{E}})$, we immediately obtain the proof of
Corollary 5.6 for $\underline{\mathcal{E}}$.

Section 6 is devoted to the proof of Theorem 6.1. 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 [BOR20]). 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 1 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 (2.78) 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 2, 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 3 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 [IU05]. 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 [Kaw19] 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, [Dol12, 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 [Li17]. 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 5.11
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 2.10)
$\mathsf{EC}_{N},\mathsf{ECVB}_{N},\mathsf{FEC},\mathsf{FECVB}$ |  various sets of exceptional collections (Definition 2.14)
$\operatorname{Br}_{N}$ (resp. $G_{N}$) |  the braid group on $N$ strands (2.55) (resp. the extension of $\operatorname{Br}_{N}$ by $\mathbb{Z}^{N}$ (2.60))
$\operatorname{\mathsf{gen}}$ |  the generalization map for exceptional collections from the central fiber to the generic fiber (2.65)
$\mathsf{numEC}_{N},\mathsf{numFEC}$ |  various sets of numerical exceptional collections (Definition 2.34)
$\mathcal{E}^{\mathsf{std}}\in\mathsf{FECVB}(\Sigma_{2})$ |  the standard full exceptional collection of $\mathbf{D}(\Sigma_{2})$ (Definition 2.36)
$\mathcal{E}^{\mathsf{std}}_{\xi}\in\mathsf{FECVB}(\mathcal{X}_{\operatorname{\mathsf{gen}}})$ |  the standard full exceptional collection of $\mathbf{D}(\mathcal{X}_{\operatorname{\mathsf{gen}}})$ (2.79)
$\mathcal{H}^{\bullet}(\mathcal{E})\ (\text{resp. }\mathcal{H}^{i}(\mathcal{E}))$ |  the total (resp. $i$-th) cohomology of $\mathcal{E}\in\mathbf{D}(X)$ with respect to the standard t-structure (3.3)
$i_{0}=i_{0}(\mathcal{E})$ |  the unique index such that $\operatorname{Supp}\mathcal{H}^{i_{0}}(\mathcal{E})=\Sigma_{2}$ (Definition 3.7)
$(R,\mathfrak{m})$ |  the complete local ring of the $A_{1}$-singularity (Notation 3.10)
$(0:I)_{M}\subseteq M$ |  the maximal submodule of $M$ annihilated by the ideal $I\subseteq R$ (3.28)
$\mathcal{I}_{C}\subset\mathcal{O}_{\Sigma_{2}}$ |  the ideal sheaf of $C\subset\Sigma_{2}$ (3.30)
$D(\ast)$ |  the dual $\mathbf{k}$-vector space of $\ast$ (3.36)
$\mathcal{H}^{i_{0}}(\mathcal{E})\simeq E(\mathcal{E})\oplus T(\mathcal{E})$ |  the canonical decomposition into an exceptional sheaf and a torsion sheaf (Lemma 3.21)
$\mathcal{T}(\mathcal{E}),\ \mathcal{F}(\mathcal{E})$ |  the torsion (resp. the torsion free) part of $\mathcal{H}^{i_{0}}(\mathcal{E})$ (Definition 3.22)
$a,s,t\in\mathbb{Z}$ |  integers specified by the irreducible decomposition of $\mathcal{T}$ (3.52) (see also Lemma 3.21)
$\ell(\ast)$ |  “length of the torsion part of $\ast$” at the generic point $\gamma$ of $C$ (Definition 3.28)
$b,r,s\in\mathbb{Z}$ |  integers specified by the irreducible decomposition of an exceptional vector bundle restricted to $C$ (Lemma 3.20)

### 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 [Hir51]. 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 [Kod63, 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, [Huy06, 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 ([IU05, 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

$\displaystyle\left(\mathcal{O}_{\Sigma_{2}}(mC)\otimes_{\mathcal{O}_{\Sigma_{2}}}-\right)\circ
T_{a}\simeq
T_{a-2m}\circ\left(\mathcal{O}_{\Sigma_{2}}(mC)\otimes_{\mathcal{O}_{\Sigma_{2}}}-\right).$
(2.19)

###### Proof.

Since $X$ is a smooth projective variety over $\mathbf{k}$, any
autoequivalence $\Psi$ of $\mathbf{D}(X)$ is a Fourier-Mukai transform; i.e.,
it is isomorphic to the integral transform by an appropriate kernel by [Orl97,
Theorem 2.2]. It then follows that $\Psi$ is isomorphic to
$\operatorname{id}_{\mathbf{D}(X)}$ if and only if $\Psi(E)\simeq E$ holds for
any $E\in\mathbf{D}(X)$. With this in mind, one can confirm the assertion by
using (2.16). ∎

The operation of taking duals is related to (inverse) spherical twists nicely.

######  Lemma 2.4.

  1. (1)

For any spherical object $\alpha\in\mathbf{D}(X)$, there exists a natural
isomorphism of functors

$\displaystyle\left(T_{\alpha}-\right)^{\vee}\simeq
T^{\prime}_{\alpha^{\vee}}\left(-^{\vee}\right).$ (2.20)

  2. (2)

When $X=\Sigma_{2}$, for any $a\in\mathbb{Z}$, there exists a natural
isomorphism of functors

$\displaystyle\left(T_{a}-\right)^{\vee}\simeq
T^{\prime}_{-2-a}\left(-^{\vee}\right).$ (2.21)

###### Proof.

There is a natural isomorphism

$\displaystyle\left(T_{\alpha}-\right)^{\vee}=\left(\Phi_{K_{\alpha}}(-)\right)^{\vee}$
$\displaystyle=\mathop{\mathbb{R}\mathcal{H}om}\nolimits_{X}\left(\mathbb{R}p_{2\ast}(p_{1}^{\ast}-\otimes^{\mathbb{L}}_{X\times_{k}X}K_{\alpha}),\mathcal{O}_{X}\right)$
(2.22)
$\displaystyle\simeq\mathop{\mathbb{R}\mathcal{H}om}\nolimits_{X}\left(\mathbb{R}p_{2\ast}\left(-\boxtimes\omega_{X}\otimes^{\mathbb{L}}_{X\times_{k}X}K_{\alpha}[\dim
X]\right),\omega_{X}[\dim X]\right)$ (2.23)
$\displaystyle\stackrel{{\scriptstyle\ast}}{{\simeq}}\mathbb{R}p_{2\ast}\mathop{\mathbb{R}\mathcal{H}om}\nolimits_{X\times_{\mathbf{k}}X}\left(-\boxtimes\omega_{X}\otimes^{\mathbb{L}}_{X\times_{k}X}K_{\alpha}[\dim
X],\omega_{X}\boxtimes\omega_{X}[2\dim X]\right)$ (2.24)
$\displaystyle\simeq\mathbb{R}p_{2\ast}\left(p_{1}^{\ast}-^{\vee}\otimes_{X\times_{\mathbf{k}}X}\left(K_{\alpha}^{\vee}\otimes
p_{1}^{\ast}\omega_{X}[\dim X]\right)\right)$ (2.25)
$\displaystyle\simeq\Phi_{K_{\alpha}^{\vee}\otimes
p_{1}^{\ast}\omega_{\Sigma_{2}}[\dim X]}(-^{\vee}),$ (2.26)

where the isomorphism $\stackrel{{\scriptstyle\ast}}{{\simeq}}$ follows from
[Har66, Chapter VII, Corollary 4.3 b)].

Comparing this with (2.14), we obtain (2.20).

The second item follows from (2.20) and the isomorphism
$\mathcal{O}_{C}(a)^{\vee}[1]\simeq\mathcal{O}_{C}(-2-a)$. ∎

######  Lemma 2.5 ([IU05, Lemma 4.15 (i) (2)]).

There is an isomorphism of autoequivalences

$\displaystyle T_{a}\circ
T_{a+1}\simeq\mathcal{O}_{\Sigma_{2}}(C)\otimes_{\mathcal{O}_{\Sigma_{2}}}-,$
(2.27)

which implies

$\displaystyle\begin{aligned} T_{a}^{\prime}\simeq
T_{a+1}\circ\mathcal{O}_{\Sigma_{2}}(-C)\otimes_{\mathcal{O}_{\Sigma_{2}}}-,\\\
{T_{a}^{\prime}}^{2}\simeq T_{a+1}\circ
T_{a+3}\circ\mathcal{O}_{\Sigma_{2}}(-2C)\otimes_{\mathcal{O}_{\Sigma_{2}}}-.\end{aligned}$
(2.28)

######  Definition 2.6.

For a triangulated category $\mathbf{D}$, we define the group
$\operatorname{Auteq}^{K_{0}-\mathsf{triv}}(\mathbf{D})$ of $K_{0}$-trivial
autoequivalences by the following exact sequence.

$\displaystyle
1\to\operatorname{Auteq}^{K_{0}-\mathsf{triv}}(\mathbf{D})\to\operatorname{Auteq}(\mathbf{D})\to\operatorname{Aut}\left(\operatorname{K_{0}}\left(\mathbf{D}\right)\right)$
(2.29)

The following well-known lemma, which follows from (2.16) and (2.17), asserts
that spherical twists are categorification of root reflections.

######  Lemma 2.7.

For any spherical object $\alpha\in\mathbf{D}(\Sigma_{2})$, it holds that

$\displaystyle
T_{\alpha}^{2},{T^{\prime}_{\alpha}}^{2}\in\operatorname{Auteq}^{K_{0}-\mathsf{triv}}(\mathbf{D}(\Sigma_{2})).$
(2.30)

######  Proposition 2.8.

For any $a,b\in\mathbb{Z}$, there exists a sequence
$a_{1},\dots,a_{\ell}\in\mathbb{Z}$ and $m\in\mathbb{Z}$ such that

$\displaystyle
T_{a}T_{b}=\mathcal{O}_{\Sigma_{2}}(mC)\otimes_{\mathcal{O}_{\Sigma_{2}}}-\circ
T_{a_{\ell}}^{\pm 2}\cdots T_{a_{1}}^{\pm 2}.$ (2.31)

###### Proof.

By (2.19), we may and will reduce the proof to that of the following claim.

######  Claim 2.9.

$T_{a}T_{b}$ is contained in the subgroup of
$\operatorname{Auteq}(\mathbf{D}(\Sigma_{2}))$ generated by
$\mathcal{O}_{\Sigma_{2}}(C)\otimes_{\mathcal{O}_{\Sigma_{2}}}-$ and
$T_{a}^{2}$ for all $a\in\mathbb{Z}$.

In the rest, we prove this claim. Consider first the case $a\leq b$. We induct
on $b-a\geq 0$. If it is $0$, we have nothing to show. In general, we have

$\displaystyle
T_{a}T_{b}\simeq\left(T_{a}T_{a+1}\right)\left({T^{\prime}_{a+1}}^{2}\right)\left(T_{a+1}T_{b}\right).$
(2.32)

By (2.27), the first term of the right hand side is isomorphic to
$\mathcal{O}_{\Sigma_{2}}(C)\otimes_{\mathcal{O}_{\Sigma_{2}}}-$. By the
induction hypothesis, the 3rd term is also contained in the subgroup mentioned
in the claim. Hence we are done with this case.

Now suppose that $a>b$. Then

$\displaystyle T_{a}T_{b}\simeq
T_{a}^{2}\left(T_{b}T_{a}\right)^{\prime}T_{b}^{2},$ (2.33)

and we know that the middle term is contained in the subgroup as shown in the
previous paragraph. ∎

######  Definition 2.10.

The subgroup of $\operatorname{Auteq}(\mathbf{D}(\Sigma_{2}))$ generated by
spherical twists will be denoted by $B$.

######  Corollary 2.11.

Fix any $a_{0}\in\mathbb{Z}$. Then elements of $B$ are exhausted by those of
the following form, where $a_{1},\dots,a_{n},m\in\mathbb{Z}$.

$\displaystyle\left(\mathcal{O}_{\Sigma_{2}}(mC)\otimes_{\mathcal{O}_{\Sigma_{2}}}-\right)\circ\left(T_{a_{n}}^{\pm
2}\circ\cdots\circ T_{a_{1}}^{\pm 2}\right)$ (2.34)
$\displaystyle\left(\mathcal{O}_{\Sigma_{2}}(mC)\otimes_{\mathcal{O}_{\Sigma_{2}}}-\right)\circ
T_{a_{0}}\circ\left(T_{a_{n}}^{\pm 2}\circ\cdots\circ T_{a_{1}}^{\pm
2}\right)$ (2.35)

Moreover, the normal subgroup

$\displaystyle B^{K_{0}-\mathsf{triv}}\coloneqq
B\cap\operatorname{Auteq}^{K_{0}-\mathsf{triv}}(\mathbf{D}(X))\triangleleft B$
(2.36)

is generated by $\\{T_{a}^{2}\mid a\in\mathbb{Z}\\}\subset
B^{K_{0}-\mathsf{triv}}$.

###### Proof.

The first assertion immediately follows from (2.28), (2.19), and Proposition
2.8.

For the second assertion, take $b\in B^{K_{0}-\mathsf{triv}}$. If $b$ is as in
(2.34), then obviously $m=0$. Also one can verify that $b$ is never like
(2.35), say, by using that it must preserve the classes
$[\mathcal{O}_{\Sigma_{2}}((a_{0}+1)f)]$ and
$[\mathcal{O}_{\Sigma_{2}}((a_{0}+2)f)]$. ∎

Lemma 2.3 implies that $B$ is a normal subgroup of
$\operatorname{Auteq}(\mathbf{D}(\Sigma_{2}))$. As we explain next, sets of
generators of $B$ are well understood.

######  Theorem 2.12.

For any $a\in\mathbb{Z}$, the group $B$ is generated by the two spherical
twists $T_{a},T_{a+1}$.

###### Proof.

The case $a=-2$ is explicitly mentioned in [IU05, Lemma 4.6]. The general case
inductively follows from this by the isomorphism of functors

$\displaystyle
T_{a-1}T_{a}\simeq\mathcal{O}_{\Sigma_{2}}(C)\otimes_{\mathcal{O}_{\Sigma_{2}}}-\simeq
T_{a}T_{a+1},$ (2.37)

which is (2.27). ∎

###  2.3. Deformation and mutation of exceptional collections

Let

$\displaystyle f\colon\mathcal{X}\to B$ (2.38)

be a smooth projective morphism of Noetherian schemes with a closed point
$0\in B$, and let

$\displaystyle X_{0}=\mathcal{X}\times_{f,B}\\{0\\}$ (2.39)

be the central fiber. Note that the properness and smoothness of $f$ implies
it is a perfect morphism, in the sense that the derived pushforward $f_{\ast}$
respects perfect complexes ([LN07, Proposition 2.1]).

######  Definition 2.13.

  1. (1)

An object $\mathcal{E}\in\operatorname{Perf}(\mathcal{X})$ is $f$-exceptional
if
$\mathbb{R}f_{\ast}\mathbb{R}\mathop{{\mathcal{H}}om}\nolimits(\mathcal{E},\mathcal{E})$
is a line bundle on $B$.

  2. (2)

A collection of $f$-exceptional objects
$\mathcal{E}_{1},\dots,\mathcal{E}_{N}\in\operatorname{Perf}\mathcal{X}$ is an
$f$-exceptional collection if
$\mathbb{R}f_{\ast}\mathbb{R}\mathop{{\mathcal{H}}om}\nolimits(\mathcal{E}_{j},\mathcal{E}_{i})=0$
for any $(i,j)$ with $1\leq i<j\leq N$.

  3. (3)

An $f$-exceptional collection as above is said to be strong if moreover
$\mathbb{R}f_{\ast}\mathbb{R}\mathop{{\mathcal{H}}om}\nolimits(\mathcal{E}_{j},\mathcal{E}_{i})$
is isomorphic to a locally free sheaf (regarded as a complex concentrated in
degree $0$) for any $(i,j)$.

  4. (4)

An $f$-exceptional collection as above is said to be full if the minimal
$B$-linear (i.e., closed under $-\otimes f^{\ast}\mathcal{F}$ by any
$\mathcal{F}\in\operatorname{Perf}B$) triangulated subcategory which contains
all of the objects in the collection is equivalent to
$\operatorname{Perf}\mathcal{X}$.

######  Definition 2.14.

We say that two $f$-exceptional collections
$\mathcal{E}_{1},\dots,\mathcal{E}_{N}$ and
$\mathcal{E}^{\prime}_{1},\dots,\mathcal{E}^{\prime}_{N}$ are _isomorphic_ if
$\mathcal{E}_{i}\simeq\mathcal{E}^{\prime}_{i}$ for each $i=1,\dots,N$. We
will let $\mathsf{EC}_{N}(f)$ (or $\mathsf{EC}_{N}(\mathcal{X})$, if $f$ is
obvious from the context) denote the set of isomorphism classes of
$f$-exceptional collections of length $N$. The set of isomorphism classes of
$f$-exceptional collections of length $N$ consisting entirely of locally free
sheaves will be denoted by $\mathsf{ECVB}_{N}(f)$ (or
$\mathsf{ECVB}(\mathcal{X})$), which comes with the obvious injection
$\mathsf{ECVB}_{N}(f)\hookrightarrow\mathsf{EC}_{N}(f)$. Similarly, the set of
equivalence classes of full $f$-exceptional collections will be denoted by
either $\mathsf{FEC}(f)$ or $\mathsf{FEC}(\mathcal{X})$, and the set of
isomorphism classes of full $f$-exceptional collections consisting entirely of
locally free sheaves will be denoted by $\mathsf{FECVB}(f)$ (or
$\mathsf{FECVB}(\mathcal{X})$), which comes with the obvious injection
$\mathsf{FECVB}(f)\hookrightarrow\mathsf{FEC}(f)$.

######  Lemma 2.15.

  1. (1)

Let $\mathcal{E}\in\operatorname{Perf}\mathcal{X}$ be an $f$-exceptional
object. Then the functor

$\displaystyle\Phi_{\mathcal{E}}\colon\operatorname{Perf}B\to\operatorname{Perf}\mathcal{X};\quad
F\mapsto f^{\ast}F\otimes\mathcal{E}$ (2.40)

is fully faithful and admits a right adjoint as follows.

$\displaystyle\Phi_{\mathcal{E}}\dashv\phi_{\mathcal{E}}^{R}\coloneqq
f_{\ast}\left(-\otimes\mathcal{E}^{\vee}\right)$ (2.41)

  2. (2)

Let
$\left(\mathcal{E}_{1},\dots,\mathcal{E}_{N}\right)\in\mathsf{EC}_{N}(\mathcal{X})$
be an $f$-exceptional collection. Then the smallest $B$-linear triangulated
subcategory of $\operatorname{Perf}\mathcal{X}$ which contains
$\mathcal{E}_{1},\dots,\mathcal{E}_{N}$ admits a $B$-linear semiorthogonal
decomposition
$\langle\Phi_{\mathcal{E}_{1}}(\operatorname{Perf}B),\dots\Phi_{\mathcal{E}_{N}}(\operatorname{Perf}B)\rangle$.

We say that a semiorthogonal decomposition
$\operatorname{Perf}\mathcal{X}=\langle\mathcal{A}_{1},\dots,\mathcal{A}_{N}\rangle$
is _$B$ -linear_ if $\mathcal{A}_{i}\otimes f^{\ast}b\subseteq\mathcal{A}_{i}$
holds for any $i=1,\dots,N$ and $b\in\operatorname{Perf}B$ (see [Kuz11,
Section 2.3]).

We will freely use the following very useful base change theorem from [Bon06,
Corollary 2.1.4]. See also [Kuz06, Section 2.4] and [Sta16, Tag 08IB] for
treatise from different points of view.

######  Lemma 2.16.

Consider the following Cartesian square of finite dimensional noetherian
schemes, where $f,g$ are perfect.

${Y}$${\mathcal{X}}$${C}$${B}$$\scriptstyle{g}$$\scriptstyle{\psi}$$\scriptstyle{f}$$\scriptstyle{\varphi}$
(2.42)

Then the standard natural transformation of functors

$\displaystyle\varphi^{\ast}\circ f_{\ast}\Rightarrow
g_{\ast}\circ\psi^{\ast}\colon\operatorname{Perf}\mathcal{X}\to\operatorname{Perf}C$
(2.43)

is an isomorphism if either $f$ or $\varphi$ is flat.

######  Corollary 2.17.

Consider a morphism of schemes as in (2.38) and a (strong) $f$-exceptional
collection
$\left(\mathcal{E}_{1},\dots,\mathcal{E}_{N}\right)\in\mathsf{EC}_{N}(\mathcal{X})$.
Take any morphism $\varphi\colon C\to B$, where $C$ is a finite dimensional
noetherian scheme, and consider the Cartesian diagram as in Lemma 2.16. Then

$\displaystyle\left(\psi^{\ast}\mathcal{E}_{1},\dots,\psi^{\ast}\mathcal{E}_{N}\right)$
(2.44)

is a (strong) $g$-exceptional collection on $Y$.

###### Proof.

As $f$ is flat, the assertion immediately follows from the following
computation. All functors are derived.

$\displaystyle
g_{\ast}\mathop{{\mathcal{H}}om}\nolimits_{Y}\left(\psi^{\ast}\mathcal{E}_{i},\psi^{\ast}\mathcal{E}_{j}\right)\simeq
g_{\ast}\psi^{\ast}\mathop{{\mathcal{H}}om}\nolimits_{\mathcal{X}}\left(\mathcal{E}_{i},\mathcal{E}_{j}\right)\stackrel{{\scriptstyle\text{Lemma~{}\ref{lm:base
change}}}}{{\simeq}}\varphi^{\ast}f_{\ast}\mathop{{\mathcal{H}}om}\nolimits_{\mathcal{X}}\left(\mathcal{E}_{i},\mathcal{E}_{j}\right)$
(2.45)

∎

######  Lemma 2.18.

Suppose that $B=\operatorname{Spec}R$ for a complete local Noetherian ring
$(R,\mathfrak{m},\mathbf{k})$. Then the natural restriction maps

$\displaystyle\mathsf{EC}_{N}(f)\to\mathsf{EC}_{N}(X_{0}),$ (2.46)
$\displaystyle\mathsf{ECVB}_{N}(f)\to\mathsf{ECVB}_{N}(X_{0})$ (2.47)

obtained in Corollary 2.17 are bijections for any $N$.

###### Proof.

Let us first show that any exceptional object
$\mathcal{E}\in\operatorname{Perf}X_{0}$ deforms to an object
$\mathcal{E}_{R}\in\operatorname{Perf}\mathcal{X}$. By definition of
exceptional object, we know that
$\operatorname{Ext}^{1}_{X_{0}}(\mathcal{E},\mathcal{E})=\operatorname{Ext}^{2}_{X_{0}}(\mathcal{E},\mathcal{E})=0$.
By the deformation theory of objects (see, say, [HT10, Corollary 3.4]), for
each $n\geq 1$ one finds the unique lift in
$\operatorname{Perf}(\mathcal{X}\otimes_{R}R/\mathfrak{m}^{n+1})$ of $E$. Then
it algebrizes uniquely to an actual object
$\mathcal{E}_{R}\in\operatorname{Perf}(\mathcal{X})$ by [Lie06, Proposition
3.6.1].

We next show that $\mathcal{E}_{R}$ is an $f$-exceptional object. This is
equivalent to the assertion $\mathcal{C}=0$, where
$\mathcal{C}\in\operatorname{Perf}B$ is defined as the cone of the following
standard morphism.

$\displaystyle\mathcal{O}_{B}\to\mathbb{R}f_{\ast}\mathbb{R}\mathop{{\mathcal{H}}om}\nolimits_{\mathcal{X}}(\mathcal{E}_{R},\mathcal{E}_{R})$
(2.48)

Consider the following Cartesian diagram.

${X_{0}}$${\mathcal{X}}$${\operatorname{Spec}\mathbf{k}}$${\operatorname{Spec}R=B}$$\scriptstyle{\iota}$$\scriptstyle{f_{0}}$${\lrcorner}$$\scriptstyle{f}$$\scriptstyle{0}$
(2.49)

By Lemma 2.16 it follows that

$\displaystyle\mathbb{L}0^{\ast}\mathbb{R}f_{\ast}\mathop{\mathbb{R}\mathcal{H}om}\nolimits_{\mathcal{X}}(\mathcal{E}_{R},\mathcal{E}_{R})\simeq\mathop{\mathbb{R}\mathrm{Hom}}\nolimits_{X_{0}}(\mathcal{E},\mathcal{E})=\mathbf{k}\operatorname{id}_{\mathcal{E}}[0],$
(2.50)

so that $\mathbb{L}0^{\ast}\mathcal{C}=0$. By Nakayama’s lemma, this implies
that $\mathcal{C}=0$. The semiorthogonality of the collection
$\mathcal{E}_{1,R},\dots,\mathcal{E}_{N,R}$ is shown by similar arguments. ∎

######  Remark 2.19.

One can similarly show that the deformation of a strong collection is also
strong. This follows from the fact that if $P\in\operatorname{Perf}B$
satisfies $0^{\ast}P\simeq\mathbf{k}^{\oplus r}[0]$ for some $r\geq 0$, then
$P\simeq R^{\oplus r}[0]$. Also, it follows from Lemma 2.15 (2) that the
deformation of a full exceptional collection is again full.

######  Definition 2.20.

Let $f$ be a morphism as in (2.38). An ($f$-)exceptional pair is an
($f$-)exceptional collection of length 2. For an $f$-exceptional pair
$\mathcal{E},\mathcal{F}$, the left mutation $L_{\mathcal{E}}\mathcal{F}$ of
$\mathcal{F}$ through $\mathcal{E}$ and the right mutation
$R_{\mathcal{F}}\mathcal{E}$ of $\mathcal{E}$ through $\mathcal{F}$ are
defined by the following distinguished triangles.

$\displaystyle
f^{\ast}\mathbb{R}f_{\ast}\mathop{\mathbb{R}\mathcal{H}om}\nolimits_{\mathcal{X}}(\mathcal{E},\mathcal{F})\otimes_{\mathcal{O}_{\mathcal{X}}}\mathcal{E}\xrightarrow{\varepsilon}\mathcal{F}\to
L_{\mathcal{E}}\mathcal{F},$ (2.51) $\displaystyle
R_{\mathcal{F}}\mathcal{E}\to\mathcal{E}\xrightarrow{\eta}f^{\ast}\mathbb{R}f_{\ast}\mathop{\mathbb{R}\mathcal{H}om}\nolimits_{\mathcal{X}}(\mathcal{E},\mathcal{F})^{\vee}\otimes_{\mathcal{O}_{\mathcal{X}}}\mathcal{F}$
(2.52)

######  Remark 2.21.

The definition of mutations given above differs by shifts from the one in
[Bon89, Section 2], but is slightly simpler in that for an orthogonal
exceptional pair, the mutations just exchange the two objects without any
shift.

By the base change theorem Lemma 2.16, one can easily verify that mutations
commute with base change.

######  Lemma 2.22.

Under the notation and the assumptions of Lemma 2.16, suppose that $f$ is flat
and hence the natural transformation (2.43) is an isomorphism. For any
$f$-exceptional pair $\left(\mathcal{E},\mathcal{F}\right)$, it follows that
$\left(\psi^{\ast}\mathcal{E},\psi^{\ast}\mathcal{F}\right)$ is an
$g$-exceptional pair and the following isomorphisms hold.

$\displaystyle
L_{\psi^{\ast}\mathcal{E}}\left(\psi^{\ast}\mathcal{F}\right)\simeq\psi^{\ast}(L_{\mathcal{E}}\mathcal{F})$
(2.53) $\displaystyle
R_{\psi^{\ast}\mathcal{F}}\left(\psi^{\ast}\mathcal{E}\right)\simeq\psi^{\ast}(R_{\mathcal{F}}\mathcal{E})$
(2.54)

Next, let us recall a group action on $\mathsf{EC}_{N}(f)$ from [BP93,
Proposition 2.1]. Let $\operatorname{Br}_{N}$ be the braid group on $N$
strands, which admits the following famous presentation by generators and
relations.

$\displaystyle\operatorname{Br}_{N}=\langle\sigma_{1},\dots,\sigma_{N-1}\mid\,$
$\displaystyle\sigma_{i}\sigma_{i+1}\sigma_{i}=\sigma_{i+1}\sigma_{i}\sigma_{i+1}\quad
i=1,\dots,N-1$ (2.55)
$\displaystyle\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i}\quad|i-j|\geq
2\rangle$ (2.56)

Consider the action

$\displaystyle\operatorname{Br}_{N}\curvearrowright\mathsf{EC}_{N}(f)$ (2.57)

given by

$\displaystyle\sigma_{i}\colon\mathcal{E}_{i},\mathcal{E}_{i+1}\mapsto\mathcal{E}_{i+1},R_{\mathcal{E}_{i+1}}\mathcal{E}_{i},$
(2.58)

so that

$\displaystyle\sigma_{i}^{-1}\colon\mathcal{E}_{i},\mathcal{E}_{i+1}\mapsto
L_{\mathcal{E}_{i}}\mathcal{E}_{i+1},\mathcal{E}_{i}.$ (2.59)

On the other hand, through the standard surjective homomorphism
$\operatorname{Br}_{N}\twoheadrightarrow\mathfrak{S}_{N};\quad\sigma_{i}\mapsto(i,i+1)$
to the symmetric group of degree $N$, the group $\operatorname{Br}_{N}$ acts
naturally on the abelian group
$\mathbb{Z}^{N}=\operatorname{\mathsf{Map}}(\left\\{1,\dots,N\right\\},\mathbb{Z})$
from the left. Let

$\displaystyle G_{N}\coloneqq\mathbb{Z}^{N}\rtimes\operatorname{Br}_{N}$
(2.60)

be the semi-direct product corresponding to the action.

One can verify that this, together with the action
$\mathbb{Z}^{N}\curvearrowright\mathsf{EC}_{N}$, where
$[a_{1},\dots,a_{N}]^{T}\in\mathbb{Z}^{N}$ sends a collection
$\left(\mathcal{E}_{1},\dots,\mathcal{E}_{N}\right)\in\mathsf{EC}_{N}$ to
$\left(\mathcal{E}_{1}[a_{1}],\dots,\mathcal{E}_{N}[a_{N}]\right)$, extends to
an action $G_{N}\curvearrowright\mathsf{EC}_{N}(f)$. In particular, one has
the following induced action.

$\displaystyle\operatorname{Br}_{N}\curvearrowright\mathsf{EC}_{N}(f)/\mathbb{Z}^{N}$
(2.61)

When $\operatorname{Perf}\mathcal{X}$ admits a full $f$-exceptional collection
of length $r$, then one similarly obtains the action
$G_{r}\curvearrowright\mathsf{FEC}(\mathcal{X})$.

######  Remark 2.23.

[BP93, Conjecture 2.2] asserts that this action should be transitive. Note
that this conjecture is equivalent to the transitivity of the action (2.61).
Theorem 6.1 below is nothing but the affirmative answer to [BP93, Conjecture
2.2] for $\mathcal{X}=\Sigma_{2}$.

Let $X$ be a smooth projective variety over the field $\mathbf{k}$. The
autoequivalences of $\mathbf{D}(X)$ and the notion of exceptional collections
are nicely compatible as we explain next.

######  Lemma 2.24.

If $\mathcal{E}_{1},\dots,\mathcal{E}_{N}\in\mathbf{D}(X)$ is an exceptional
collection and $\Phi\in\operatorname{Auteq}(\mathbf{D}(X))$, then so is
$\Phi(\mathcal{E}_{1}),\dots,\Phi(\mathcal{E}_{N})\in\mathbf{D}(X)$. In
particular, there is the natural action

$\displaystyle\operatorname{Auteq}(\mathbf{D}(X))\curvearrowright\mathsf{EC}_{N}(X).$
(2.62)

The following lemma is easy to verify and plays an important role in this
paper.

######  Lemma 2.25.

The actions (2.57) and (2.62) commute.

###  2.4. Results obtained via deformation of $\Sigma_{2}$ to
$\mathbb{P}^{1}\times\mathbb{P}^{1}$

Consider the (isotrivial) degeneration (2.2) over an algebraically closed
field $\mathbf{k}$. Consider the discrete valuation ring
$R=\left(\mathbf{k}[[t]],(t),\mathbf{k}\right)$ and take the base change by
$B=\operatorname{Spec}R\to\mathbb{A}^{1}_{t}$ of the family. We write

$\displaystyle f\colon\mathcal{X}=\mathcal{X}\times_{\mathbb{A}^{1}}B\to B$
(2.63)

by abuse of notation. The central fiber $\mathcal{X}_{0}$ of $f$ is isomorphic
to $\Sigma_{2}$. Also, let

$\displaystyle\xi\colon{\overline{K}}\coloneqq\overline{\mathbf{k}((t))}\to B$
(2.64)

be the geometric generic point of $B$. The isotriviality of the family (2.2)
outside the origin implies that the geometric generic fiber
$\mathcal{X}_{\xi}\to\operatorname{Spec}{\overline{K}}$ is isomorphic to
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ over ${\overline{K}}$. Throughout this
section, we freely use the symbols introduced in this paragraph.

Since the generic fiber $\mathbb{P}^{1}\times\mathbb{P}^{1}$ is a del Pezzo
surface, the properties of exceptional collections on it is very well known by
[KO94]. We list the known properties.

######  Theorem 2.26.

  1. (1)

Any exceptional object on $\mathbb{P}^{1}\times\mathbb{P}^{1}$ is isomorphic
to a shift of an exceptional vector bundle, so that the natural map
$\mathsf{EC}_{N}(\mathbb{P}^{1}\times\mathbb{P}^{1})\hookrightarrow\mathsf{ECVB}_{N}(\mathbb{P}^{1}\times\mathbb{P}^{1})$
is a bijection.

  2. (2)

$\mathsf{EC}_{4}(\mathbb{P}^{1}\times\mathbb{P}^{1})=\mathsf{FEC}(\mathbb{P}^{1}\times\mathbb{P}^{1})$,
and the action
$G_{4}\curvearrowright\mathsf{FEC}(\mathbb{P}^{1}\times\mathbb{P}^{1})$ is
transitive.

  3. (3)

Any exceptional collection on $\mathbb{P}^{1}\times\mathbb{P}^{1}$ can be
extended to a full exceptional collection.

The non-triviality of the group
$\operatorname{Auteq}^{K_{0}-\mathsf{triv}}(\mathbf{D}(\Sigma_{2}))$ implies
that an exceptional object on $\Sigma_{2}$ is not uniquely determined by its
class in $K_{0}$, even modulo shifts by $\mathbb{Z}[2]$. However, if one
considers only exceptional vector bundles, then it is the case:

######  Lemma 2.27 ($=$a weaker version of [OU15, Lemma 3.5]).

Let $\mathcal{E},\mathcal{E}^{\prime}$ be exceptional vector bundles on
$\Sigma_{2}$ such that
$[\mathcal{E}]=[\mathcal{E}^{\prime}]\in\operatorname{K_{0}}\left(\Sigma_{2}\right)$.
Then $\mathcal{E}\simeq\mathcal{E}^{\prime}$.

The degeneration (2.63) allows one to compare various invariants of
$\Sigma_{2}$ to those of $\mathbb{P}^{1}\times\mathbb{P}^{1}$. Recall that
$\mathcal{X}_{0}\simeq\Sigma_{2}$.

######  Definition 2.28.

For an exceptional object $\mathcal{E}\in\mathbf{D}(\mathcal{X}_{0})$, let
$\mathcal{E}_{R},\operatorname{\mathsf{gen}}(\mathcal{E})$ denote the unique
deformation of $\mathcal{E}$ to $\mathcal{X}$ and its restriction to the
geometric generic fiber $\mathcal{X}_{\xi}$, respectively. For an exceptional
collection $\underline{\mathcal{E}}$ on the central fiber, we will similarly
write
$\underline{\mathcal{E}}_{R},\operatorname{\mathsf{gen}}(\underline{\mathcal{E}})$
to mean its (unique) deformation to $\mathcal{X}$ and its restriction to
$\mathcal{X}_{\xi}$, respectively.

Let

$\displaystyle\operatorname{\mathsf{gen}}\colon\mathsf{EC}_{4}(\mathcal{X}_{0})\to\mathsf{FEC}(\mathcal{X}_{\xi})$
(2.65)

be the map which sends (an isomorphism class of) an exceptional collection
$\underline{\mathcal{E}}$ of length 4 on $\Sigma_{2}$ to
$\operatorname{\mathsf{gen}}(\underline{\mathcal{E}})\in\mathsf{FEC}(\mathcal{X}_{\xi})$,
which is obtained by restricting the deformation of $\underline{\mathcal{E}}$
to the $f$-exceptional collection, whose existence and uniqueness is
guaranteed by Lemma 2.18, to the geometric generic fiber (see Corollary 2.17).
See Corollary 2.37 below for the surjectivity of
$\operatorname{\mathsf{gen}}$.

One similarly defines the map
$\operatorname{\mathsf{gen}|_{\mathsf{ECVB}_{4}(\mathcal{X}_{0})}}\colon\mathsf{ECVB}_{4}(\mathcal{X}_{0})\to\mathsf{FECVB}(\mathcal{X}_{\xi})$,
to obtain the following diagram.

${\mathsf{EC}_{4}(\mathcal{X}_{0})}$${\mathsf{FEC}(\mathcal{X}_{\xi})}$${\mathsf{ECVB}_{4}(\mathcal{X}_{0})}$${\mathsf{FECVB}(\mathcal{X}_{\xi})}$$\scriptstyle{\operatorname{\mathsf{gen}}}$$\scriptstyle{\operatorname{\mathsf{gen}|_{\mathsf{ECVB}_{4}(\mathcal{X}_{0})}}}$
(2.66)

We next compare $\operatorname{K_{0}}\left(\operatorname{Perf}\right)$ of the
surfaces. Note that we have the following diagram of schemes (the labels of
the arrows in the diagram will be freely used).

${\mathcal{X}_{0}}$${\mathcal{X}}$${\mathcal{X}_{\xi}}$${\operatorname{Spec}\mathbf{k}}$${\operatorname{Spec}R}$${\operatorname{Spec}{\overline{K}}}$$\scriptstyle{i}$$\scriptstyle{f_{0}}$${\lrcorner}$$\scriptstyle{f}$$\scriptstyle{j}$$\scriptstyle{f_{\xi}}$${\llcorner}$$\scriptstyle{\iota}$$\scriptstyle{{\overline{\jmath}}}$
(2.67)

Applying the functor
$\operatorname{K_{0}}\left(\operatorname{Perf}(-)\right)$, we obtain the first
two rows of Figure 2.68, which is a commutative diagram of commutative rings
with units.

${\operatorname{K_{0}}\left(\mathcal{X}_{0}\right)}$${\operatorname{K_{0}}\left(\mathcal{X}\right)}$${\operatorname{K_{0}}\left(\mathcal{X}_{\xi}\right)}$${\operatorname{K_{0}}\left(\mathbf{k}\right)}$${\operatorname{K_{0}}\left(R\right)}$${\operatorname{K_{0}}\left({\overline{K}}\right)}$${\mathbb{Z}}$${\mathbb{Z}}$${\mathbb{Z}}$$\scriptstyle{f_{0\ast}}$$\scriptstyle{\simeq}$$\scriptstyle{i^{\ast}}$$\scriptstyle{j^{\ast}}$$\scriptstyle{\simeq}$$\scriptstyle{f_{\ast}}$$\scriptstyle{f_{\xi\ast}}$$\scriptstyle{\simeq}$$\scriptstyle{\iota^{\ast}}$$\scriptstyle{{\overline{\jmath}}^{\ast}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{=}$$\scriptstyle{=}$
(2.68)

The derived dual ∨ defined in (2.6) induces an automorphism of commutative
rings

$\displaystyle\operatorname{K_{0}}\left(\operatorname{Perf}\mathcal{X}\right)=\operatorname{K_{0}}\left(\left(\operatorname{Perf}\mathcal{X}\right)^{\mathrm{op}}\right)\xrightarrow[\simeq]{\vee}\operatorname{K_{0}}\left(\operatorname{Perf}\mathcal{X}\right);\quad[\mathcal{E}]\mapsto[\mathcal{E}^{\vee}].$
(2.69)

The _Euler pairing_ on
$\operatorname{K_{0}}\left(\operatorname{Perf}\mathcal{X}\right)=\operatorname{K_{0}}\left(\mathcal{X}\right)$
(note that $\mathcal{X}$ is a regular scheme) is the following bilinear
pairing.

$\displaystyle\chi_{\mathcal{X}}\colon\operatorname{K_{0}}\left(\mathcal{X}\right)\times\operatorname{K_{0}}\left(\mathcal{X}\right)\to\operatorname{K_{0}}\left(R\right);\quad(v,w)\mapsto
f_{\ast}\left(v^{\vee}\cdot w\right)$ (2.70)

We can similarly define

$\displaystyle\chi_{\mathcal{X}_{0}}\colon\operatorname{K_{0}}\left(\mathcal{X}_{0}\right)\times\operatorname{K_{0}}\left(\mathcal{X}_{0}\right)\to\operatorname{K_{0}}\left(\mathbf{k}\right);\quad(v,w)\mapsto
f_{0\ast}\left(v^{\vee}\cdot w\right),$ (2.71)
$\displaystyle\chi_{\mathcal{X}_{\xi}}\colon\operatorname{K_{0}}\left(\mathcal{X}_{\xi}\right)\times\operatorname{K_{0}}\left(\mathcal{X}_{\xi}\right)\to\operatorname{K_{0}}\left({\overline{K}}\right);\quad(v,w)\mapsto
f_{\xi\ast}\left(v^{\vee}\cdot w\right).$ (2.72)

######  Lemma 2.29.

$\displaystyle\iota^{\ast}\circ\chi_{\mathcal{X}}=\chi_{\mathcal{X}_{0}}\circ(i^{\ast}\times
i^{\ast}),$ (2.73)
$\displaystyle{\overline{\jmath}}^{\ast}\circ\chi_{\mathcal{X}}=\chi_{\mathcal{X}_{\xi}}\circ(j^{\ast}\times
j^{\ast})$ (2.74)

Let $\underline{\mathcal{E}}$ be a full exceptional collection of
$\mathcal{X}_{0}$, and
$\underline{\mathcal{E}}_{R},\underline{\mathcal{E}}_{\xi},$ be the
deformation of $\underline{\mathcal{E}}$ to $\mathcal{X}$ and the restriction
of $\underline{\mathcal{E}}_{R}$ to $\mathcal{X}_{\xi}$ as defined in
Definition 2.28. As pointed out in Remark 2.19, both
$\underline{\mathcal{E}}_{R},\underline{\mathcal{E}}_{\xi}$ are full
exceptional collections.

######  Lemma 2.30.

$\operatorname{K_{0}}\left(\underline{\mathcal{E}}\right),\operatorname{K_{0}}\left(\underline{\mathcal{E}}_{R}\right).\operatorname{K_{0}}\left(\underline{\mathcal{E}}_{\xi}\right)$
are bases of
$\operatorname{K_{0}}\left(\mathcal{X}_{0}\right),\operatorname{K_{0}}\left(\mathcal{X}\right),\operatorname{K_{0}}\left(\mathcal{X}_{\xi}\right)$,
respectively. In particular, the horizontal maps $i^{\ast},j^{\ast}$ in the
first row of (2.68) are isomorphisms.

###### Proof.

Immediately follows from the fact that the collections
$\underline{\mathcal{E}},\underline{\mathcal{E}}_{R},\underline{\mathcal{E}}_{\xi}$
are full exceptional collections of the triangulated categories over the base
of length $4$. ∎

On the other hand we obtain the bottom row of the diagram Figure 2.68, where
the vertical maps to the bottom row are isomorphisms of rings. Let

$\displaystyle\operatorname{K_{0}}\left(\operatorname{\mathsf{gen}}\right)\colon\operatorname{K_{0}}\left(\mathcal{X}_{0}\right)\to\operatorname{K_{0}}\left(\mathcal{X}_{\xi}\right)$
(2.75)

be the isomorphism of abelian groups obtained from the diagram Figure 2.68.
Also, regard $\chi_{\mathcal{X}_{0}},\chi_{\mathcal{X}_{\xi}}$ as
$\mathbb{Z}$-valued bilinear pairings by the diagram Figure 2.68. With all the
preparations above, we can show the desired properties of the map
$\operatorname{K_{0}}\left(\operatorname{\mathsf{gen}}\right)$.

######  Proposition 2.31.

The isomorphism $\operatorname{K_{0}}\left(\operatorname{\mathsf{gen}}\right)$
of (2.75) respects the pairings
$\chi_{\mathcal{X}_{0}},\chi_{\mathcal{X}_{\xi}}$ on the source and the target
abelian groups. Moreover, it fits in the following commutative diagram.

${\mathsf{EC}_{1}(\mathcal{X}_{0})}$${\mathsf{EC}_{1}(\mathcal{X}_{\xi})}$${\operatorname{K_{0}}\left(\mathcal{X}_{0}\right)}$${\operatorname{K_{0}}\left(\mathcal{X}_{\xi}\right)}$$\scriptstyle{\operatorname{\mathsf{gen}}}$$\scriptstyle{\operatorname{K_{0}}\left(\right)}$$\scriptstyle{\operatorname{K_{0}}\left(\right)}$$\scriptstyle{\operatorname{K_{0}}\left(\operatorname{\mathsf{gen}}\right)}$$\scriptstyle{\simeq}$

######  Proposition 2.32.

Let $\mathcal{E},\mathcal{E}^{\prime}\in\mathbf{D}(\mathcal{X}_{0})$ be
exceptional objects. The following conditions are equivalent.

  1. (1)

$[\mathcal{E}]=[\mathcal{E}^{\prime}]\in\operatorname{K_{0}}\left(\mathcal{X}_{0}\right)$.

  2. (2)

$\operatorname{\mathsf{gen}}(\mathcal{E})\simeq\operatorname{\mathsf{gen}}(\mathcal{E}^{\prime})[2m]$
for some $m\in\mathbb{Z}$.

###### Proof.

(2) $\Rightarrow$ (1) is an consequence of the commutative diagram above and
that $\operatorname{K_{0}}\left(\operatorname{\mathsf{gen}}\right)$ is an
isomorphism.

Conversely, assume (1). It then follows that
$[\operatorname{\mathsf{gen}}(\mathcal{E})]=[\operatorname{\mathsf{gen}}(\mathcal{E}^{\prime})]\in\operatorname{K_{0}}\left(\mathcal{X}_{\xi}\right)$,
which in turn implies (2) by Theorem 1.1 (2). ∎

For exceptional vector bundles on $\Sigma_{2}\simeq\mathcal{X}_{0}$, we have
the following reconstruction result. This is an immediate corollary of Lemma
2.27.

######  Lemma 2.33.

For any $N=1,\dots,4$, the map
$\mathsf{ECVB}_{N}(\Sigma_{2})\xrightarrow{\operatorname{K_{0}}\left(\right)}\mathsf{numEC}_{N}(\Sigma_{2})$
is injective.

See Definition 2.34 for the definition of the set
$\mathsf{numEC}_{N}(\Sigma_{2})$.

######  Definition 2.34.

Let $S$ be a smooth projective variety, and for simplicity let us assume that
$\operatorname{K_{0}}\left(S\right)$ is isomorphic to the numerical
Grothendieck group; i.e., the Euler pairing $\chi$ is non-degenerate. This in
particular implies that $\operatorname{K_{0}}\left(S\right)$ is a free abelian
group of finite rank.

An exceptional vector is an element $e\in\operatorname{K_{0}}\left(S\right)$
such that $e^{2}=\chi_{S}(e,e)=1$.

A _numerical exceptional collection_ on $S$ is a sequence of exceptional
vectors $e_{1},\dots,e_{N}\in\operatorname{K_{0}}\left(S\right)$ such that
$\chi_{S}(e_{j},e_{i})=0$ for any $1\leq i<j\leq N$. A numerical exceptional
collection is said to be _full_ if it is a basis of
$\operatorname{K_{0}}\left(S\right)$; i.e, when
$N=\operatorname{rank}\operatorname{K_{0}}\left(S\right)$.

The set of numerical exceptional collections of length $N$ (resp. full) on $S$
will be denoted by $\mathsf{numEC}_{N}(S)$ and $\mathsf{numFEC}(S)$,
respectively.

For a numerical exceptional collection $(e,f)$ of length $2$, which will also
be called a numerical exceptional pair, its right and left mutations are the
new numerical exceptional pairs defined and denoted as follows.

$\displaystyle\left(f,R_{f}(e)\coloneqq e-\chi_{S}(e,f)f\right)$ (2.76)
$\displaystyle\left(L_{e}(f)\coloneqq f-\chi_{S}(e,f)e,e\right)$ (2.77)

By similar arguments for the action $G_{N}\curvearrowright\mathsf{EC}_{N}$,
one can verify that for a surface $S$ with
$\operatorname{K_{0}}\left(S\right)\simeq\mathbb{Z}^{N}$ there is an action
$G_{N}\curvearrowright\mathsf{numFEC}(S)$, where the subgroup
$\operatorname{Br}_{N}$ acts by the mutations (2.76) (2.77) and
$\mathbb{Z}^{N}$ by the change of signs (hence the action descends to the
quotient
$\left(\mathbb{Z}/2\mathbb{Z}\right)^{N}\rtimes\operatorname{Br}_{N}$).

At last we claim that everything goes together.

######  Proposition 2.35.

There exists the following commutative diagram of sets equipped with the
action of the group $G_{4}=\mathbb{Z}^{4}\rtimes\operatorname{Br}_{4}$.

${\mathsf{EC}_{4}(\mathcal{X}_{0})}$${\mathsf{FEC}(\mathcal{X}_{\xi})}$${\mathsf{numFEC}(\mathcal{X}_{0})}$${\mathsf{numFEC}(\mathcal{X}_{\xi})}$$\scriptstyle{\operatorname{\mathsf{gen}}}$$\scriptstyle{\operatorname{K_{0}}\left(\right)}$$\scriptstyle{\operatorname{K_{0}}\left(\right)}$$\scriptstyle{\operatorname{K_{0}}\left(\operatorname{\mathsf{gen}}\right)}$$\scriptstyle{\simeq}$

Let us now introduce a particular ($f$-)exceptional collection of invertible
sheaves, which will be called _the standard collection_ and serve as the base
point of the sets $\mathsf{FEC}$ and $\mathsf{FECVB}$ in this paper.

######  Definition 2.36.

The _standard full exceptional collection_ of invertible sheaves on
$\Sigma_{2}\simeq\mathcal{X}_{0}$ is defined as follows.

$\displaystyle\mathcal{E}^{\mathsf{std}}\coloneqq\left(\mathcal{O}_{\Sigma_{2}},\mathcal{O}_{\Sigma_{2}}(f),\mathcal{O}_{\Sigma_{2}}(C+2f),\mathcal{O}_{\Sigma_{2}}(C+3f)\right)\in\mathsf{FECVB}(\Sigma_{2}).$
(2.78)

By Lemma 2.18 and Remark 2.19, it uniquely deforms to a full $f$-exceptional
collection of invertible sheaves. We will write it
$\mathcal{E}^{\mathsf{std}}_{R}$, and its pullback to $\mathcal{X}_{\xi}$ will
be denoted by $\mathcal{E}^{\mathsf{std}}_{\xi}$. Using the deformation
invariance of the intersection numbers, one can easily confirm that

$\displaystyle\mathcal{E}^{\mathsf{std}}_{\xi}=\left(\mathcal{O}_{\mathcal{X}_{\xi}},\mathcal{O}_{\mathcal{X}_{\xi}}(1,0),\mathcal{O}_{\mathcal{X}_{\xi}}(1,1),\mathcal{O}_{\mathcal{X}_{\xi}}(2,1)\right)$
(2.79)

under an isomorphism
$\mathcal{X}_{\xi}\simeq\mathbb{P}^{1}\times\mathbb{P}^{1}$.

######  Corollary 2.37.

The map
$\operatorname{\mathsf{gen}}\colon\mathsf{EC}_{4}(\mathcal{X}_{0})\to\mathsf{FEC}(\mathcal{X}_{\xi})$
is surjective.

###### Proof.

By Theorem 2.26 (2), we know that
$\mathsf{FEC}(\mathcal{X}_{\xi})=G_{4}\cdot\mathcal{E}^{\mathsf{std}}_{\xi}$.
Since the diagram of Proposition 2.35 is $G_{4}$-equivariant and
$\operatorname{\mathsf{gen}}(\mathcal{E}^{\mathsf{std}})=\mathcal{E}^{\mathsf{std}}_{\xi}$,
we obtain the conclusion. ∎

######  Corollary 2.38.

For any $\underline{\mathcal{E}}\in\mathsf{EC}_{4}(\mathcal{X}_{0})$, there
exists $\sigma\in G_{4}$ such that
$\operatorname{K_{0}}\left(\sigma(\underline{\mathcal{E}})\right)=\operatorname{K_{0}}\left(\mathcal{E}^{\mathsf{std}}\right)\in\mathsf{numFEC}(\mathcal{X}_{0})$.

###### Proof.

Again by Theorem 2.26 (2), one can find an element $\sigma\in G_{4}$ such that
$\sigma\operatorname{\mathsf{gen}}(\underline{\mathcal{E}})=\mathcal{E}^{\mathsf{std}}_{\xi}$.
Again by the equivariance, it follows that
$\operatorname{\mathsf{gen}}(\sigma(\underline{\mathcal{E}}))=\mathcal{E}^{\mathsf{std}}_{\xi}=\operatorname{\mathsf{gen}}(\mathcal{E}^{\mathsf{std}})$,
which means that
$\operatorname{K_{0}}\left(\sigma(\underline{\mathcal{E}})\right)=\operatorname{K_{0}}\left(\mathcal{E}^{\mathsf{std}}\right)$
by Proposition 2.32. ∎

##  3\. Twisting exceptional objects down to exceptional vector bundles

The purpose of this section is to show the following theorem.

######  Theorem 3.1.

For any exceptional object $\mathcal{E}\in\mathbf{D}(\Sigma_{2})$, there
exists an exceptional vector bundle $\mathcal{F}$ and a sequence of integers
$a_{1},\dots,a_{n},m$ such that

$\displaystyle\mathcal{E}\simeq\left(T_{a_{n}}\circ\cdots\circ
T_{a_{1}}\right)(\mathcal{F})[m].$ (3.1)

The similar result for spherical objects is given in [IU05, Proposition 1.6].
In fact, we prove Theorem 3.1 by suitably modifying the proof of [IU05,
Proposition 1.6].

######  Notation 3.2.

Let $X$ be an integral noetherian scheme. For
$\mathcal{E}\in\operatorname{coh}X$, we define

$\displaystyle\operatorname{Supp}\mathcal{E}\coloneqq\operatorname{Spec}\operatorname{Im}\left(\mathcal{O}_{X}\to\mathop{{\mathcal{E}}nd}\nolimits_{X}(\mathcal{E})\right)\subset
X$ (3.2)

and call it the _schematic support_ of $\mathcal{E}$. It is universal among
the closed subschemes $\iota\colon Z\hookrightarrow X$ which admits a coherent
sheaf $\mathcal{E}^{\prime}\in\operatorname{coh}Z$ such that
$\iota_{\ast}\mathcal{E}^{\prime}\simeq\mathcal{E}$. The underlying closed
subset of $X$, or equivalently the reduced closed subscheme
$(\operatorname{Supp}\mathcal{E})_{\operatorname{\mathsf{red}}}\subset X$, is
called the _reduced support_ of $\mathcal{E}$. Also we let
$\operatorname{tors}\mathcal{E}\subset\mathcal{E}$ be the maximum torsion
subsheaf of $\mathcal{E}$. For an object $\mathcal{E}\in\mathbf{D}(X)$, we use
the following notation.

$\displaystyle\mathcal{H}^{\bullet}(\mathcal{E})\coloneqq\bigoplus_{i\in\mathbb{Z}}\mathcal{H}^{i}(\mathcal{E})\quad\operatorname{Supp}\mathcal{E}\coloneqq\bigcup_{i\in\mathbb{Z}}\left(\operatorname{Supp}\mathcal{H}^{i}(\mathcal{E})\right)_{\operatorname{\mathsf{red}}}$
(3.3)

###  3.1. First properties of $\mathcal{H}^{\bullet}(\mathcal{E})$

As the first step toward the proof of Theorem 3.1, in this subsection we prove
some basic properties of $\mathcal{H}^{\bullet}(\mathcal{E})$. Part of them
concern
$\left(\operatorname{Supp}\mathcal{H}^{i}(\mathcal{E})\right)_{\operatorname{\mathsf{red}}}$,
which can be summarized as follows.

> There exists the unique integer $i_{0}\in\mathbb{Z}$ with the following
> properties.
>
>   * •
>
> $\operatorname{Supp}\mathcal{H}^{i_{0}}(\mathcal{E})=\Sigma_{2}$
>
>   * •
>
>
> $\left(\operatorname{Supp}\operatorname{tors}\mathcal{H}^{i_{0}}(\mathcal{E})\right)_{\operatorname{\mathsf{red}}}=C$
>
>   * •
>
>
> $\left(\operatorname{Supp}\mathcal{H}^{i}(\mathcal{E})\right)_{\operatorname{\mathsf{red}}}=C$
> for $\forall i\neq i_{0}$
>
>

Based on this, the similar statements for the _schematic_ supports will be
proved in the next subsection; namely, we will remove
$\operatorname{\mathsf{red}}$ from the second and the third items. In fact,
this step has been the main obstacle for the project.

The similar results for spherical objects appear in [IU05, Lemma 4.8], where
it is rather easily shown that the schematic support of the cohomology sheaves
of a spherical object coincides with $C$. However, unfortunately, the proof of
[IU05, Lemma 4.8] does not immediately apply to our situation. The fact
$\operatorname{Supp}\mathcal{E}=\Sigma_{2}$ prevents us from studying the
problem locally around the curve $C$.

######  Lemma 3.3.

The cohomology sheaves of an exceptional object
$\mathcal{E}\in\mathbf{D}(\Sigma_{2})$ enjoy the following properties.

  1. (1)

$\mathcal{H}^{\bullet}(\mathcal{E})$ is rigid ($\iff E_{2}^{1,q}=0\ \forall
q\in\mathbb{Z}$ in the spectral sequence (3.4)).

  2. (2)

There exists a unique integer $i_{0}\in\mathbb{Z}$ such that

     * •

$\operatorname{Supp}\mathcal{H}^{i_{0}}(\mathcal{E})=\Sigma_{2}$.

     * •

$\mathcal{H}^{i}(\mathcal{E})$ for $i\neq i_{0}$ and
$\operatorname{tors}\mathcal{H}^{i_{0}}(\mathcal{E})$ are pure sheaves with
$\left(\operatorname{Supp}(-)\right)_{\operatorname{\mathsf{red}}}=C$, unless
$=0$.

  3. (3)

$\operatorname{tors}(\mathcal{H}^{i_{0}}(\mathcal{E}))$ is rigid, and
$\mathcal{H}^{i_{0}}(\mathcal{E})/\operatorname{tors}(\mathcal{H}^{i_{0}}(\mathcal{E}))$
is an exceptional vector bundle.

See the following definition for the notion of rigidity.

######  Definition 3.4.

An object $\mathcal{E}\in\mathbf{D}(X)$ is said to be _rigid_ if
$\operatorname{Ext}_{X}^{1}(\mathcal{E},\mathcal{E})=0$.

We will freely use the following standard fact on rigid objects.

######  Lemma 3.5.

Let $\mathcal{E}\in\mathbf{D}(X)$ be a rigid object. Then
$g\mathcal{E}\simeq\mathcal{E}$ for any
$g\in\operatorname{Aut}^{0}_{X/\mathbf{k}}$. In particular,
$\operatorname{Supp}g\mathcal{E}=\operatorname{Supp}\mathcal{E}\subset X$ for
any such $g$.

We need the following spectral sequence, which also plays the central role for
the study of spherical objects in [IU05].

######  Lemma 3.6.

For any object $\mathcal{E}\in\mathbf{D}(X)$, there exists the following
spectral sequence.

$\displaystyle
E_{2}^{p,q}=\bigoplus_{i}\operatorname{Ext}_{\Sigma_{2}}^{p}(\mathcal{H}^{i}(\mathcal{E}),\mathcal{H}^{i+q}(\mathcal{E}))\Rightarrow
E^{p+q}=\operatorname{Hom}_{\Sigma_{2}}^{p+q}(\mathcal{E},\mathcal{E})$ (3.4)

Moreover, using the classes
$e^{i}=e^{i}(\mathcal{E})\in\operatorname{Ext}_{\Sigma_{2}}^{2}(\mathcal{H}^{i}(\mathcal{E}),\mathcal{H}^{i-1}(\mathcal{E}))$
canonically determined by $\mathcal{E}$, the $d_{2}$ maps of (3.4) are given
by

$\displaystyle
d_{2}^{p,q}\colon({\phi_{i}})_{i}\mapsto\left((-1)^{p+q}\phi_{i-1}\circ
e^{i}-e^{i+q}\circ\phi_{i}\right)_{i}.$ (3.5)

###### Proof.

See [IU05, Section 4.1], in particular [IU05, Proposition 4.1]. ∎

###### Proof of Lemma 3.3.

Throughout the proof, we consider the spectral sequence (3.4) for the
exceptional object $\mathcal{E}$. We prove the four items one by one.

(1) The exceptionality of $\mathcal{E}$ is translated into the following
conditions.

$\displaystyle
E^{n}=\begin{cases}\mathbf{k}\operatorname{id}_{\mathcal{E}}&n=0\\\ 0&n\neq
0\end{cases}$ (3.6)

Since $\Sigma_{2}$ is a smooth projective surface, $E_{2}^{p,q}=0$ unless
$0\leq p\leq 2$. Hence (3.4) is $E_{3}$-degenerate, and moreover is
$E_{2}$-degenerate at $(1,q)$ for any $q\in\mathbb{Z}$. Combined with (3.6),
this implies that $E_{2}^{1,q}=0$ for any $q\neq-1$. In the next paragraph we
also confirm $E_{2}^{1,-1}=0$, thereby concluding the rigidity of
$\mathcal{H}^{\bullet}(\mathcal{E})$.

It follows from the explicit description (3.5) of $d_{2}$ maps that
$0\neq\sum_{i\in\mathbb{Z}}\operatorname{id}_{\mathcal{H}^{i}(\mathcal{E})}\in
E_{2}^{0,0}$ is in fact contained in $\operatorname{Ker}d_{2}^{0,0}\simeq
E_{3}^{0,0}\simeq E_{\infty}^{0,0}$, which implies that $E_{\infty}^{0,0}\neq
0$. Combined with the isomorphism $E^{0}\simeq\mathbf{k}$ from (3.6) and the
epimorphism $E^{0}\twoheadrightarrow E^{0}/F^{1}E^{0}\simeq E_{\infty}^{0,0}$,
this implies that $E_{\infty}^{0,0}\simeq E^{0}(\simeq\mathbf{k})$ and hence
$E_{2}^{1,-1}\simeq E_{\infty}^{1,-1}\simeq F^{1}E^{0}/F^{2}E^{0}=0$.

(2) We further obtain the following equalities from (3.6).

$\displaystyle e_{2}^{0,q}$ $\displaystyle=e_{2}^{2,q-1}\quad(\text{if $q\neq
0$})$ (3.7) $\displaystyle e_{2}^{0,0}$ $\displaystyle=e_{2}^{2,-1}+1$ (3.8)

To see (3.7) for $q=-1$, note that the arguments in the previous paragraph
implies $0=E_{\infty}^{2,-2}\simeq E_{3}^{2,-2}$. Moreover, (1) implies that
either
$\operatorname{Supp}\left(\operatorname{tors}\mathcal{H}^{i}(\mathcal{E})\right)_{\operatorname{\mathsf{red}}}=\Sigma_{2},C,\
\text{or}\ \emptyset$. This follows from Lemma 3.5 and the orbit decomposition
$\Sigma_{2}=\left(\Sigma_{2}\setminus C\right)\coprod C$ for the action
$\operatorname{Aut}_{\Sigma_{2}/\mathbf{k}}\curvearrowright\Sigma_{2}$.

Suppose for a contradiction that
$\operatorname{Supp}\mathcal{H}^{i}(\mathcal{E})_{\operatorname{\mathsf{red}}}\neq\Sigma_{2}$
for every $i$. Then $\operatorname{Supp}\mathcal{E}$ coincides with $C$ and
hence there is an isomorphism
$\mathcal{E}\otimes\omega_{\Sigma_{2}}\simeq\mathcal{E}$, from which we deduce
$\operatorname{ext}_{\Sigma_{2}}^{2}(\mathcal{E},\mathcal{E})=\operatorname{ext}_{\Sigma_{2}}^{0}(\mathcal{E},\mathcal{E})$
by the Serre duality. This contradicts (3.6) and hence there must be at least
one integer $i_{0}$ with
$\operatorname{Supp}\mathcal{H}^{i_{0}}(\mathcal{E})_{\operatorname{\mathsf{red}}}=\Sigma_{2}$.

Now consider the Serre duality

$\displaystyle(E_{2}^{2,q})^{\vee}\simeq\bigoplus_{i}\operatorname{Hom}_{\Sigma_{2}}(\mathcal{H}^{i}(\mathcal{E}),\mathcal{H}^{i-q}(\mathcal{E})\otimes\omega_{\Sigma_{2}}).$
(3.9)

Fix a non-trivial morphism
$s\colon\omega_{\Sigma_{2}}\hookrightarrow\mathcal{O}_{\Sigma_{2}}$ such that
the support of its cokernel is disjoint from the set of associated points of
$\mathcal{H}^{i}(\mathcal{E})$ for all $i\in\mathbb{Z}$. This is possible, as
the linear system $|-K_{\Sigma_{2}}|$ is base point free and there are only
finitely many (schematic) points to be avoided (see [HL10, p. 8]). The
property which we required for $s$ implies that for each $i\in\mathbb{Z}$ the
natural morphism

$\displaystyle\mathcal{H}^{i-q}(\mathcal{E})\otimes\omega_{\Sigma_{2}}\stackrel{{\scriptstyle\operatorname{id}\otimes
s}}{{\to}}\mathcal{H}^{i-q}(\mathcal{E})$ (3.10)

is injective. Thus we obtain an injection of vector spaces

$\phi_{q}\colon(E_{2}^{2,q})^{\vee}\hookrightarrow E_{2}^{0,-q}$

and hence an inequality

$\displaystyle e_{2}^{2,q}\leq e_{2}^{0,-q}$ (3.11)

for any $q$.

Next we prove that $\phi_{0}$ is not surjective on the direct summands indexed
by those $i$ with
$\operatorname{Supp}\mathcal{H}^{i}(\mathcal{E})=\Sigma_{2}$. To see this, (by
slight abuse of notation) let $i_{0}$ be one of such indices and apply the
functor $\operatorname{Hom}(\mathcal{H}^{i_{0}}(\mathcal{E}),-)$ to the
following short exact sequence.

$\displaystyle
0\to\mathcal{H}^{i_{0}}(\mathcal{E})\otimes\omega_{\Sigma_{2}}\xrightarrow[]{\operatorname{id}\otimes
s}\mathcal{H}^{i_{0}}(\mathcal{E})\to\mathcal{H}^{i_{0}}(\mathcal{E})|_{Z(s)}\to
0.$ (3.12)

The rigidity of $\mathcal{H}^{i_{0}}(\mathcal{E})$, which we confirmed in (1),
implies that we have the following short exact sequence.

$\displaystyle
0\to\operatorname{Hom}\left(\mathcal{H}^{i_{0}}(\mathcal{E}),\mathcal{H}^{i_{0}}(\mathcal{E})\otimes\omega_{\Sigma_{2}}\right)\to\operatorname{Hom}\left(\mathcal{H}^{i_{0}}(\mathcal{E}),\mathcal{H}^{i_{0}}(\mathcal{E})\right)\to\operatorname{Hom}\left(\mathcal{H}^{i_{0}}(\mathcal{E}),\mathcal{H}^{i_{0}}(\mathcal{E})|_{Z(s)}\right)\to
0$ (3.13)

The 3rd term is not $0$ by the assumption
$\operatorname{Supp}\mathcal{H}^{i_{0}}(\mathcal{E})=\Sigma_{2}$ and hence
$\operatorname{id}\otimes s$, which is identified with the direct summand of
$\phi_{0}$ of interest, is not surjective.

Since we showed above that there is at least one such index $i_{0}$, we have
confirmed the non-surjectivity of $\phi_{0}$ and hence the following
inequality.

$\displaystyle e_{2}^{2,0}<e_{2}^{0,0}$ (3.14)

Summarizing the results so far, we obtain the following sequence of
(in)equalities.

$e_{2}^{2,-1}\stackrel{{\scriptstyle\eqref{eq:e_2^{2,q} <= e_2^{0,
-q}}(q=-1)}}{{\leq}}e_{2}^{0,1}\stackrel{{\scriptstyle\eqref{eq:e^{0,q} =
e^{2,q-1}}(q=1)}}{{=}}e_{2}^{2,0}\stackrel{{\scriptstyle\eqref{eq:e_2^{2,0} <
e_2^{0,0}}}}{{=}}e_{2}^{0,0}-\operatorname{rank}\phi_{0}\stackrel{{\scriptstyle\eqref{eq:e^{0,0}
= e^{2,-1}+1}}}{{=}}e_{2}^{2,-1}+1-\operatorname{rank}\phi_{0}\leq
e_{2}^{2,-1},$

which implies that

$\displaystyle\operatorname{rank}\phi_{0}=e_{2}^{0,0}-e_{2}^{2,0}=1.$ (3.15)

This means that the number of the indices $i_{0}$ with
$\operatorname{Supp}\mathcal{H}^{i_{0}}(\mathcal{E})=\Sigma_{2}$ is exactly
one.

Finally, recall that rigid sheaf with one-dimensional support is pure by
[Kul97, Corollary 2.2.3]. We already confirmed the rigidity of
$\mathcal{H}^{i}(\mathcal{E})$ for $i\neq i_{0}$ above, and for
$\operatorname{tors}\mathcal{H}^{i_{0}}(\mathcal{E})$ it is proven below.

(3) Put $\mathcal{H}\coloneqq\mathcal{H}^{i_{0}}(\mathcal{E})$,
$\mathcal{T}\coloneqq\operatorname{tors}(\mathcal{H})$ and
$\mathcal{F}\coloneqq\mathcal{H}/\mathcal{T}$. Consider the spectral sequence

$\displaystyle
E_{1}^{p,q}=\bigoplus_{j}\operatorname{Ext}_{\Sigma_{2}}^{p+q}(G_{j},G_{j+p})\Rightarrow
E^{p+q}=\operatorname{Ext}_{\Sigma_{2}}^{p+q}(\mathcal{H},\mathcal{H})$ (3.16)

arising from the short exact sequence $0\to G_{2}\to\mathcal{H}\to G_{1}\to
0$, where $G_{1}=\mathcal{F}$ and $G_{2}=\mathcal{T}$. For obvious reasons we
see $E_{1}^{p,q}=0$ unless $0\leq p+q\leq 2$ and $-1\leq p\leq 1$.

On the other hand, since $\mathcal{H}$ is rigid, it is stable under the action
of
$\operatorname{Aut}_{\Sigma_{2}/\mathbf{k}}=\operatorname{Aut}_{\Sigma_{2}/\mathbf{k}}^{0}$
by Lemma 3.5. Since the torsion part of a coherent sheaf is uniquely
determined by the sheaf, it follows that $\mathcal{T}$ is also stable under
the same group action. Hence it follows that
$\left(\operatorname{Supp}\mathcal{T}\right)_{\operatorname{\mathsf{red}}}\subset
C$, so that

$\displaystyle\mathcal{T}\otimes\omega_{\Sigma_{2}}\simeq\mathcal{T}.$ (3.17)

Combined with the Serre duality, this implies the equality

$\displaystyle e_{1}^{p,q}=e_{1}^{-p,2-q}$ (3.18)

for $p\neq 0$.

It then follows that $0=E_{1}^{-1,1}=E_{1}^{1,1}$, where the first equality is
the consequence of the fact that $G_{2}$ is torsion and $G_{1}$ is torsion
free, and the second equality is the case $(p,q)=(1,1)$ of (3.18).

Thus we have confirmed that the spectral sequence (3.16) is $E_{1}$-degenerate
at $(0,1)$. Hence $E_{1}^{0,1}\simeq E_{\infty}^{0,1}\simeq E^{1}=0$, where
the last vanishing is nothing but the rigidity of $\mathcal{H}$, which we
confirmed in (1). Thus we have shown the rigidity of $\mathcal{T}$ and
$\mathcal{F}$.

Again in the spectral sequence (3.16), the vanishing $E^{1}=0$ implies
$0=E_{\infty}^{1,0}=E_{2}^{1,0}$. Hence $d_{1}^{0,0}$ is surjective. Also the
vanishing $E^{1}=0$ implies $E_{\infty}^{-1,2}=0$. Since
$0=E_{1}^{1,1}=E_{2}^{1,1}$, the spectral sequence is $E_{2}$-degenerate at
$(-1,2)$ and hence $E_{\infty}^{-1,2}\simeq E_{2}^{-1,2}$. This implies that
$d_{1}^{-1,2}$ is injective. Thus we obtain

$\displaystyle\hom(\mathcal{H},\mathcal{H})$ $\displaystyle=e_{1}^{1,-1}+\ker
d_{1}^{0,0}$ $\displaystyle=e_{1}^{1,-1}+e_{1}^{0,0}-e_{1}^{1,0}$ (3.19)
$\displaystyle\operatorname{ext}^{2}(\mathcal{H},\mathcal{H})$
$\displaystyle=e_{1}^{-1,3}+\operatorname{coker}d_{1}^{-1,2}$
$\displaystyle=e_{1}^{-1,3}+e_{1}^{0,2}-e_{1}^{-1,2}.$ (3.20)

Therefore substituting (3.19) and (3.20) into

$\displaystyle\eqref{eq:the difference is
1}\iff\hom(\mathcal{H},\mathcal{H})=\operatorname{ext}^{2}(\mathcal{H},\mathcal{H})+1$
(3.21)

and using (3.18), we obtain

$1=\hom(\mathcal{H},\mathcal{H})-\operatorname{ext}^{2}(\mathcal{H},\mathcal{H})=e_{1}^{0,0}-e_{1}^{0,2}\stackrel{{\scriptstyle\eqref{eq:cT
is stable under Serre functor up to
shift}}}{{=}}\hom(\mathcal{F},\mathcal{F})-\operatorname{ext}^{2}(\mathcal{F},\mathcal{F}).$

Since $\mathcal{F}$ is rigid, the same proof as in [OU15, Lemma 2.2] shows
that $\mathcal{F}$ is an exceptional vector bundle. ∎

######  Definition 3.7.

For an exceptional object $\mathcal{E}\in\mathbf{D}(\Sigma_{2})$, we will
write $i_{0}(\mathcal{E})=i_{0}\in\mathbb{Z}$ for the unique integer such that
$\operatorname{Supp}\mathcal{H}^{i_{0}}(\mathcal{E})=\Sigma_{2}$.

Let
$\operatorname{rank}\colon\operatorname{K_{0}}\left(\Sigma_{2}\right)\to\mathbb{Z}$
be the rank function. If we let
$\iota\colon\operatorname{Spec}\mathbf{k}(\Sigma_{2})\to\Sigma_{2}$ denote the
embedding of the generic point, $\operatorname{rank}$ is concisely defined as
the composition of the map $\operatorname{K_{0}}\left(\iota^{\ast}\right)$ and
the isomorphism
$\operatorname{K_{0}}\left(\operatorname{Spec}\mathbf{k}(\Sigma_{2})\right)\simeq\mathbb{Z}$
which sends the class of $\mathbf{k}(\Sigma_{2})$ to $1$. As a corollary of
Lemma 3.3, we obtain the following

######  Corollary 3.8.

The equality

$\displaystyle\operatorname{rank}\mathcal{E}=(-1)^{i_{0}}\operatorname{rank}\mathcal{H}^{i_{0}}(\mathcal{E})$
(3.22)

holds for any exceptional object $\mathcal{E}\in\mathbf{D}(\Sigma_{2})$. In
particular, $\operatorname{rank}\mathcal{E}\neq 0$.

###  3.2. Properties of the schematic support of
$\operatorname{tors}\mathcal{H}^{\bullet}(\mathcal{E})$

The aim of this subsection is to prove Proposition 3.18, which asserts that
the _schematic_ support of
$\operatorname{tors}\mathcal{H}^{\bullet}(\mathcal{E})$ coincides with $C$.
This is the most technical part of this paper.

######  Remark 3.9.

In this paper, Proposition 3.18 will be used only as a black box. Hence one
can first assume Proposition 3.18 to understand the rest of the paper and then
later come back to its proof.

To describe the schematic support of the cohomology sheaves of $\mathcal{E}$,
we consider the anti-canonical morphism $f\colon\Sigma_{2}\to P(1,1,2)$ to the
weighted projective plane of weight $(1,1,2)$, which contracts the
$(-2)$-curve $C$ to the singularity $(0:0:1)$.

######  Notation 3.10.

Let $(R,\mathfrak{m})$ denote the local ring of $P(1,1,2)$ at the singular
point. It is isomorphic to $\mathbf{k}[x,y,z]_{(x,y,z)}/(z^{2}-xy)$, the
$A_{1}$ singularity.

Our first goal is to give an $R$-module structure on $E_{2}^{p,q}$ for
$(p,q)\neq(0,0)$ of the spectral sequence (3.4) with respect to which the
differentials of the spectral sequence are $R$-linear. The similar fact for
spherical objects is used in [IU05], in which case the existence of such an
$R$-module structure is trivial. In fact, since the support of a spherical
object is concentrated in $C$, one can use the pushforward along
$f|_{\operatorname{Spec}R}$.

######  Remark 3.11.

For any $\mathcal{M}\in\operatorname{coh}\Sigma_{2}$ with
$\left(\operatorname{Supp}\mathcal{M}\right)_{\operatorname{\mathsf{red}}}\subset
C$, there is a natural homomorphism of $\mathbf{k}$-algebras
$R\to\operatorname{End}_{\Sigma_{2}}(\mathcal{M})$ which kills
$\mathfrak{m}^{\ell}$ for some $\ell>0$. Hence for $i\in\mathbb{Z}$ and
$\mathcal{N}\in\operatorname{coh}\Sigma_{2}$, there are natural $R$-module
structures of finite length on
$\operatorname{Ext}_{\Sigma_{2}}^{i}(\mathcal{M},\mathcal{N})\simeq\operatorname{Hom}_{\mathbf{D}(\Sigma_{2})}(\mathcal{M},\mathcal{N}[i])$
and
$\operatorname{Ext}_{\Sigma_{2}}^{i}(\mathcal{N},\mathcal{M})\simeq\operatorname{Hom}_{\mathbf{D}(\Sigma_{2})}(\mathcal{N},\mathcal{M}[i])$
given by precomposition and postcomposition, respectively. If $\mathcal{N}$ is
also supported in $C$, we may use
$R\to\operatorname{End}_{\Sigma_{2}}(\mathcal{N})$ instead. However, we end up
with the same $R$-module structures defined via
$R\to\operatorname{End}_{\Sigma_{2}}(\mathcal{M})$.

For $q\neq 0$, the reduced support of either $\mathcal{H}^{i}(\mathcal{E})$ or
$\mathcal{H}^{i+q}(\mathcal{E})$ is contained in $C$ and therefore
$E_{2}^{p,q}$ has a canonical $R$-module structure by Remark 3.11. For $q=0$
and $p\neq 0,2$, $E_{2}^{p,0}=0$.

For $p=2$, we have

$E_{2}^{2,0}\simeq\bigoplus_{i\neq
i_{0}}\operatorname{Ext}_{\Sigma_{2}}^{2}(\mathcal{H}^{i}(\mathcal{E}),\mathcal{H}^{i}(\mathcal{E}))\oplus\operatorname{Ext}_{\Sigma_{2}}^{2}(\mathcal{H}^{i_{0}}(\mathcal{E}),\mathcal{H}^{i_{0}}(\mathcal{E})).$

Since the reduced support of $\mathcal{H}^{i}(\mathcal{E})$ for $i\neq i_{0}$
is contained in $C$, all the direct summands but the last one have canonical
$R$-module structures again by Remark 3.11.

Recall the following short exact sequence from the proof of Lemma 3.3 (3).

$\displaystyle
0\to\mathcal{T}\coloneqq\operatorname{tors}\mathcal{H}^{i_{0}}(\mathcal{E})\stackrel{{\scriptstyle\iota}}{{\to}}\mathcal{H}^{i_{0}}(\mathcal{E})\to\mathcal{F}\coloneqq\mathcal{H}^{i_{0}}(\mathcal{E})/\mathcal{T}\to
0$ (3.23)

Take $\operatorname{Hom}_{X}(-,\mathcal{H}^{i_{0}}(\mathcal{E}))$ to obtain
the following exact sequence

$\operatorname{Ext}_{\Sigma_{2}}^{2}(\mathcal{F},\mathcal{H}^{i_{0}}(\mathcal{E}))\to\operatorname{Ext}_{\Sigma_{2}}^{2}(\mathcal{H}^{i_{0}}(\mathcal{E}),\mathcal{H}^{i_{0}}(\mathcal{E}))\stackrel{{\scriptstyle\iota^{\ast}}}{{\to}}\operatorname{Ext}_{\Sigma_{2}}^{2}(\mathcal{T},\mathcal{H}^{i_{0}}(\mathcal{E}))\to
0,$

where

$\displaystyle\begin{aligned}
\operatorname{Ext}_{\Sigma_{2}}^{2}(\mathcal{H}^{i_{0}}(\mathcal{E})&,\mathcal{H}^{i_{0}}(\mathcal{E}))&&\to&\operatorname{Ext}_{\Sigma_{2}}^{2}(&\mathcal{T},\mathcal{H}^{i_{0}}(\mathcal{E}))\\\
&x&&\mapsto&&x\circ\iota\end{aligned}$ (3.24)

is the precomposition (think of $x$ as a morphism
$\mathcal{H}^{i_{0}}(\mathcal{E})\to\mathcal{H}^{i_{0}}(\mathcal{E})[2]$ in
$\mathbf{D}(\Sigma_{2})$). The following vanishing follows from the
exceptionality of $\mathcal{F}$.

$\operatorname{Ext}_{\Sigma_{2}}^{2}(\mathcal{F},\mathcal{H}^{i_{0}}(\mathcal{E}))\simeq\operatorname{Ext}_{\Sigma_{2}}^{2}(\mathcal{F},\mathcal{F})=0$

Hence we conclude that the map $\iota^{\ast}$ is an isomorphism of
$\mathbf{k}$-vector spaces. Since the reduced support of $\mathcal{T}$ is
contained in $C$, the right hand side of (3.24) has a canonical $R$-module
structure.

######  Definition 3.12.

We transfer the $R$-module structure on
$\operatorname{Ext}_{\Sigma_{2}}^{2}(\mathcal{T},\mathcal{H}^{i_{0}}(\mathcal{E}))$
to
$\operatorname{Ext}_{\Sigma_{2}}^{2}(\mathcal{H}^{i_{0}}(\mathcal{E}),\mathcal{H}^{i_{0}}(\mathcal{E}))$
via the isomorphism $\iota^{\ast}$ of (3.24), thereby giving an $R$-module
structure on $E_{2}^{2,0}$.

######  Lemma 3.13.

For $q\neq 0$, the maps $d_{2}^{0,q}\colon E_{2}^{0,q}\to E_{2}^{2,q-1}$ in
the spectral sequence (3.4) are $R$-linear.

###### Proof.

For $q\neq 0,1$, as already explained, the $R$-module structures on the source
and the target of $d_{2}^{0,q}$ are naturally defined and hence the
$R$-linearity of $d_{2}^{0,q}$ are rather obvious.

Let us show the $R$-linearity of $d_{2}^{0,1}$. Take an arbitrary element

$\displaystyle(\phi_{i})_{i}\in
E_{2}^{0,1}=\bigoplus_{i}\operatorname{Hom}_{\Sigma_{2}}(\mathcal{H}^{i}(\mathcal{E}),\mathcal{H}^{i+1}(\mathcal{E})).$
(3.25)

It suffices to show that the maps as follows, which appear in the description
of $d_{2}^{0,1}$ given in Lemma 3.6, are $R$-linear.

$\displaystyle\phi_{i_{0}-1}$ $\displaystyle\mapsto\phi_{i_{0}-1}\circ
e^{i_{0}}\in\operatorname{Ext}_{\Sigma_{2}}^{2}(\mathcal{H}^{i_{0}}(\mathcal{E}),\mathcal{H}^{i_{0}}(\mathcal{E}))$
(3.26) $\displaystyle\phi_{i_{0}}$ $\displaystyle\mapsto
e^{i_{0}+1}\circ\phi_{i_{0}}\in\operatorname{Ext}_{\Sigma_{2}}^{2}(\mathcal{H}^{i_{0}}(\mathcal{E}),\mathcal{H}^{i_{0}}(\mathcal{E}))$
(3.27)

Under the isomorphism $\iota^{\ast}$ of (3.24), the map (3.26) is identified
with

$\phi_{i_{0}-1}\mapsto\phi_{i_{0}-1}\circ
e^{i_{0}}\circ\iota\in\operatorname{Ext}_{\Sigma_{2}}^{2}(\mathcal{T},\mathcal{H}^{0}(\mathcal{E})),$

and, similarly, the map (3.27) is identified with

$\phi_{i_{0}}\mapsto
e^{i_{0}+1}\circ\phi_{i_{0}}\circ\iota\in\operatorname{Ext}_{\Sigma_{2}}^{2}(\mathcal{T},\mathcal{H}^{i_{0}}(\mathcal{E})).$

These maps are $R$-linear, for the reason that the $R$-module structures given
in Remark 3.11 are by means of precompositions. As the $R$-module structure on
$\operatorname{Ext}_{\Sigma_{2}}^{2}(\mathcal{H}^{i_{0}}(\mathcal{E}),\mathcal{H}^{i_{0}}(\mathcal{E}))$
is given in Definition 3.12 by transferring the $R$-module structure on
$\operatorname{Ext}_{\Sigma_{2}}^{2}(\mathcal{T},\mathcal{H}^{i_{0}}(\mathcal{E}))$
via $\iota^{\ast}$, this is exactly what we had to prove. ∎

Suppose $\mathcal{M}$ is a pure coherent sheaf on $\Sigma_{2}$ with
$\left(\operatorname{Supp}\mathcal{M}\right)_{\operatorname{\mathsf{red}}}=C$.
Then by Remark 3.11, $M\coloneqq
H^{0}(\Sigma_{2},\mathcal{M})\simeq\operatorname{Hom}_{\Sigma_{2}}(\mathcal{O}_{\Sigma_{2}},\mathcal{M})$
has a standard $R$-module structure of finite length and hence there is an
integer $\ell$ such that $\mathfrak{\mathfrak{m}}^{\ell}M=0$. In general, for
an $R$-module $M$ and an ideal $I\subset R$, we let $(0:I)_{M}$ denote the
_annihilator_

$\displaystyle(0:I)_{M}\coloneqq\\{x\in M\mid Ix=0\\}\subseteq M.$ (3.28)

Geometrically speaking, this is the maximum submodule of $M$ which is
“supported on the closed subscheme $\operatorname{Spec}R/I$”.

######  Lemma 3.14.

For a pure coherent sheaf $\mathcal{M}$ on $\Sigma_{2}$ with
$\left(\operatorname{Supp}\mathcal{M}\right)_{\operatorname{\mathsf{red}}}=C$,
assume

$n\coloneqq\min\\{\ell\mid\mathfrak{\mathfrak{m}}^{\ell}M=0\\}>1,$

where $M=H^{0}(\Sigma_{2},\mathcal{M})\in\operatorname{mod}R$ as above. Then
the following _strict_ inequality holds.

$\displaystyle\dim_{\mathbf{k}}\frac{M}{(0:\mathfrak{m}^{n-1})_{M}}<\dim_{\mathbf{k}}\mathfrak{m}^{n-1}M$
(3.29)

###### Proof.

Without loss of generality, we may assume that
$f^{\ast}f_{\ast}\mathcal{M}\stackrel{{\scriptstyle\eta}}{{\to}}\mathcal{M}$
is surjective; i.e., $\mathcal{M}$ is $f$-globally generated. In fact, if
$\eta$ is not surjective, we can replace $\mathcal{M}$ with the image
$\operatorname{Im}\eta$. By standard arguments on the adjoint pair
$f^{\ast}\dashv f_{\ast}$, there is a canonical isomorphism
$f_{\ast}\operatorname{Im}\eta\xrightarrow{\sim}f_{\ast}\mathcal{M}=M$ and
$\operatorname{Im}\eta$ is automatically $f$-globally generated.

The following fact about the $A_{1}$-singularity $R$ is known well: for each
$i>0$, the $i$-th powers of the ideal sheaf

$\displaystyle\mathcal{I}_{C}=\mathcal{O}_{\Sigma_{2}}(-C)\subset\mathcal{O}_{\Sigma_{2}}$
(3.30)

of $C$ and $\mathfrak{m}$ are related to each other as follows.

$\displaystyle\begin{aligned}
\mathfrak{m}^{i}\mathcal{O}_{\Sigma_{2}}=\mathcal{O}_{\Sigma_{2}}(-iC)=\mathcal{I}_{C}^{i}\subseteq\mathcal{O}_{\Sigma_{2}}\\\
\left(f|_{\operatorname{Spec}R}\right)_{\ast}\mathcal{I}_{C}^{i}=\mathfrak{m}^{i}\subseteq
R\end{aligned}$ (3.31)

The vanishing $\mathfrak{m}^{n}M=0$ means that $M=f_{\ast}\mathcal{M}$ is a
sheaf on $\operatorname{Spec}(R/\mathfrak{m}^{n})$, from which we deduce that
$f^{*}f_{*}\mathcal{M}$ and hence its quotient $\mathcal{M}$ are supported on
$f^{-1}(\operatorname{Spec}(R/\mathfrak{m}^{n}))\stackrel{{\scriptstyle\eqref{eq:powers
of cO ( - C ) vs frakm}}}{{=}}nC$. This means
$\mathcal{I}_{C}^{n}\mathcal{M}=0$.

On the other hand, note that the image of the canonical morphism

$\displaystyle
H^{0}\left(\Sigma_{2},\mathcal{I}_{C}^{n-1}\right)\otimes_{\mathbf{k}}H^{0}(\Sigma_{2},\mathcal{M})\to
H^{0}\left(\Sigma_{2},\mathcal{I}_{C}^{n-1}\mathcal{M}\right),$ (3.32)

as a submodule of $H^{0}(\Sigma_{2},\mathcal{M})=M$, is $\mathfrak{m}^{n-1}M$
by (3.31). Thus we see
$H^{0}\left(\Sigma_{2},\mathcal{I}_{C}^{n-1}\mathcal{M}\right)\supset\mathfrak{m}^{n-1}M\neq
0$, hence $\mathcal{I}_{C}^{n-1}\mathcal{M}\neq 0$. Therefore we conclude

$n=\min\\{\ell\mid\mathcal{I}_{C}^{\ell}\mathcal{M}=0\\}.$

Since $\mathcal{I}_{C}(\mathcal{I}_{C}^{n-1}\mathcal{M})=0$, we can naturally
think of $\mathcal{I}_{C}^{n-1}\mathcal{M}$ as an object of
$\operatorname{coh}C$. Since we assumed that $\mathcal{M}$ is pure, so is its
subsheaf $\mathcal{I}_{C}^{n-1}\mathcal{M}$. Hence
$\mathcal{I}_{C}^{n-1}\mathcal{M}$ is a vector bundle on $C$ of positive rank.

Likewise $\frac{\mathcal{M}}{(0:\mathcal{I}_{C}^{n-1})_{\mathcal{M}}}$ can be
thought of an object of $\operatorname{coh}C$, where
$(0:\mathcal{I}_{C}^{n-1})_{\mathcal{M}}$ is the sheaf version of the
annihilator (3.28); i.e., the maximum subsheaf of $\mathcal{M}$ whose
schematic support is contained in $(n-1)C$. We claim that there is an
isomorphism

$\displaystyle\varphi\colon\mathcal{O}_{C}(-(n-1)C)\otimes_{\mathcal{O}_{C}}\frac{\mathcal{M}}{(0:\mathcal{I}_{C}^{n-1})_{\mathcal{M}}}\xrightarrow{\sim}\mathcal{I}_{C}^{n-1}\mathcal{M}$
(3.33)

of vector bundles on $C$. Indeed $\varphi$ is induced from the product
morphism
$\mathcal{I}_{C}^{n-1}\otimes_{\mathcal{O}_{\Sigma_{2}}}\mathcal{M}\to\mathcal{I}_{C}^{n-1}\mathcal{M}$,
whose surjectivity implies that of $\varphi$. At the generic point $\xi$ of
$C$, the stalk $(\mathcal{I}_{C})_{\xi}$ is the maximal ideal of the discrete
valuation ring $\mathcal{O}_{\Sigma_{2},\xi}$ and $\mathcal{M}_{\xi}$ is a
finite length
$\mathcal{O}_{\Sigma_{2},\xi}/(\mathcal{I}_{C}^{n})_{\xi}$-module. Hence one
sees that $\varphi$ is an isomorphism at $\xi$ (use the structure theorem for
finitely generated modules over a discrete valuation ring). Therefore it is
enough to show that the left hand side of (3.33) is pure; i.e., there is no
zero-dimensional subsheaf.

Assume for a contradiction that
$\mathcal{M}/(0:\mathcal{I}_{C}^{n-1})_{\mathcal{M}}$ is not pure. Then there
is a subsheaf
$(0:\mathcal{I}_{C}^{n-1})_{\mathcal{M}}\subsetneq\mathcal{S}\subset\mathcal{M}$
such that
$\operatorname{Supp}\left(\mathcal{S}/(0:\mathcal{I}_{C}^{n-1})_{\mathcal{M}}\right)$
is zero-dimensional. Then $\mathcal{I}_{C}^{n-1}\mathcal{S}\subset\mathcal{M}$
is non-zero and zero-dimensional, which contradicts the purity of
$\mathcal{M}$. To see $\dim\mathcal{I}_{C}^{n-1}\mathcal{S}=0$, note that
there is an epimorphism

$\displaystyle\mathcal{I}_{C}^{n-1}\otimes_{\mathcal{O}_{\Sigma_{2}}}\mathcal{S}/(0:\mathcal{I}_{C}^{n-1})_{\mathcal{M}}\twoheadrightarrow\mathcal{I}_{C}^{n-1}\mathcal{S}$
(3.34)

and that $\dim\mathcal{S}/(0:\mathcal{I}_{C}^{n-1})_{\mathcal{M}}=0$.

Now consider the following $R$-module.

$V\coloneqq\frac{M}{H^{0}\left(\Sigma_{2},(0:\mathcal{I}_{C}^{n-1})_{\mathcal{M}}\right)}=\frac{M}{(0:\mathfrak{m}^{n-1})_{M}}$

To see the equality, note that the denominator of the left hand side, as a
submodule of $M=H^{0}(\Sigma_{2},\mathcal{M})$, coincides with the following
subset.

$\\{s\in M\mid
s_{P}\in(0:\mathcal{I}_{C,P}^{n-1})_{\mathcal{M}_{P}}\quad\forall P\in C\\}.$

Since $\mathcal{I}_{C,P}^{n-1}=\mathfrak{m}^{n-1}\mathcal{O}_{\Sigma_{2},P}$,
this coincides with the denominator of the right hand side.

If we put $L\coloneqq\mathcal{O}_{C}(-(n-1)C)$ and
$E\coloneqq\mathcal{M}/(0:\mathcal{I}_{C}^{n-1})_{\mathcal{M}}$, then (3.33)
is rewritten as

$L\otimes_{\mathcal{O}_{C}}E\simeq\mathcal{I}_{C}^{n-1}\mathcal{M}.$

Noting $H^{0}(C,L)\simeq\mathfrak{m}^{n-1}/\mathfrak{m}^{n}$, we see that
$\mathfrak{m}^{n-1}M$ is the image of the product map

$\psi\colon H^{0}(C,L)\otimes_{\mathbf{k}}V\to
H^{0}(C,\mathcal{I}_{C}^{n-1}\mathcal{M}).$

Then the assertion follows from Lemma 3.15 below. ∎

######  Lemma 3.15.

Let $E$ and $L$ be a vector bundle and a line bundle on a smooth projective
curve $C$, respectively. Suppose $\dim H^{0}(C,L)>1$ and let $V\subset
H^{0}(C,E)$ be a non-zero linear subspace. Then we have a strict inequality

$\dim V<\operatorname{rank}\psi,$

where $\psi\colon H^{0}(C,L)\otimes_{\mathbf{k}}V\to
H^{0}(C,L\otimes_{\mathcal{O}_{C}}E)$ denotes the product map.

###### Proof.

Without loss of generality, we may replace $E$ with the subsheaf generated by
$V$. Take a pair of linearly independent global sections $s,t\in H^{0}(C,L)$.
It follows that $\mathcal{O}_{C}\cdot t\not\subseteq\mathcal{O}_{C}\cdot s$ as
subsheaves of $L$. Since $E$ is locally free, this implies that
$(\mathcal{O}_{C}\cdot
t)\otimes_{\mathcal{O}_{C}}E\not\subseteq(\mathcal{O}_{C}\cdot
s)\otimes_{\mathcal{O}_{C}}E$ as subsheaves of $L\otimes_{\mathcal{O}_{C}}E$.
Since $(\mathcal{O}_{C}\cdot t)\otimes_{\mathcal{O}_{C}}E$ and
$(\mathcal{O}_{C}\cdot s)\otimes_{\mathcal{O}_{C}}E$ are the subsheaves of
$L\otimes_{\mathcal{O}_{C}}E$ generated by $\psi(\mathbf{k}t\otimes V)$ and
$\psi(\mathbf{k}s\otimes V)$ respectively, we conclude
$\psi(\mathbf{k}t\otimes V)\not\subseteq\psi(\mathbf{k}s\otimes V)$. Thus we
see

$\displaystyle V\simeq\psi(\mathbf{k}s\otimes
V)\subsetneq\psi(\mathbf{k}s\otimes V)+\psi(\mathbf{k}t\otimes
V)\subseteq\operatorname{Im}\psi.$ (3.35)

Taking $\dim_{\mathbf{k}}$, we obtain the assertion. ∎

Let $\operatorname{mod}_{0}R\subset\operatorname{mod}R$ be the category of
Artinian ($\iff\dim_{\mathbf{k}}<\infty$) $R$-modules. There is an
(anti-)involution of categories

$D\colon\left(\operatorname{mod}_{0}R\right)^{\mathrm{op}}\xrightarrow{\sim}\operatorname{mod}_{0}R$

which is defined as follows.

$\displaystyle DM\coloneqq\operatorname{Hom}_{\mathbf{k}}(M,\mathbf{k})$
(3.36)

######  Corollary 3.16.

Under the same notation and assumption as in Lemma 3.14, put $W\coloneqq DM$.
Then the following strict inequality holds.

$\dim_{\mathbf{k}}\frac{W}{(0:\mathfrak{m}^{n-1})_{W}}>\dim_{\mathbf{k}}\mathfrak{m}^{n-1}W$

###### Proof.

To obtain the assertion, apply the anti-involution $D$ to both sides of the
equality (3.29) (before applying $\dim_{\mathbf{k}}$) and then use the
following Lemma 3.17. Note that $D$ preserves dimensions. ∎

######  Lemma 3.17.

For $M\in\operatorname{mod}_{0}R$ and integer $\ell\geq 0$, there are
isomorphisms as follows.

$\displaystyle D(\mathfrak{m}^{\ell}M)$ $\displaystyle\simeq
DM/(0:\mathfrak{m}^{\ell})_{DM}$ (3.37) $\displaystyle
D\left(M/(0:\mathfrak{m}^{\ell})_{M}\right)$
$\displaystyle\simeq\mathfrak{m}^{\ell}DM$ (3.38)

###### Proof.

To obtain the second isomorphism, replace $M$ with $DM$ in the first
isomorphism and then use $D^{2}\simeq\operatorname{id}$. To see the first
isomorphism, consider the following short exact sequence. The inclusion $i$ is
the canonical one.

$\displaystyle 0\to(0:\mathfrak{m}^{\ell})_{DM}\stackrel{{\scriptstyle
i}}{{\to}}DM\to C\coloneqq\operatorname{coker}i\to 0$ (3.39)

By applying the anti-involution $D$ to this, we obtain the following short
exact sequence.

$\displaystyle 0\to DC\to M\to D(0:\mathfrak{m}^{\ell})_{DM}\to 0$ (3.40)

Then, as a submodule of $M$, $DC$ is computed as follows.

$\displaystyle DC=\\{x\in M\mid y\in(0:\mathfrak{m}^{\ell})_{DM}\Rightarrow
i(y)(x)=0\\}=\\{x\in M\mid y\in DM,\mathfrak{m}^{\ell}y=0\Rightarrow
y(x)=0\\}.$ (3.41)

Note that $y\in DM$ satisfies $\mathfrak{m}^{\ell}y=0$ if and only if
$y(\mathfrak{m}^{\ell}M)=0$; i.e., $y\in
D(M/\mathfrak{m}^{\ell}M)\stackrel{{\scriptstyle
q^{\ast}}}{{\hookrightarrow}}DM$, where $q\colon M\twoheadrightarrow
M/\mathfrak{m}^{\ell}M$ is the quotient map. Hence

$\displaystyle DC=\\{x\in M\mid y\in D(M/\mathfrak{m}^{\ell}M)\Rightarrow
y(x)=0\\}=D(M/\mathfrak{m}^{\ell}M)^{\perp}=\mathfrak{m}^{\ell}M,$ (3.43)

so that $C\simeq D(\mathfrak{m}^{\ell}M)$. ∎

Now we are ready to prove the following

######  Proposition 3.18.

For an exceptional object $\mathcal{E}\in\mathbf{D}(\Sigma_{2})$, it holds
that

$\operatorname{Supp}\operatorname{tors}\mathcal{H}^{i_{0}}(\mathcal{E})=\operatorname{Supp}\mathcal{H}^{i}(\mathcal{E})\
(i\neq i_{0})=C\text{ or }\emptyset.$

###### Proof.

We first discuss $\operatorname{Supp}\mathcal{H}^{i}(\mathcal{E})$ for $i\neq
i_{0}$. We may and will assume $\bigoplus_{i\neq
i_{0}}\mathcal{H}^{i}(\mathcal{E})\neq 0$, since otherwise there is nothing to
prove. Put

$\displaystyle n\coloneqq\min\bigcap_{i\neq
i_{0}}\left\\{\ell\mid\mathcal{I}_{C}^{\ell}\mathcal{H}^{i}(\mathcal{E})=0\right\\}\geq
1.$ (3.44)

In the spectral sequence (3.4) for the exceptional object $\mathcal{E}$, we
have the isomorphism

$\displaystyle d_{2}^{0,1}\colon
E_{2}^{0,1}=\bigoplus_{i}\operatorname{Hom}_{\Sigma_{2}}(\mathcal{H}^{i}(\mathcal{E}),\mathcal{H}^{i+1}(\mathcal{E}))\xrightarrow{\sim}E_{2}^{2,0}=\bigoplus_{i}\operatorname{Ext}_{\Sigma_{2}}^{2}(\mathcal{H}^{i}(\mathcal{E}),\mathcal{H}^{i}(\mathcal{E}))$
(3.45)

of _$R$ -modules_ by Lemma 3.13. Note that $\mathfrak{m}^{n}E_{2}^{0,1}=0$,
since for each $i$ either $\mathcal{I}_{C}^{n}\mathcal{H}^{i}(\mathcal{E})=0$
or $\mathcal{I}_{C}^{n}\mathcal{H}^{i+1}(\mathcal{E})=0$ holds.

On the other hand, for each $i\neq i_{0}$ there is an isomorphism of
$R$-modules given by the Serre duality:

$\operatorname{Ext}^{2}_{\Sigma_{2}}(\mathcal{H}^{i}(\mathcal{E}),\mathcal{H}^{i}(\mathcal{E}))\simeq
D\operatorname{Hom}(\mathcal{H}^{i}(\mathcal{E}),\mathcal{H}^{i}(\mathcal{E}))$

This implies that $\mathfrak{m}^{n-1}E_{2}^{2,0}\neq 0$. Combining this with
the isomorphism of $R$-modules (3.45), we obtain

$n=\min\\{\ell\mid\mathfrak{m}^{\ell}E_{2}^{0,1}=0\\}=\min\\{\ell\mid\mathfrak{m}^{\ell}E_{2}^{2,0}=0\\}.$

Now assume for a contradiction that $n>1$. Let

$\displaystyle\mathcal{M}\coloneqq\bigoplus_{i}\mathop{{\mathcal{H}}om}\nolimits_{\Sigma_{2}}(\mathcal{H}^{i}(\mathcal{E}),\mathcal{H}^{i+1}(\mathcal{E})),$
(3.46)

so that $E_{2}^{0,1}\simeq f_{\ast}\mathcal{M}$. We can apply Lemma 3.14 to
$\mathcal{M}$, to obtain the strict inequality

$\displaystyle\dim_{\mathbf{k}}\frac{E_{2}^{0,1}}{(0:\mathfrak{m}^{n-1})_{E_{2}^{0,1}}}<\dim_{\mathbf{k}}\mathfrak{m}^{n-1}E_{2}^{0,1}.$
(3.47)

On the other hand, let

$\displaystyle\mathcal{M}^{\prime}\coloneqq\bigoplus_{i\neq
i_{0}}\mathop{{\mathcal{E}}nd}\nolimits_{\Sigma_{2}}(\mathcal{H}^{i}(\mathcal{E}))\oplus\mathop{{\mathcal{H}}om}\nolimits_{\Sigma_{2}}(\mathcal{H}^{i_{0}}(\mathcal{E}),\mathcal{T}).$
(3.48)

By Lemma 3.13 and the Serre duality, it follows that there is an isomorphism
of $R$-modules $E_{2}^{2,0}\simeq Df_{\ast}\mathcal{M}^{\prime}$. We can apply
Corollary 3.16 to $\mathcal{M}^{\prime}$, to obtain the strict inequality

$\displaystyle\dim_{\mathbf{k}}\frac{E_{2}^{2,0}}{(0:\mathfrak{m}^{n-1})_{E_{2}^{2,0}}}>\dim_{\mathbf{k}}\mathfrak{m}^{n-1}E_{2}^{2,0}.$
(3.49)

The strict inequalities (3.47) and (3.49) contradict the isomorphism (3.45).
Hence we obtain $n=1$, which means that $\mathcal{H}^{i}(\mathcal{E})$ is an
$\mathcal{O}_{C}$-module for any $i\neq i_{0}$. In fact, by rigidity,
$\mathcal{H}^{i}(\mathcal{E})$ is a vector bundle on $C$ for any $i\neq
i_{0}$.

Finally, to investigate $\operatorname{tors}\mathcal{H}^{i_{0}}(\mathcal{E})$,
we consider the derived dual $\mathcal{E}^{\vee}$. As we show in Lemma 3.25
below, there is an isomorphism as follows.

$\mathcal{H}^{-i_{0}+1}(\mathcal{E}^{\vee})\simeq(\operatorname{tors}\mathcal{H}^{i_{0}}(\mathcal{E}))^{\vee}[1]$

Since $-i_{0}+1\neq-i_{0}\stackrel{{\scriptstyle\text{Lemma~{}\ref{lm:dual
exceptional object}}}}{{=}}i_{0}(\mathcal{E}^{\vee})$, by applying what we
have just proved to the exceptional object $\mathcal{E}^{\vee}$, we see that
the left hand side, hence the right hand side, is a vector bundle on $C$.
Hence so is $\operatorname{tors}\mathcal{H}^{i_{0}}(\mathcal{E})$. ∎

######  Remark 3.19.

Proposition 3.18 will _not_ be used in the proof of Lemma 3.25, so that it is
harmless to use Lemma 3.25 in the proof of Proposition 3.18 (as we did in the
last paragraph). Actually, in the proof of Lemma 3.25, we only use Lemma 3.3
and some standard facts on homological algebra.

###  3.3. More on the structure of $\mathcal{H}^{\bullet}(\mathcal{E})$

In this subsection we give a structure theorem for
$\mathcal{H}^{i_{0}}(\mathcal{E})$ in Lemma 3.21. It is then used to give a
structure theorem for $\operatorname{tors}\mathcal{H}^{\bullet}(\mathcal{E})$
in Corollary 3.23.

Below is repeatedly used in this paper.

######  Lemma 3.20 ($=$[Kul97, Remark 2.3.4]).

Let $\mathcal{E}\in\mathbf{D}(\Sigma_{2})$ be an exceptional vector bundle of
$\operatorname{rank}\mathcal{E}=r$. Then there is an isomorphism

$\displaystyle\mathcal{E}|_{C}\simeq\mathcal{O}_{C}(b)^{\oplus
s}\oplus\mathcal{O}_{C}(b+1)^{\oplus r-s}$ (3.50)

for some $b\in\mathbb{Z}$ and $s\in\mathbb{N}$ such that $1\leq s\leq r$.

Note that the integers $b$ and $s$ in Lemma 3.20 are uniquely determined by
$\mathcal{E}$.

######  Lemma 3.21.

Let $\mathcal{E}\in\mathbf{D}(\Sigma_{2})$ be an exceptional object. Then the
unique non-torsion cohomology sheaf $\mathcal{H}^{i_{0}}(\mathcal{E})$
decomposes as

$\mathcal{H}^{i_{0}}(\mathcal{E})\simeq E\oplus T,$

where $E$ is an exceptional sheaf and $T$ is a vector bundle on $C$. Moreover,
if $E$ is not locally free, then there is an integer $a\in\mathbb{Z}$ such
that

  * •

The torsion part $\operatorname{tors}E$ is a direct sum of copies of
$\mathcal{O}_{C}(a)$.

  * •

$T$ is a direct sum of copies of $\mathcal{O}_{C}(a)$ and
$\mathcal{O}_{C}(a+1)$.

######  Definition 3.22.

For an exceptional object $\mathcal{E}\in\mathbf{D}(\Sigma_{2})$, let
$E=E(\mathcal{E})$ denote the exceptional sheaf $E$ of Lemma 3.21. Also let
$\mathcal{F}=\mathcal{F}(\mathcal{E})$ denote the torsion free part of the
sheaf $\mathcal{H}^{i_{0}}(\mathcal{E})$, which is known to be an exceptional
vector bundle by Lemma 3.3 (3).

###### Proof of Lemma 3.21.

Consider the following standard short exact sequence.

$\displaystyle
0\to\mathcal{T}\coloneqq\operatorname{tors}\mathcal{H}^{i_{0}}(\mathcal{E})\to\mathcal{H}^{i_{0}}(\mathcal{E})\to\mathcal{F}\coloneqq\mathcal{H}^{i_{0}}(\mathcal{E})/\mathcal{T}\to
0$ (3.51)

Lemma 3.3 (3) asserts that $\mathcal{F}$ is an exceptional vector bundle.
Hence we assume that $\mathcal{T}\neq 0$, since otherwise there is nothing to
prove.

$\mathcal{T}$ is a rigid sheaf again by Lemma 3.3 (3). Combined with
Proposition 3.18, this implies that there are $a\in\mathbb{Z},s>0,t\geq 0$
such that

$\displaystyle\mathcal{T}\simeq\mathcal{O}_{C}(a)^{\oplus
s}\oplus\mathcal{O}_{C}(a+1)^{\oplus t}.$ (3.52)

By Lemma 3.20, $\mathcal{F}|_{C}$ is also rigid and hence there are
$b\in\mathbb{Z},s^{\prime}>0,t^{\prime}\geq 0$ such that

$\mathcal{F}|_{C}\simeq\mathcal{O}_{C}(b)^{\oplus
s^{\prime}}\oplus\mathcal{O}_{C}(b+1)^{\oplus t^{\prime}}.$

If $\operatorname{Ext}_{\Sigma_{2}}^{1}(\mathcal{F},\mathcal{T})=0$, then the
short exact sequence (3.51) splits and we are done. Therefore we assume
$\operatorname{Ext}_{\Sigma_{2}}^{1}(\mathcal{F},\mathcal{T})\neq 0$, which
implies

$\displaystyle a\leq b-1$ (3.53)

In order to make the argument conceptual, fix a $\mathbf{k}$-vector space $V$
of dimension $s$ and replace $\mathcal{O}_{C}(a)^{\oplus s}$ with
$V\otimes_{\mathbf{k}}\mathcal{O}_{C}(a)$. Let

$\displaystyle\mathcal{H}^{\prime}\coloneqq\mathcal{H}^{i_{0}}(\mathcal{E})/\mathcal{O}_{C}(a+1)^{\oplus
t}$ (3.54)

be the quotient by the subsheaf $\mathcal{O}_{C}(a+1)^{\oplus
t}\subset\mathcal{T}\subset\mathcal{H}^{i_{0}}(\mathcal{E})$, which fits in
the following short exact sequence.

$\displaystyle 0\to
V\otimes_{\mathbf{k}}\mathcal{O}_{C}(a)\to\mathcal{H}^{\prime}\to\mathcal{F}\to
0$ (3.55)

From this we see

$\operatorname{Hom}_{\Sigma_{2}}(\mathcal{O}(a+1)^{\oplus
t},\mathcal{H}^{\prime})=0,$

which implies that $\mathcal{H}^{\prime}$ is rigid by Mukai’s lemma [Kul97,
Lemma 2.1.4. 2.(a)], (which is obtained from the spectral sequence of the form
(3.16)).

Let

$\displaystyle[f\colon\operatorname{Ext}_{\Sigma_{2}}^{1}(\mathcal{F},\mathcal{O}_{C}(a))^{\vee}\to
V]\in\operatorname{Hom}_{\mathbf{k}}(\operatorname{Ext}_{\Sigma_{2}}^{1}(\mathcal{F},\mathcal{O}_{C}(a))^{\vee},V)\simeq\operatorname{Ext}_{\Sigma_{2}}^{1}(\mathcal{F},V\otimes_{\mathbf{k}}\mathcal{O}_{C}(a))$
(3.56)

correspond to the extension (3.55). We will show that $f$ is injective. In the
long exact sequence obtained by applying
$\operatorname{Hom}_{\Sigma_{2}}(\mathcal{H}^{\prime},-)$ to (3.55),

  * •

the map
$\operatorname{Hom}_{\Sigma_{2}}(\mathcal{H}^{\prime},\mathcal{H}^{\prime})\to\operatorname{Hom}_{\Sigma_{2}}(\mathcal{H}^{\prime},\mathcal{F})$
is surjective since
$\operatorname{Hom}_{\Sigma_{2}}(\mathcal{H}^{\prime},\mathcal{F})\cong\operatorname{Hom}_{\Sigma_{2}}(\mathcal{F},\mathcal{F})=\mathbf{k}\cdot\operatorname{id}_{\mathcal{F}}$
by the exceptionality of $\mathcal{F}$ and

  * •

$\operatorname{Ext}^{1}(\mathcal{H}^{\prime},\mathcal{H}^{\prime})=0$ by the
rigidity of $\mathcal{H}^{\prime}$.

Hence we obtain

$\operatorname{Ext}^{1}_{\Sigma_{2}}(\mathcal{H}^{\prime},\mathcal{O}_{C}(a))=0.$

Next we apply $\operatorname{Hom}_{\Sigma_{2}}(-,\mathcal{O}_{C}(a))$ to
(3.55) to obtain a surjective map

$V^{\vee}\cong\operatorname{Hom}_{\Sigma_{2}}(V\otimes\mathcal{O}_{C}(a),\mathcal{O}_{C}(a))\to\operatorname{Ext}^{1}_{\Sigma_{2}}(\mathcal{F},\mathcal{O}_{C}(a)).$
(3.57)

For any $\varphi\in V^{\vee}$, the map (3.57) sends $\varphi$ to
$(\varphi\otimes\operatorname{id}_{\mathcal{O}_{C}(a)})\circ e$, where
$e\in\operatorname{Ext}_{\Sigma_{2}}^{1}(\mathcal{F},V\otimes_{\mathbf{k}}\mathcal{O}_{C}(a))$
denotes the extension class (3.55). Take a basis $\\{v_{1},\dots,v_{s}\\}$ of
$V$ and decompose $e$ as $\sum_{i}e_{i}\otimes v_{i}$ under the isomorphism
$\operatorname{Ext}_{\Sigma_{2}}^{1}(\mathcal{F},V\otimes_{\mathbf{k}}\mathcal{O}_{C}(a))\simeq\operatorname{Ext}_{\Sigma_{2}}^{1}\left(\mathcal{F},\mathcal{O}_{C}(a)\right)\otimes_{\mathbf{k}}V$.
Then one can confirm that (3.57) sends $\varphi$ to
$\sum_{i}\varphi(v_{i})e_{i}$ and that $f$ sends a linear form
$\xi\in\operatorname{Ext}_{\Sigma_{2}}^{1}(\mathcal{F},\mathcal{O}_{C}(a))^{\vee}$
to $\sum_{i}\xi(e_{i})v_{i}$. Hence the surjectivity of the map (3.57) implies
that $e_{1},\dots,e_{s}$ generates
$\operatorname{Ext}_{\Sigma_{2}}^{1}\left(\mathcal{F},\mathcal{O}_{C}(a)\right)$,
which in turn is equivalent to the injectivity of $f$.

Now consider the universal extension of $\mathcal{F}$ by $\mathcal{O}_{C}(a)$.

$\displaystyle
0\to\operatorname{Ext}_{\Sigma_{2}}^{1}(\mathcal{F},\mathcal{O}_{C}(a))^{\vee}\otimes_{\mathbf{k}}\mathcal{O}_{C}(a)\to
E\to\mathcal{F}\to 0$ (3.58)

The inequality (3.53) implies
$\mathop{\mathbb{R}\mathrm{Hom}}\nolimits_{\Sigma_{2}}(\mathcal{F},\mathcal{O}_{C}(a))\simeq\operatorname{Ext}_{\Sigma_{2}}^{1}(\mathcal{F},\mathcal{O}_{C}(a))[-1]$,
so that the distinguished triangle which (3.58) yields is isomorphic to the
defining distinguished triangle (2.17) for the inverse spherical twist
$T^{\prime}_{a}(\mathcal{F})$. In particular, there is an isomorphism $E\simeq
T^{\prime}_{a}(\mathcal{F})$ and hence $E$ is an exceptional sheaf. Moreover,
the injectivity of $f$ and the basic properties of universal extensions imply
that there is an isomorphism

$\displaystyle\mathcal{H}^{\prime}\simeq
E\oplus\left(\operatorname{coker}f\otimes_{\mathbf{k}}\mathcal{O}_{C}(a)\right).$
(3.59)

In what follows let
$d\coloneqq\dim_{\mathbf{k}}\operatorname{Ext}_{\Sigma_{2}}^{1}(\mathcal{F},\mathcal{O}_{C}(a))$,
so that

$\mathcal{H}^{\prime}\simeq E\oplus\mathcal{O}_{C}(a)^{\oplus s-d}.$

If $d=0$, then $E\simeq\mathcal{F}$ is a vector bundle and we are done. So, in
the rest of the proof, we assume $d>0$; i.e., we assume that
$\operatorname{tors}E\neq 0$.

Let us prove

$\displaystyle\operatorname{Ext}_{\Sigma_{2}}^{1}(\mathcal{H}^{\prime},\mathcal{O}_{C}(a+1))=0,$
(3.60)

which together with (3.54) implies

$\displaystyle\mathcal{H}^{i_{0}}(\mathcal{E})\simeq\mathcal{H}^{\prime}\oplus\mathcal{O}_{C}(a+1)^{\oplus
t}\simeq E\oplus\mathcal{O}_{C}(a)^{\oplus
s-d}\oplus\mathcal{O}_{C}(a+1)^{\oplus t}.$ (3.61)

Since $\operatorname{tors}E\simeq\mathcal{O}_{C}(a)^{\oplus d}$, this is the
desired conclusion.

(3.60) follows from the local-to-global spectral sequence and the following
vanishings.

$\displaystyle
H^{1}(\Sigma_{2},\mathop{{\mathcal{H}}om}\nolimits_{\Sigma_{2}}(\mathcal{H}^{\prime},\mathcal{O}_{C}(a+1)))$
$\displaystyle=0$ (3.62) $\displaystyle
H^{0}(\Sigma_{2},\mathop{{\mathcal{E}}xt}\nolimits_{\Sigma_{2}}^{1}(\mathcal{H}^{\prime},\mathcal{O}_{C}(a+1)))$
$\displaystyle=0$ (3.63)

(3.62), in turn, follows from
$\mathcal{H}^{\prime}=E\oplus\mathcal{O}_{C}(a)^{\oplus s-d}$ and [OU15,
Theorem 1.4(1)], which says that an exceptional sheaf $E$ whose torsion part
is a non-zero direct sum of copies of $\mathcal{O}_{C}(a)$ satisfies

$\displaystyle E|_{C}\simeq\mathcal{O}_{C}(a+1)^{\oplus r}$ (3.64)

for some $r$.

Finally, (3.63) follows from the following isomorphisms.

$\mathop{{\mathcal{E}}xt}\nolimits_{\Sigma_{2}}^{1}(\mathcal{H}^{\prime},\mathcal{O}_{C}(a+1))\stackrel{{\scriptstyle\eqref{eq:cH'-extension}}}{{\simeq}}\mathop{{\mathcal{E}}xt}\nolimits_{\Sigma_{2}}^{1}(V\otimes_{\mathbf{k}}\mathcal{O}_{C}(a),\mathcal{O}_{C}(a+1))\simeq
V^{\vee}\otimes\mathcal{O}_{C}(-1)$

∎

From Lemma 3.21 we immediately obtain

######  Corollary 3.23.

Suppose that $\mathcal{E}$ is an exceptional object with
$\operatorname{tors}E\neq 0$. Then, with the notation of Lemma 3.21, it holds
that

$\displaystyle\operatorname{tors}\mathcal{H}^{\bullet}(\mathcal{E})=\bigoplus_{i\neq
i_{0}}\mathcal{H}^{i}(\mathcal{E})\oplus
T\oplus\operatorname{tors}E\simeq\mathcal{O}_{C}(a)^{\oplus
s}\oplus\mathcal{O}_{C}(a+1)^{\oplus\ell(\mathcal{E})-s}$ (3.65)

for some $1\leq s\leq\ell(\mathcal{E})$.

######  Remark 3.24.

This is analogous to [IU05, Corollary 4.10] (see also [IU05, Section 5])

###### Proof.

If $i\neq i_{0}$, then $\operatorname{Supp}\mathcal{H}^{i}(\mathcal{E})=C$ and
hence
$\mathcal{H}^{i}(\mathcal{E})\otimes\omega_{\Sigma_{2}}\simeq\mathcal{H}^{i}(\mathcal{E})$.
Thus we see

$\operatorname{Ext}_{\Sigma_{2}}^{1}(\mathcal{H}^{i_{0}}(\mathcal{E}),\mathcal{H}^{i}(\mathcal{E}))\stackrel{{\scriptstyle\text{Serre
duality}}}{{\simeq}}\operatorname{Ext}_{\Sigma_{2}}^{1}(\mathcal{H}^{i}(\mathcal{E}),\mathcal{H}^{i_{0}}(\mathcal{E}))^{\vee}\stackrel{{\scriptstyle\text{Lemma~{}\ref{lm:properties
of cohomology sheaves} \eqref{it:cohomology(E) is rigid}}}}{{=}}0.$

Since $E$ is a direct summand of $\mathcal{H}^{i_{0}}(\mathcal{E})$, this
implies
$\operatorname{Ext}_{\Sigma_{2}}^{1}(E,\mathcal{H}^{i}(\mathcal{E}))=0,\operatorname{Ext}_{\Sigma_{2}}^{1}(\mathcal{H}^{i}(\mathcal{E}),E)=0$.
Moreover, since $\dim\operatorname{Supp}\mathcal{H}^{i}(\mathcal{E})=1$, the
local to global spectral sequence for Ext groups implies

$\displaystyle\begin{aligned}
H^{1}(\Sigma_{2},\mathop{{\mathcal{H}}om}\nolimits_{\Sigma_{2}}(E,\mathcal{H}^{i}(\mathcal{E})))=0,\\\
H^{1}(\Sigma_{2},\mathop{{\mathcal{H}}om}\nolimits_{\Sigma_{2}}(\mathcal{H}^{i}(\mathcal{E}),E))=0.\end{aligned}$
(3.66)

On the other hand, by Proposition 3.18 and the rigidity, there is a vector
bundle $\mathcal{V}$ on $C$ such that
$\mathcal{H}^{i}(\mathcal{E})\simeq\iota_{\ast}\mathcal{V}$. Hence there are
isomorphisms as follows.

$\displaystyle\begin{aligned}
\mathop{{\mathcal{H}}om}\nolimits_{\Sigma_{2}}(E,\mathcal{H}^{i}(\mathcal{E}))\simeq\iota_{\ast}\mathop{{\mathcal{H}}om}\nolimits_{C}(E|_{C},\mathcal{V})\stackrel{{\scriptstyle\eqref{eq:E|C
= OC(a+1) oplus
ell}}}{{\simeq}}\iota_{\ast}\mathop{{\mathcal{H}}om}\nolimits_{C}(\mathcal{O}_{C}(a+1),\mathcal{V})^{\oplus
r}\\\
\mathop{{\mathcal{H}}om}\nolimits_{\Sigma_{2}}(\mathcal{H}^{i}(\mathcal{E}),E)\stackrel{{\scriptstyle\eqref{eq:universal
extension}}}{{\simeq}}\mathop{{\mathcal{H}}om}\nolimits_{\Sigma_{2}}(\mathcal{H}^{i}(\mathcal{E}),\mathcal{O}_{C}(a))^{\oplus
d}\simeq\iota_{\ast}\mathop{{\mathcal{H}}om}\nolimits_{C}(\mathcal{V},\mathcal{O}_{C}(a))^{\oplus
d}\end{aligned}$ (3.67)

Combining these isomorphisms with (3.66), we obtain the following vanishings.

$\displaystyle\begin{aligned}
H^{1}(C,\mathop{{\mathcal{H}}om}\nolimits_{C}(\mathcal{O}_{C}(a+1),\mathcal{V})=0\\\
H^{1}(C,\mathop{{\mathcal{H}}om}\nolimits_{C}(\mathcal{V},\mathcal{O}_{C}(a))=0\end{aligned}$
(3.68)

From this we deduce that $\mathcal{V}$ is of the form

$\mathcal{V}\simeq\mathcal{O}_{C}(a)^{\oplus
s_{i}}\oplus\mathcal{O}_{C}(a+1)^{\oplus t_{i}}$

for some $s_{i}$ and $t_{i}$, concluding the proof. ∎

###  3.4. Derived dual of exceptional objects

Let $\mathcal{E}$ be an exceptional object which is not isomorphic to a shift
of a vector bundle, and let $E=E(\mathcal{E})$ be the exceptional sheaf in
Definition 3.22. In what follows we will mainly discuss the case where
$\operatorname{tors}E\neq 0$. If $\operatorname{tors}E=0$ ($\iff$ $E$ is a
vector bundle), we will replace $\mathcal{E}$ with its derived dual
$\mathcal{E}^{\vee}$ and reduce the problem to the main case. What we mean by
this will be made precise by Corollary 3.27.

######  Lemma 3.25.

For an exceptional object $\mathcal{E}$ on $\Sigma_{2}$, the cohomology
sheaves of the derived dual $\mathcal{E}^{\vee}$ are related to those of
$\mathcal{E}$ as follows:

  * •

$i_{0}(\mathcal{E}^{\vee})=-i_{0}$.

  * •

If $i\neq-i_{0}$, then
$\mathcal{H}^{i}(\mathcal{E}^{\vee})\simeq\mathop{{\mathcal{E}}xt}\nolimits_{\Sigma_{2}}^{1}(\mathcal{H}^{-i+1}(\mathcal{E}),\mathcal{O}_{\Sigma_{2}})$.

  * •

For $i=-i_{0}$, $\mathcal{H}^{-i_{0}}(\mathcal{E}^{\vee})$ fits into an exact
sequence

$0\to\mathop{{\mathcal{E}}xt}\nolimits_{\Sigma_{2}}^{1}(\mathcal{H}^{i_{0}+1}(\mathcal{E}),\mathcal{O}_{\Sigma_{2}})\to\mathcal{H}^{-i_{0}}(\mathcal{E}^{\vee})\to\mathop{{\mathcal{H}}om}\nolimits_{\Sigma_{2}}(\mathcal{H}^{i_{0}}(\mathcal{E}),\mathcal{O}_{\Sigma_{2}})\to
0.$

  * •

For $i=-i_{0}+1$, the cohomology sheaf can also be written as

$\mathcal{H}^{-i_{0}+1}(\mathcal{E}^{\vee})\simeq\mathop{{\mathcal{E}}xt}\nolimits_{\Sigma_{2}}^{1}(\operatorname{tors}\mathcal{H}^{i_{0}}(\mathcal{E}),\mathcal{O}_{\Sigma_{2}})\simeq\left(\operatorname{tors}\mathcal{H}^{i_{0}}(\mathcal{E})\right)^{\vee}[1].$

###### Proof.

Consider the following spectral sequence.

$E_{2}^{p,q}=\mathop{{\mathcal{E}}xt}\nolimits_{\Sigma_{2}}^{p}(\mathcal{H}^{-q}(\mathcal{E}),\mathcal{O}_{\Sigma_{2}})\Rightarrow\mathcal{H}^{p+q}(\mathcal{E}^{\vee})$

Since $\Sigma_{2}$ is a smooth projective surface, $E_{2}^{p,q}=0$ if $p<0$ or
$p>2$. Since $\mathcal{H}^{i}(\mathcal{E})$ is torsion for $i\neq i_{0}$ by
Lemma 3.3 (2), $E_{2}^{0,q}=0$ for $q\neq-i_{0}$. Moreover, for any $i\neq
i_{0}$, $\mathcal{H}^{i}(\mathcal{E})$ is pure by Lemma 3.3 (2). Furthermore,
$\operatorname{tors}\mathcal{H}^{i_{0}}(\mathcal{E})$ is pure again by Lemma
3.3 (2) and
$\mathcal{H}^{i_{0}}(\mathcal{E})/\operatorname{tors}\mathcal{H}^{i_{0}}(\mathcal{E})$
is locally free by Lemma 3.3 (3). Hence by [HL10, Theorem 1.1.10,
$1)\Rightarrow 2)$], $E_{2}^{2,q}=0$ for any $q\in\mathbb{Z}$.

Summing up, we see that $E_{2}^{p,q}\neq 0$ only if $(p,q)=(0,-i_{0})$ or
$p=1$. In particular, this spectral sequence is $E_{2}$-degenerate. All
assertions follow from these observations. ∎

######  Lemma 3.26.

Let $\mathcal{E}$ be an exceptional object such that both $E=E(\mathcal{E})$
and $E(\mathcal{E}^{\vee})$ are vector bundles. Then $\mathcal{E}\simeq
E[-i_{0}]$.

###### Proof.

We consider the spectral sequence

$E_{2}^{p,q}=\operatorname{Ext}_{\Sigma_{2}}^{p}(E,\mathcal{H}^{q}(\mathcal{E}))\Rightarrow\operatorname{Hom}^{p+q}(E,\mathcal{E}).$

As $\Sigma_{2}$ is a smooth surface, $E_{2}^{p,q}\neq 0$ only if $0\leq p\leq
2$. In particular, it is $E_{3}$-degenerate. Since
$E_{2}^{2,i_{0}-1}\simeq\operatorname{Hom}_{\Sigma_{2}}(\mathcal{H}^{i_{0}-1}(\mathcal{E}),E)^{\vee}=0$
for $\mathcal{H}^{i_{0}-1}(\mathcal{E})$ being torsion and $E$ being torsion
free, it follows that $d_{2}^{0,i_{0}}=0$ and hence

$\displaystyle E_{2}^{0,i_{0}}=\ker d_{2}^{0,i_{0}}\simeq
E_{3}^{0,i_{0}}\simeq E_{\infty}^{0,i_{0}}.$ (3.69)

Take any

$[\varphi\colon
E[-i_{0}]\to\mathcal{E}]\in\operatorname{Ext}^{i_{0}}(E,\mathcal{E})$

whose image under the surjection
$\operatorname{Ext}^{i_{0}}(E,\mathcal{E})\twoheadrightarrow
E_{\infty}^{0,i_{0}}$ of the spectral sequence, which in fact is
$\mathcal{H}^{i_{0}}(\varphi)$, corresponds to the natural inclusion
$E\hookrightarrow\mathcal{H}^{i_{0}}(\mathcal{E})$ under the isomorphisms
(3.69). Then the derived dual

$\varphi^{\vee}\colon\mathcal{E}^{\vee}\to E^{\vee}[i_{0}]$

induces the surjection

$\mathcal{H}^{-i_{0}}(\mathcal{E}^{\vee})\to\mathop{{\mathcal{H}}om}\nolimits_{\Sigma_{2}}(\mathcal{H}^{i_{0}}(\mathcal{E}),\mathcal{O}_{\Sigma_{2}})\simeq
E^{\vee}$

in Lemma 3.25. As we assumed that $E(\mathcal{E}^{\vee})\simeq E^{\vee}$ is
torsion free, by applying what we have just confirmed to $\mathcal{E}^{\vee}$,
we similarly obtain a morphism

$\psi\colon E^{\vee}[i_{0}]\to\mathcal{E}^{\vee}$

such that $\mathcal{H}^{-i_{0}}(\psi)$ is the inclusion $E^{\vee}\simeq
E(\mathcal{E}^{\vee})\hookrightarrow\mathcal{H}^{-i_{0}}(\mathcal{E}^{\vee})$
as a direct summand. Then it follows that the composite
$\varphi^{\vee}\circ\psi$ is an automorphism of $E^{\vee}[i_{0}]$ and
therefore $\mathcal{E}^{\vee}$ splits as a direct sum of $E^{\vee}[i_{0}]$ and
$\operatorname{Cone}\psi$. Since the exceptional object $\mathcal{E}^{\vee}$
is indecomposable, this implies $\operatorname{Cone}\psi=0$ and hence
$\mathcal{E}^{\vee}\simeq E^{\vee}[i_{0}]$, which is equivalent to
$\mathcal{E}\simeq E[-i_{0}]$. ∎

Thus we immediately obtain the following

######  Corollary 3.27.

Let $\mathcal{E}$ be an exceptional object which is not isomorphic to a shift
of a vector bundle. If $E=E(\mathcal{E})$ is torsion free, then
$\operatorname{tors}E(\mathcal{E}^{\vee})\neq 0$.

###  3.5. Length of the torsion part

Now we introduce the notion of length, which measures for an exceptional
object the distance from a shift of a vector bundle. The proof of Theorem 3.1
is reduced to the assertion Theorem 3.32 that one can always reduce the length
by an appropriate spherical twist.

######  Definition 3.28.

Let $\gamma$ be the generic point of $C$ and $\mathcal{O}_{\Sigma_{2},\gamma}$
the local ring of $\Sigma_{2}$ at $\gamma$. For a coherent sheaf
$\mathcal{H}$, put

$\ell(\mathcal{H})=\operatorname{length}_{\mathcal{O}_{\Sigma_{2},\gamma}}\operatorname{tors}\mathcal{H}_{\gamma}$

where $\mathcal{H}_{\gamma}$ is the stalk of $\mathcal{H}$ at $\gamma$ and
$\operatorname{length}_{\mathcal{O}_{\gamma}}\operatorname{tors}\mathcal{H}_{\gamma}$
is the length of its torsion part.

For an object $\mathcal{E}\in D(\Sigma_{2})$, we define

$\ell(\mathcal{E})\coloneqq\sum_{i}\ell(\mathcal{H}^{i}(\mathcal{E})).$

Let us give a bit more concrete description for $\ell(\mathcal{E})$. Let
$\mathcal{E}_{\gamma}\in\mathbf{D}(\mathcal{O}_{\Sigma_{2},\gamma})$ be the
pull-back of $\mathcal{E}$ by the flat morphism
$\operatorname{Spec}\mathcal{O}_{\Sigma_{2},\gamma}\to\Sigma_{2}$. The length
function $\ell$ is defined for objects in
$\mathbf{D}(\mathcal{O}_{\Sigma_{2},\gamma})$ as well, and it immediately
follows from the exactness of the (underived) pull-back functor that
$\ell(\mathcal{E}_{\gamma})=\ell(\mathcal{E})$. On the other hand, since
$\mathcal{O}_{\Sigma_{2},\gamma}$ is a DVR, for
$\mathcal{M}\in\mathbf{D}(\mathcal{O}_{\Sigma_{2},\gamma})$ there is an
isomorphism
$\mathcal{M}\simeq\bigoplus_{i\in\mathbb{Z}}\mathcal{H}^{i}(\mathcal{M})[-i]$
and hence the equality
$\ell(\mathcal{M})=\sum_{i\in\mathbb{Z}}\ell(\mathcal{H}^{i}(\mathcal{M}))$.
By the structure theorem for finitely generated modules over a DVR,
$\mathcal{H}^{i}(\mathcal{M})$ is a direct sum of finite copies of
$\mathcal{O}_{\Sigma_{2},\gamma}$ and
$\mathcal{O}_{\Sigma_{2},\gamma}/(t^{p})$ for various $p\geq 1$, where $t$ is
a generator of the maximal ideal. Based on this, we obtain the following
invariance.

######  Lemma 3.29.

For any object $\mathcal{E}\in\mathbf{D}(\Sigma_{2})$,
$\ell(\mathcal{E}^{\vee})=\ell(\mathcal{E})$.

###### Proof.

Note first that
$\ell(\mathcal{E}^{\vee})=\ell((\mathcal{E}^{\vee})_{\gamma})=\ell((\mathcal{E}_{\gamma})^{\vee})$.
Hence it is enough to show $\ell(\mathcal{M}^{\vee})=\ell(\mathcal{M})$ for
all $\mathcal{M}\in\mathbf{D}(\mathcal{O}_{\Sigma_{2},\gamma})$. By the
explicit descriptions of $\mathcal{M}$ we gave above, it is enough to show
this for $\mathcal{M}=\mathcal{O}_{\Sigma_{2},\gamma}/(t^{p})$. In this case
one can easily confirm $\mathcal{M}^{\vee}\simeq\mathcal{M}[-1]$, so we are
done. ∎

If $\mathcal{E}$ is an exceptional object with
$\operatorname{Supp}(\mathcal{H}^{i_{0}})=\Sigma_{2}$, then Proposition 3.18
implies

$\displaystyle\begin{aligned}
\ell(\mathcal{E})&=\sum_{i}\operatorname{rank}_{C}\operatorname{tors}(\mathcal{H}^{i}(\mathcal{E}))\\\
&=\sum_{i\neq
i_{0}}\operatorname{rank}_{C}\mathcal{H}^{i}(\mathcal{E})+\operatorname{rank}_{C}\operatorname{tors}\mathcal{H}^{i_{0}}(\mathcal{E}),\end{aligned}$
(3.70)

where $\operatorname{rank}_{C}$ denotes the rank of a coherent sheaf on $C$.
Note that thus defined $\ell(\mathcal{E})$ is the same as $\ell(\mathcal{E})$
in Corollary 3.23. From this we immediately obtain the following
characterization of exceptional vector bundles among exceptional objects.

######  Lemma 3.30.

An exceptional object $\mathcal{E}$ is isomorphic to a shift of a vector
bundle if and only if $\ell(\mathcal{E})=0$.

The following (sub)additivity of the length function with respect to short
exact sequences will be useful later.