# Exceptional collections and phantoms of special Dolgachev surfaces

Ilya Karzhemanov
Laboratory of AGHA, Moscow Institute of Physics and Technology, 9 Institutskiy
per., Dolgoprudny, Moscow Region, 141701, Russia
E-mail address<EMAIL_ADDRESS>and  Ludmil Katzarkov
University of Miami, Coral Gables, FL
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad.
Georgi Bonchev Str., Block 8, Sofia, 1113, Bulgaria
National Research University Higher School of Economics, Laboratory of Mirror
Symmetry, NRU HSE, 6 Usacheva str., Moscow, 119048, Russia
E-mail address<EMAIL_ADDRESS>
###### Abstract.

We provide an explicit description of exceptional collection of maximal length
in the derived category $D^{b}(Y)$ for a particular class of elliptic surfaces
$Y$. The existence of non - trivial semiorthogonal complement (a ‘‘ phantom
’’) of this collection is also established.

MS 2010 classification: 14J27, 18E30, 14J81

Key words: elliptic surface, derived category, phantom

##  1\. Introduction

Let $Y$ be a smooth projective surface over $\mathbb{C}$.1)1)1) We will use
standard notions and results (although we recall some of them in the text)
from [32], [7], [11] concerning compact complex surfaces. Assume that
$q(Y)=p_{g}(Y)=0$ (i. e. irregularity and geometric genus of $Y$ are trivial).
Consider the derived category $D^{b}(Y)$ of coherent sheaves on $Y$. Then
Theorem 6 in [53] asserts that any numerically exceptional collection (if
exists) $A_{1},\ldots,A_{N}$ of maximal length $N=c_{2}(Y)$ of objects
$A_{i}\in D^{b}(Y)$ (see 2.4 and 2.8 below for a setup) yields a similar
collection of _line bundles_ (again denoted $A_{i}$) on $Y$, so that the Chow
lattice $N^{1}(Y)$ of $1$ \- cycles on $Y$ is generated by $A_{i}$ and is
unimodular with respect to the intersection pairing. The aim of the present
paper is to apply this fact in order to

$\begin{array}[]{c}\textit{find \emph{explicit} exceptional collections of
maximal length on certain (minimal) elliptic surfaces $Y$.}\end{array}$

Our candidate for $Y$, allowing an algebro - geometric construction, will be
members of a family of the so - called _Dolgachev surfaces_ (see 2.1). In this
case, direct calculation yields a numerically exceptional collection of
maximal length on $Y$ (see 2.4 and Theorem 2.10), and an additional geometric
reasoning shows that the given collection is actually exceptional (see 2.8).
We apply these constructions in order to find a _phantom category_ in
$D^{b}(Y)$ (see Section 3). To our best knowledge this is the most explicit
description of the phantom phenomenon for $D^{b}$ of algebraic surfaces having
positive Kodaira dimension (compare with [21], [22], [23], [17] and [49]).

Section 4 generalizes the preceding algebro - geometric considerations and
treats the problem of maximal length exceptional collections on Dolgachev
surfaces $\mathfrak{S}$ in _transcendental_ terms (Corollary 4.15 together
with [15, Lemma 4.3] also prove the existence of phantoms in
$D^{b}(\mathfrak{S})$ for a particular class of $\mathfrak{S}$ related to
$Y$). Appendix at the end contains a non - technical discussion of relations
with other work and some further generalizations (yet this section is a bit ‘‘
heavy ’’ for Introduction and an interested reader may turn to it, if need be,
after reading the rest of the paper).

##  2\. Explicit constructions

###  2.1. Dolgachev’s construction

Firstly, one starts with two lines $L_{1},L_{2}\subset\mathbb{P}^{2}$ and a
plane quartic curve $Q$ (to be specified). In projective coordinates
$[x:y:z]$, let the equations of $L_{1}$ and $L_{2}$ be $z=0$ and $y=0$,
respectively. Further, one considers the points

$q_{1}:=[1:0:0],\qquad q_{5}:=[0:0:1],\qquad q_{9}:=[1:0:1]$

and some additional points

$q_{1}>q_{2}>q_{3}>q_{4},\qquad q_{5}>q_{6}>q_{7}>q_{8},\qquad q_{9},$

where ‘‘ $q_{i}>q_{i+1}$ ’’ signifies that the point $q_{i+1}$ is infinitely
close to $q_{i}$. Then $Q$ is chosen to have $q_{5},q_{6},q_{7}$ as double
points and pass through all other $q_{i}$ (see Figure 1). Furthermore, $L_{1}$
passes through $q_{1},q_{2},q_{3}$, while $L_{2}$ passes through
$q_{1},q_{5},q_{9}$. The equation of $Q$ can be taken in the form

$(xz+y^{2})^{2}-x^{3}z+ax^{2}yz+axy^{3}=0$

for any $a\in\mathbb{C}$. Finally, one picks a smooth cubic curve $C$, which
passes through all $q_{i}$, with equation being

$xz^{2}-x^{2}z+a^{\prime}xyz+y^{2}z+ay^{3}=0,$

where $a,a^{\prime}$ satisfy the relation
$(a^{\prime}-a)^{2}-a(a^{\prime}-a)+1=0$.

Figure 1.

From the data just presented one constructs a (Halphen) pencil

(2.2) $\lambda
yz((xz+y^{2})^{2}-x^{3}z+ax^{2}yz+axy^{3})+\mu(xz^{2}-x^{2}z+a^{\prime}xyz+y^{2}z+ay^{3})^{2}=0,\qquad\lambda,\mu\in\mathbb{C},$

of plane curves. Let $X$ be the surface obtained by blowing up
$\mathbb{P}^{2}$ at nine points $q_{i}$. Then (2.2) yields an elliptic
fibration $p_{\scriptscriptstyle X}:X\longrightarrow\mathbb{P}^{1}$. It has
unique multiple fiber $2C$ (we identify the cubic $C\subset\mathbb{P}^{2}$
with its proper transform on $X$). Also, if $R_{1},\ldots,R_{9}$ are the
exceptional curves of the contraction $X\longrightarrow\mathbb{P}^{2}$, then
(the proper transforms of) $Q,L_{1}$ and $L_{2}$ together with
$R_{1},R_{2},R_{3},R_{5},R_{6},R_{7}$ form a singular fiber of type $I_{9}$
(cf. [7, V.7, Table 3] and Figure 2 below). Let us determine the rest of the
singular fibers of $p_{\scriptscriptstyle X}$.

Denote by $p:\operatorname{Jac}X\longrightarrow\mathbb{P}^{1}$ the Jacobian
fibration associated with $p_{\scriptscriptstyle X}$.

######  Lemma 2.3.

$p$ has at least $4$ singular fibers.

###### Proof.

Suppose first that $p$ has at most $2$ singular fibers. In this case, the
global monodromy
$\Gamma:=\Gamma_{\operatorname{Jac}X}\subset\operatorname{SL}_{2}(\mathbb{Z})$
(cf. 3.5 below) is isomorphic to $\mathbb{Z}$, which implies that the index
$[\operatorname{SL}_{2}(\mathbb{Z}):\Gamma]$ is infinite. However, since $p$
is (algebraic and) Jacobian, one has
$[\operatorname{SL}_{2}(\mathbb{Z}):\Gamma]<\infty$ (see e. g. [11, Section
3]), a contradiction.

Further, if $F_{1},F_{2},F_{3}$ are the only singular fibers of $p$, then let
$g_{i}\in\operatorname{SL}_{2}(\mathbb{Z})$ be the images of loops around
$p(F_{i})$. Now, since the fundamental group
$\pi_{1}(\mathbb{P}^{1}\setminus\\{p(F_{3})\\})$ is trivial, it follows that
$\Gamma$ is generated by $g_{1},g_{2}$ subject to the relation $g_{1}\cdot
g_{2}=1$, which is again impossible. $\Box$

It follows from Lemma 2.3 that the remaining singular fibers of
$p_{\scriptscriptstyle X}$ are all of type $I_{1}$ (cf. Figure 2). Indeed,
logarithmic transformation from $\operatorname{Jac}X$ to $X$ induces an
analytic isomorphism away from the unique multiple fiber on $X$ (see [7,
V.13]), where the latter fiber is supported on a smooth curve isomorphic to
$C$ (cf. (2.2)). Then from $c_{2}(X)=c_{2}(\operatorname{Jac}X)=12$ we obtain

$12=c_{2}(X)=\sum_{b\in\mathbb{P}^{1}\,:\,p^{-1}(p)=p_{\scriptscriptstyle
X}^{-1}(b):=F_{b}\,\text{is singular}}\chi_{\text{top}}(F_{b}),$

with the fiber $I_{9}$ having $\chi_{\text{top}}=9$ and
$\chi_{\text{top}}(F_{b})>0$ for at least _three more_ $b$, so that all the
corresponding $F_{b}$ are of type $I_{1}$ and one gets exactly three of them.

Figure 2.

Fix some general fiber $F$ of $p_{\scriptscriptstyle X}$. One computes the
divisor

$D:=F+(2L_{1}+R_{3})+(2R_{1}+L_{2})+(2R_{6}+R_{7})$

is divisible by $3$ in $\operatorname{Pic}X$. Then we can form a $3:1$ cyclic
covering $X^{\prime}\longrightarrow X$ with $D$ as its ramification locus.
Furthermore, the curves $L_{1},R_{3},\ldots,R_{7}$ in $D$ can be contracted by
some $\phi:X\longrightarrow W$ to three $\displaystyle\frac{1}{3}(1,2)$ \-
points $O_{1},O_{2},O_{3}\in W$.2)2)2) This is standard via the construction
of a nef and big divisor $\mathcal{L}$ on $S$ satisfying $\mathcal{L}\cdot
L_{1}=\mathcal{L}\cdot R_{3}=\ldots=\mathcal{L}\cdot R_{7}=0$, so that the
linear system $|m\mathcal{L}|$ is basepoint - free, $m\gg 1$ (compare with
[31, Section 3]). The surface $W$ is algebraic and $X^{\prime}\longrightarrow
X$ descends to a $3:1$ cyclic covering $\psi:Y\longrightarrow W$ ramified at
$O_{i}$ and $F$ (we denote every cycle $\psi^{*}\phi_{*}Z$ on $Y$, where
$Z\in\operatorname{Pic}X$, again by $Z$). Namely, we have
$X^{\prime}={\text{Spec
}}_{X}\displaystyle\bigoplus_{i=0}^{2}\mathcal{O}_{X}(-iD)$, whereas
$Y={\text{Spec
}}_{W}\displaystyle\bigoplus_{i=0}^{2}\phi_{*}\mathcal{O}_{X}(-iD)$ for the
coherent sheaves $\phi_{*}\mathcal{O}_{X}(-iD)$ on $W$ (cf. Figure 3).

Figure 3.

###  2.4. The Picard lattice of $Y$

Let $\mu_{3}$ be the Galois group of the above covering $\psi:Y\longrightarrow
W$. Note that the surface $Y$ is smooth because
$\phi_{*}F\cap\operatorname{Sing}W=\emptyset$ and locally analytically over
every $O_{i}\in W$, for $\operatorname{Sing}W=\\{O_{1},O_{2},O_{3}\\}$,
morphism $\psi$ coincides with the quotient
$\mathbb{C}^{2}\longrightarrow\mathbb{C}^{2}/\mu_{3}$. Construction in 2.1
also provides an evident elliptic fibration $p_{\scriptscriptstyle
Y}:Y\longrightarrow\mathbb{P}^{1}$, with fibers $\Phi_{1},\ldots,\Phi_{4}$ of
type $I_{3}$ and two multiple fibers supported on the smooth elliptic curves
$C$ and $F$, respectively, such that $2C\equiv 3F\equiv\Phi_{i}$ (numerically
on $Y$) — these are the only singular fibers of $p_{\scriptscriptstyle Y}$
(cf. Lemma 2.3). Recall that

(2.5) $K_{Y}=2F-C\equiv\frac{1}{6}\,\Phi_{i}$

(see e. g. [7, V.12]).

######  Proposition 2.6.

Let $R_{2},R_{5},\ldots,G_{2}$ be the curves as on Figure 3. Then their
intersection data is given by Table A below.

###### Proof.

Note that the cycle

$\gamma:=R_{8}+\frac{2}{3}\,R_{7}+\frac{1}{3}\,R_{6}$

on $X$ is relatively trivial with respect to $\phi:X\longrightarrow W$ (for
example, $\gamma\cdot L_{2}=0$ because $L_{2}\cdot R_{8}=L_{2}\cdot
R_{7}=L_{2}\cdot R_{6}=0$, and so on). Then we have

$(R_{8}^{2}):=3(\gamma^{2})=-1$

on $Y$ by definition. It also follows from (2.5) that

$K_{Y}\cdot R_{8}:=3(\frac{1}{6}\,\Phi_{i}\cdot\gamma)=1.$

Similarly, the cycle

$\delta:=R_{5}+\frac{2}{3}\,L_{2}+\frac{1}{3}\,R_{1}$

is trivial over $W$, which gives

$R_{5}\cdot R_{8}:=3(\delta\cdot\gamma)=1$

on $Y$. In particular, the curve $R_{8}\subset Y$ is irreducible, which
implies that $R_{8}\cdot H_{1}=R_{8}\cdot H_{2}=2$ because $R_{8}$ is
$\mu_{3}$ \- invariant and $\mu_{3}$ permutes $H_{i}$. Other intersections are
treated in exactly the same way and we skip their computation. $\Box$

| $K_{Y}$ | $R_{2}$ | $R_{5}$ | $R_{8}$ | $H_{1}$ | $H_{2}$ | $F_{1}$ | $F_{2}$ | $G_{1}$ | $G_{2}$
---|---|---|---|---|---|---|---|---|---|---
$K_{Y}$ | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0
$R_{2}$ | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$R_{5}$ | 0 | 1 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0
$R_{8}$ | 1 | 0 | 1 | -1 | 2 | 2 | 2 | 2 | 2 | 2
$H_{1}$ | 0 | 0 | 0 | 2 | -2 | 1 | 0 | 0 | 0 | 0
$H_{2}$ | 0 | 0 | 0 | 2 | 1 | -2 | 0 | 0 | 0 | 0
$F_{1}$ | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 1 | 0 | 0
$F_{2}$ | 0 | 0 | 0 | 2 | 0 | 0 | 1 | -2 | 0 | 0
$G_{1}$ | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 1
$G_{2}$ | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 1 | -2

Table A

######  Notation.

In what follows, for notation’s simplicity, we will be usually neglecting the
use of $\equiv$ , thus identifying any element from $\operatorname{Pic}Y$ with
its class in $H^{2}(Y,\mathbb{Z})$. However, in some cases with a possibility
of confusion, we will distinguish between $\equiv$ and $=$. We will also
specify sometimes the linear equivalence $\sim$ between Cartier divisors.

######  Lemma 2.7.

The classes

$\begin{array}[]{ccc}\vskip 12.0pt plus 4.0pt minus 4.0ptK_{Y},\ R_{2},\
R_{5},\ R_{8},\ H_{1},\ H_{2},\
F_{1},-\frac{1}{3}\,K_{Y}-\frac{1}{3}\,R_{2}+\frac{1}{3}\,R_{5}-\frac{1}{3}\,F_{1}+\frac{1}{3}\,F_{2}-\frac{1}{3}\,H_{1}+\frac{1}{3}\,H_{2},\\\
\vskip 12.0pt plus 4.0pt minus
4.0pt-\frac{1}{3}\,K_{Y}-\frac{1}{3}\,R_{2}+\frac{1}{3}\,R_{5}+\frac{1}{3}\,F_{1}-\frac{1}{3}\,F_{2}+\frac{1}{3}\,G_{1}-\frac{1}{3}\,G_{2},\
G_{2}\end{array}$

form a basis of the lattice $\operatorname{Pic}Y$.

###### Proof.

Note that the Gramm matrix $A$ of the basis
$\left\\{K_{Y},R_{2},R_{5},R_{8},H_{1},H_{2},F_{1},F_{2},G_{1},G_{2}\right\\}$
of $\operatorname{Pic}Y\otimes\mathbb{R}$, given by Table A, has determinant
$-81$. In turn, the matrix $M$, transforming this basis to the one proposed by
lemma, is

$\left(\begin{array}[]{cccccccccc}1&0&0&0&0&0&0&-\frac{1}{3}&-\frac{1}{3}&0\\\
0&1&0&0&0&0&0&-\frac{1}{3}&-\frac{1}{3}&0\\\
0&0&1&0&0&0&0&\frac{1}{3}&\frac{1}{3}&0\\\ 0&0&0&1&0&0&0&0&0&0\\\
0&0&0&0&1&0&0&-\frac{1}{3}&0&0\\\ 0&0&0&0&0&1&0&\frac{1}{3}&0&0\\\
0&0&0&0&0&0&1&-\frac{1}{3}&\frac{1}{3}&0\\\
0&0&0&0&0&0&0&\frac{1}{3}&-\frac{1}{3}&0\\\ 0&0&0&0&0&0&0&0&\frac{1}{3}&0\\\
0&0&0&0&0&0&0&0&-\frac{1}{3}&1\\\ \end{array}\right)$

and has $\det M=1/9$.3)3)3) All linear algebra computations in the paper are
carried via PARI/GP. Then it is immediate to check that
$M^{t}AM\in\text{SL}_{10}(\mathbb{Z})$ for the transposed matrix $M^{t}$ of
$M$. This means the claimed basis yields a type I unimodular lattice of
signature $(1,9)$. Finally, the latter coincides with $\operatorname{Pic}Y$ by
uniqueness of such lattices, which concludes the proof. $\Box$

###  2.8. One exceptional collection on $Y$

Consider the bounded derived category $D^{b}(Y)$ of coherent sheaves on $Y$
(for the definitions that follow one may take $Y$ to be any smooth projective
variety). This is a triangulated category, so that the notions of exceptional
object/collection, semiorthogonal decomposition, etc. make sense (see [14] and
[13] for foundations).

Recall that an object $A\in D^{b}(Y)$ is called _exceptional_ if
$\operatorname{Hom}(A,A)=\mathbb{C}$ and $\operatorname{Ext}^{i}(A,A)=0$ for
all $i\neq 0$. More generally, a collection $A_{1},\ldots,A_{N}$ of
exceptional objects is called exceptional if
$\operatorname{Ext}^{{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}}(A_{i},A_{j})=0$
for all $i>j$. Given any such collection, one has a _semiorthogonal
decomposition (SOD)_

$D^{b}(Y)=\left<\mathcal{A},A_{1},\ldots,A_{N}\right>,$

where $\mathcal{A}:=\left<A_{1},\ldots,A_{N}\right>^{\bot}:=\left\\{A\in
D^{b}(Y)\ |\
\operatorname{Ext}^{{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}}(A_{i},A)=0,1\leqslant
i\leqslant N\right\\}$. The collection is called _full_ if $\mathcal{A}=0$.
(N. B. In the previous setting, $A_{i}$ actually denotes the _category_
generated by the corresponding object, $1\leqslant i\leqslant N$. The notions
of SOD and of being full extend verbatim to the case of arbitrary strictly
full triangulated subcategories $A_{i}$.)

######  Remark 2.9 (cf. [22, Section 2] or [47]).

Taking the _Chern character_ ch of any $A\in D^{b}(Y)$ delivers a certain
numerical obstruction for being exceptional. Namely, the category $D^{b}(Y)$
becomes replaced by the Abelian group
$K_{0}(Y):=K_{0}(D^{b}(Y))\ni\text{ch}(A)$ (the _K - group_ of $Y$) and the
complex
$\operatorname{Ext}^{{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}}(\star,\star)$
is replaced by
$\chi(\star,\star)=\displaystyle\sum_{k}(-1)^{k}\dim\operatorname{Ext}^{k}(\star,\star)$,
which is a bilinear form on $K_{0}(Y)$. Then the above objects
$A_{1},\ldots,A_{N}$ yield a _numerically exceptional collection_
$\text{ch}(A_{1}),\ldots,\text{ch}(A_{N})$, so that the $N\times N$ \- matrix
with entries $\chi(A_{i},A_{j})$ is upper - triangular. Furthermore, if the
collection is (numerically) full, then $K_{0}(Y)$ is a free group of rank
$N=\dim HH_{{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}}(D^{b}(Y))=c_{2}(Y)$,
$\text{ch}(A_{i})$ generate $K_{0}(Y)$ and the form $\chi(\star,\star)$ is
unimodular. (Note also that our surface $Y$ admits a numerically exceptional
collection of _maximal length_ $c_{2}(Y)=12$ according to [53, Theorem 6]. The
latter is constructed in [53, 3.1] by first finding a diagonal basis
$A_{2},\ldots,A_{11}$ for the intersection form on $\operatorname{Pic}Y$ (cf.
Lemma 2.7 above) such that $K_{Y}=\displaystyle\sum_{i=2}^{10}A_{i}-3A_{11}$,
and then taking $A_{1}:=\mathcal{O}_{Y}$, $A_{12}:=2A_{11}$ to be the rest of
the pertinent collection. Finally, we observe that if the collection
$A_{1},\ldots,A_{12}$ is indeed exceptional, then the HKR - isomorphism
$HH_{k}(Y)\simeq\bigoplus_{q-p=k}H^{p,q}(Y)$ yields
$HH_{{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}}(\mathcal{A})=0$ for higher
Hochschield homology, whereas the equality $K_{0}(\mathcal{A})=0$ is clear.)

######  Theorem 2.10.

The line bundles

$\begin{array}[]{ccc}\vskip 12.0pt plus 4.0pt minus
4.0ptA_{1}:=\mathcal{O}_{Y},\
A_{2}:=-\frac{8}{3}\,K_{Y}-\frac{2}{3}\,R_{2}-\frac{1}{3}\,R_{5}-R_{8}-\frac{2}{3}\,H_{1}+\frac{2}{3}\,H_{2}-\frac{2}{3}\,F_{1}+\frac{2}{3}\,F_{2},\\\
\vskip 12.0pt plus 4.0pt minus
4.0ptA_{3}:=\frac{2}{3}\,K_{Y}-\frac{1}{3}\,R_{2}+\frac{1}{3}\,R_{5}-R_{8}-H_{1}-\frac{2}{3}\,F_{1}+\frac{2}{3}\,F_{2}-\frac{2}{3}\,G_{1}-\frac{1}{3}\,G_{2},\\\
\vskip 12.0pt plus 4.0pt minus
4.0ptA_{4}:=\frac{5}{3}\,K_{Y}-\frac{1}{3}\,R_{2}-\frac{2}{3}\,R_{5}-R_{8}-H_{1}-\frac{2}{3}\,F_{1}+\frac{2}{3}\,F_{2}-\frac{2}{3}\,G_{1}-\frac{1}{3}\,G_{2},\\\
\vskip 12.0pt plus 4.0pt minus
4.0ptA_{5}:=-R_{8}-\frac{1}{3}\,H_{1}+\frac{1}{3}\,H_{2}-\frac{2}{3}\,F_{1}+\frac{2}{3}\,F_{2}-\frac{1}{3}\,G_{1}-\frac{2}{3}\,G_{2},\\\
\vskip 12.0pt plus 4.0pt minus
4.0ptA_{6}:=-\frac{5}{3}\,K_{Y}-\frac{2}{3}\,R_{2}-\frac{1}{3}\,R_{5}-R_{8}-\frac{1}{3}\,H_{1}+\frac{1}{3}\,H_{2}-F_{1}+F_{2}-\frac{2}{3}\,G_{1}-\frac{1}{3}\,G_{2},\\\
\vskip 12.0pt plus 4.0pt minus
4.0ptA_{7}:=-\frac{5}{3}\,K_{Y}-\frac{2}{3}\,R_{2}-\frac{1}{3}\,R_{5}-R_{8}-\frac{1}{3}\,H_{1}+\frac{1}{3}\,H_{2}+F_{2}-\frac{2}{3}\,G_{1}-\frac{1}{3}\,G_{2},\\\
\vskip 12.0pt plus 4.0pt minus
4.0ptA_{8}:=2K_{Y}+\frac{1}{3}\,H_{1}-\frac{1}{3}\,H_{2}-\frac{1}{3}\,F_{1}-\frac{2}{3}\,F_{2}+\frac{1}{3}\,G_{1}+\frac{2}{3}\,G_{2},\\\
\vskip 12.0pt plus 4.0pt minus
4.0ptA_{9}:=-2K_{Y}-R_{8}-\frac{1}{3}\,H_{1}+\frac{1}{3}\,H_{2}-\frac{2}{3}\,F_{1}+\frac{2}{3}\,F_{2}-\frac{1}{3}\,G_{1}+\frac{1}{3}\,G_{2},\\\
\vskip 12.0pt plus 4.0pt minus
4.0ptA_{10}:=\frac{1}{3}\,K_{Y}-\frac{2}{3}\,R_{2}-\frac{1}{3}\,R_{5}-R_{8}-\frac{2}{3}\,H_{1}-\frac{1}{3}\,H_{2}-\frac{2}{3}\,F_{1}+\frac{2}{3}\,F_{2},\\\
\vskip 12.0pt plus 4.0pt minus 4.0ptA_{11}:=6K_{Y}+R_{8}+H_{1}-2F_{2},\\\
\vskip 12.0pt plus 4.0pt minus
4.0ptA_{12}:=\frac{7}{3}\,K_{Y}-\frac{2}{3}\,R_{2}-\frac{1}{3}\,R_{5}-R_{8}-\frac{1}{3}\,H_{1}+\frac{1}{3}\,H_{2}-F_{1}-\frac{2}{3}\,G_{1}-\frac{1}{3}\,G_{2}\end{array}$

form an exceptional collection in $D^{b}(Y)$ of length $12$.4)4)4) The fact
that this collection is _numerically_ exceptional follows by direct
computation with PARI/GP. Note however that in our special situation this
construction of a maximal length numerically exceptional collection is
different from that described in Remark 2.9.

###### Proof.

Recall that
$\operatorname{Ext}^{{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}}(A_{i},A_{j}):=H^{{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}}(Y,A_{i}^{*}\otimes
A_{j})$. Here $A_{i}^{*}$ denotes the dual, or negative, of $A_{i}$ (we will
not distinguish between divisors and the corresponding line bundles). On
proving the claimed vanishings for all $i>j$ we will proceed in a case - by -
case manner starting with $j=1$.

One easily checks via Proposition 2.6 that

$\chi(A_{i},A_{1}):=\chi(Y,A_{i}^{*})=0$

for all $1<i\leqslant 12$. Hence in order for
$\operatorname{Ext}^{{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}}(A_{i},A_{1})$
to vanish it suffices to show that $H^{0}(Y,A_{i}^{*})=H^{2}(Y,A_{i}^{*})=0$.

######  Lemma 2.11.

$H^{2}(Y,A_{i}^{*})=0$ for all $i\neq 11$ and
$H^{0}(Y,A_{8}^{*})=H^{0}(Y,A_{11}^{*})=0$.

###### Proof.

Observe that $H^{2}(Y,A_{i}^{*})=H^{0}(Y,K_{Y}+A_{i})$ by Serre duality. Now,
if $H^{2}(Y,A_{i}^{*})\neq 0$ for some $i\neq 8,11$, then there exists a curve
$Z\sim K_{Y}+A_{i}$ on $Y$. At the same time, we have $Z\cdot K_{Y}=-1$ by
definition of $A_{i}$ (cf. Proposition 2.6), a contradiction. Same argument
shows that $H^{0}(Y,A_{11}^{*})=0$.

Further, if $i=8$ and $H^{2}(Y,A_{8}^{*})\neq 0$, then there exists a curve

$Z\sim
K_{Y}+A_{8}=3K_{Y}+\frac{1}{3}\,H_{1}-\frac{1}{3}\,H_{2}-\frac{1}{3}\,F_{1}-\frac{2}{3}\,F_{2}+\frac{1}{3}\,G_{1}+\frac{2}{3}\,G_{2}.$

Note that $Z\cdot H_{1}=Z\cdot G_{2}=-1$ and hence $Z_{1}:=Z-H_{1}-G_{2}$ is
another curve. Moreover, we have $Z_{1}\cdot K_{Y}=0$, which implies that
$Z_{1}$ belongs to fibers of the morphism $p_{\scriptscriptstyle
Y}:Y\longrightarrow\mathbb{P}^{1}$ (cf. 2.4). On the other hand, from

$Z_{1}\cdot R_{8}=3K_{Y}\cdot R_{8}-H_{1}\cdot R_{8}-G_{2}\cdot R_{8}=-1$

we get $R_{8}\subseteq Z_{1}$, a contradiction.

Finally, for $A_{8}^{*}\cdot R_{8}=-2$ and $A_{8}^{*}\cdot K_{Y}=0$ one
obtains $H^{0}(Y,A_{8}^{*})=0$ exactly as above, which concludes the proof.
$\Box$

Fix some $i\neq 11$ and suppose that $H^{0}(Y,A_{i}^{*})\neq 0$ (cf. Lemma
2.11). Let $Z\sim A_{i}^{*}$ be a curve. We compute $Z\cdot K_{Y}=1$ and
$(Z^{2})=-1$ by construction. Then there exists a _unique_ irreducible
component $Z_{h}\subseteq Z$ not contracted by the morphism
$p_{\scriptscriptstyle Y}$.

Let us establish a couple of supplementary results:

######  Lemma 2.12.

In the previous setting, if $Z_{h}\neq Z$, then the curve $Z_{v}:=Z-Z_{h}$
belongs to _reducible_ fibers of $p_{\scriptscriptstyle Y}$ and satisfies
$(Z_{v}^{2})<0$, $Z_{v}\cdot Z\leqslant 0$. Furthermore, we have
$\deg\omega_{\scriptscriptstyle Z_{h}}\geqslant-2$ for the dualizing sheaf of
$Z_{h}$, with equality holding if and only if $Z_{h}\simeq\mathbb{P}^{1}$.

###### Proof.

The fact that $Z_{v}$ belongs to the fibers of $p_{\scriptscriptstyle Y}$
follows from (2.5) and $Z_{v}\cdot K_{Y}=0$ for $Z_{h}\cdot K_{Y}=1$. Suppose
next that $Z_{v}\cdot Z>0$. Then, since the intersection form is even and
positive semi - definite on the components of the fibers of
$p_{\scriptscriptstyle Y}$, we obtain $(Z_{v}^{2})\leqslant 0$,

$(Z_{h}^{2})=(Z^{2})-2Z\cdot Z_{v}+(Z_{v}^{2})<-3$

and

$\deg\omega_{\scriptscriptstyle Z_{h}}=(K_{Y}+Z_{h})\cdot Z_{h}<-2.$

On the other hand, we have $\deg\omega_{\scriptscriptstyle
Z_{h}}=2h^{1}(Z_{h},\mathcal{O}_{Z_{h}})-2$ by Riemann – Roch and Serre
duality (see e. g. [7, II.3, II.6]), so that $\deg\omega_{\scriptscriptstyle
Z_{h}}\geqslant-2$, a contradiction (this also proves the last assertion of
lemma).

Finally, if $(Z_{v}^{2})=0$, then $Z_{v}$ consists of (scheme) fibers of
$p_{\scriptscriptstyle Y}$ (cf. [7, III.8, Lemma 8.2 (10)]) and we get
contradiction with $Z_{v}\cdot Z\leqslant 0$ for $Z_{h}$ being a horizontal
component. $\Box$

######  Proposition 2.13.

Normalization of the curve $Z_{h}$ can not be rational or elliptic.

###### Proof.

Let $\mu_{3}$ be the Galois group of the covering $\psi:Y\longrightarrow W$
from 2.1. Then the curves $Z$ and $Z_{h}$ are not $\mu_{3}$ \- invariant,
since $R_{8}$ _is_ $\mu_{3}$ \- invariant and there is no non - trivial linear
relation between the classes of $R_{l},H_{i},F_{h},G_{k}$. Hence the cycle
$\psi^{*}\psi(Z_{h})$ splits on $Y$.

Suppose first that $Z_{h}\simeq\mathbb{P}^{1}$ and $\psi(Z_{h})$ is also
smooth. Associate with the morphism $p_{\scriptscriptstyle
Y}:Y\longrightarrow\mathbb{P}^{1}$ a rational $\mu_{3}$ \- invariant function
$f\in\mathbb{C}(Y)$ in the usual way. This yields the corresponding (elliptic)
morphism $p_{\scriptscriptstyle W}:W\longrightarrow\mathbb{P}^{1}$ such that
$p_{\scriptscriptstyle Y}\big{|}_{Z_{h}}=p_{\scriptscriptstyle
W}\big{|}_{\psi(Z_{h})}$ via $Z_{h}\simeq\psi(Z_{h})$ induced by $\psi$. Note
that $Z_{h}\cdot 3F=6$ by (2.5) and the divisor $(p_{\scriptscriptstyle
W}\big{|}_{\psi(Z_{h})})^{*}(p_{\scriptscriptstyle W}(F))$ is divisible by $3$
because $\psi^{*}\psi(Z_{h})$ splits. Hence $p_{\scriptscriptstyle
Y}\big{|}_{Z_{h}}$ corresponds to the rational function

(2.14)
$f\big{|}_{Z_{h}}=\frac{(t-a)^{3}(t-b)^{3}}{\displaystyle\prod_{i=1}^{3}(t-a_{i})^{2}}$

for some projective coordinate $t$ and $a,b\in F$, $a_{i}\in C$. Let
$Z^{\prime}_{h}\subset\psi^{*}\psi(Z_{h})$ be another irreducible component.
We have $Z^{\prime}_{h}\cap Z_{h}\cap F=\\{a,b\\}$ because $\psi$ (totally)
ramifies on $F$. In particular, $a,b$ are fixed by $\mu_{3}$. We also have

(2.15)
$f\big{|}_{Z^{\prime}_{h}}=\frac{(t^{\prime}-a)^{3}(t^{\prime}-b)^{3}}{\displaystyle\prod_{i=1}^{3}(t^{\prime}-a_{i})^{2}}=f\big{|}_{Z_{h}}$

by construction for some projective coordinate $t^{\prime}$ and $a,b,a_{i}$ as
above. Furthermore, the Möbius transformation $t\mapsto t^{\prime}$ preserves
two points $a,b$ and has the order dividing $3$, which implies that
$t^{\prime}=\varepsilon t$ for $\varepsilon:=\sqrt[3]{1}$. Then from (2.14)
and (2.15) we obtain that $a=0$, $b=\infty$ and $\varepsilon$ permutes
$a_{1},a_{2},a_{3}$. In particular, the set of points $a_{1},a_{2},a_{3}\in C$
is $\mu_{3}$ \- invariant, which implies that $a_{i}$ form an orbit of length
$3$ (cf. Proposition 3.3 below). But then we get
$\psi(a_{1})=\psi(a_{2})=\psi(a_{3})=:c$ by definition of $\psi$, which
implies that

$f\big{|}_{\psi(Z_{h})}=\frac{(t-a)^{3}(t-b)^{3}}{(t-c)^{6}},$

contradicting the fact that $f\big{|}_{Z_{h}}$ is not the $3^{\text{rd}}$
power of a rational function.

Suppose next that $Z_{h}\simeq\mathbb{P}^{1}$ and $\psi(Z_{h})$ is singular.
Then both $Z_{h}$ and $Z^{\prime}_{h}$ are normalizations of $\psi(Z_{h})$.
Again from $Z_{h}\cdot 3F=6$ we deduce the expressions (2.14), (2.15), which
gives $f=[3^{\text{rd}}\ \text{power}]$ exactly as above, a contradiction.
Finally, if $Z_{h}$ is rational and singular, then we have
$\deg\eta^{*}\mathcal{O}_{Z_{h}}(3F)=Z_{h}\cdot 3F=6$ for the normalization
$\eta:Z^{\text{norm}}_{h}\longrightarrow Z_{h}$ (cf. [20, Example 1.2.3]).
Hence all the preceding arguments apply verbatim to the normalization of
$Z_{h}$ and $Z^{\prime}_{h}$.

We now turn to the elliptic case. Let us treat only smooth $Z_{h}$ because the
general case is reduced to this one via the normalization exactly as for
rational $Z_{h}$. Similarly as above, we obtain

(2.16) $3a+3b\sim 2a_{1}+2a_{2}+2a_{3}$

on $Z_{h}$ for some $a,b\in F$, $a_{i}\in C$. Regard $Z_{h}$ as a cubic in
$\mathbb{P}^{2}$. Applying an automorphism one may assume $a\in Z_{h}$ to be
an inflection point. Let $L$ be a linear form defining the tangent line to
$Z_{h}$ at $a$. In addition, denote by $L_{i}$ a linear form defining the
tangent at $a_{i}$, $1\leqslant i\leqslant 3$. Then it follows from (2.16)
that $f\big{|}_{Z_{h}}$ coincides with the rational function
$LQ/(\displaystyle\prod_{i=1}^{3}L_{i})$ on $Z_{h}$, where
$(Q=0)\subset\mathbb{P}^{2}$ is some conic intersecting $Z_{h}$ at $b$ (with
multiplicity $3$) and at the points $Z_{h}\cap L_{i}\setminus\\{a_{i}\\}$,
$1\leqslant i\leqslant 3$. Now, arguing exactly as in the case of rational
$Z_{h}$, we get

$f\big{|}_{Z^{\prime}_{h}}=L^{\prime}Q^{\prime}/(\displaystyle\prod_{i=1}^{3}L^{\prime}_{i})=f\big{|}_{Z_{h}}$

for another irreducible component $Z^{\prime}_{h}\subset\psi^{*}\psi(Z_{h})$.
Here the forms $L^{\prime},Q^{\prime}$ and $L^{\prime}_{i}$ are related with
$L,Q$ and $L_{i}$, respectively, by some projective transformation of
$\mathbb{P}^{2}$ of order $3$, which also fixes $a$ and $b$. In particular,
the set $\\{a_{1},a_{2},a_{3}\\}$ is again a $\mu_{3}$ \- orbit of length $3$,
so that $L_{1}=L_{2}=L_{3}$ as previously (recall that $\psi\big{|}_{Z_{h}}$
is a birational morphism), i. e. $a_{1}=a_{2}=a_{3}$, a contradiction.

Proposition 2.13 is completely proved. $\Box$

######  Corollary 2.17.

The curve $Z$ is _reducible_.

###### Proof.

Indeed, otherwise $Z=Z_{h}$ has $\deg\omega_{\scriptscriptstyle Z}=0$ by
adjunction, contradicting Proposition 2.13. $\Box$

We proceed with our case - by - case analysis excluding the possibilities for
the curve $Z\sim A_{i}^{*}$.

######  Lemma 2.18.

Let $Z_{h}\neq Z$. Then $i\neq 2$.

###### Proof.

Assume the contrary. One checks via Proposition 2.6 the intersections of
$Z\sim A_{2}^{*}$ with all fiber components of the morphism
$p_{\scriptscriptstyle Y}$ (cf. 2.4) are _positive_ , except for $Z\cdot
R_{2}=-1$, $Z\cdot H_{1}=Z\cdot F_{1}=0$. Note also that $Z\cdot R_{8}=2$. Let
us show that only $R_{2},H_{1},F_{1}$ can belong to $Z_{v}$.

Indeed, if $Z$ has $H_{2}$, say, as its irreducible component, so that
$Z-H_{2}$ is again a curve, then we get

$(Z-H_{2})\cdot R_{8}=0=Z_{h}\cdot R_{8}+(Z_{v}-H_{2})\cdot R_{8}.$

This implies that either $Z_{h}=R_{8}$ and $Z_{v}-H_{2}=R_{5}+mR_{2}$ or
$Z_{h}\cdot R_{8}=0$ and $Z_{v}-H_{2}=mR_{2}$ for some (positive)
$m\in\mathbb{Z}$. In both cases, we have

$0=Z\cdot H_{1}\geqslant Z_{v}\cdot H_{1}=1,$

a contradiction. Similarly, replacing $H_{2}$ by the curve
$\Phi_{2}-H_{1}-H_{2}$ (cf. Figure 3), we get contradiction. Exactly the same
reasoning also shows that the curves $F_{2}$ and $\Phi_{3}-F_{1}-F_{2}$ are
not among the components of $Z$.

Further, if $G_{1}\subset Z$, then we have $(Z-G_{1})\cdot R_{8}=0$, which
implies that $Z=R_{8}+G_{1}+R_{5}+mR_{2}$ or $Z_{h}+G_{1}+mR_{2}$, similarly
as above. In the former case, we obtain

$-1=(Z^{2})=1+2m-2m^{2},$

a contradiction. On the other hand, if $Z=Z_{h}+G_{1}+mR_{2}$, then

$(Z_{h}^{2})=(Z-G_{1}-mR_{2})^{2}=-3-2Z\cdot G_{1}+2m-2m^{2}<-3$

for $Z\cdot G_{1}>0$, which contradicts Lemma 2.12. Same applies to the curves
$G_{2}$ and $\Phi_{4}-G_{1}-G_{2}$.

Now, if $R^{\prime}\subseteq Z_{v}$ for $R^{\prime}:=\Phi_{1}-R_{2}-R_{5}$,
then $-3=(Z-R^{\prime})\cdot R_{8}\geqslant-1$ as above, a contradiction.
Finally, if $R_{5}\subseteq Z$, then from $(Z-R_{5})\cdot R_{8}=1$ and the
preceding results we deduce the following possibilities for $Z$:

$R_{8}+R_{5}+H_{1}+mR_{2},\qquad R_{8}+R_{5}+F_{1}+mR_{2},\qquad
Z_{h}+R_{5}+mR_{2}$

($Z_{h}\cdot R_{8}=1$ in the last case). In the first two cases, we get

$-1=(Z^{2})=1+2m-2m^{2},$

a contradiction. In turn, if $Z=Z_{h}+R_{5}+mR_{2}$, then

$(Z_{h}^{2})=(Z-R_{5}-mR_{2})^{2}=-3-2Z\cdot R_{5}+4m-2m^{2}\leqslant-1$

for $Z\cdot R_{5}\geqslant 0$, which contradicts Lemma 2.12 and Proposition
2.13.

Thus we obtain

$Z=Z_{h}+aR_{2}+bH_{1}+cF_{1}$

(as $1$ \- cycles) by construction and Lemma 2.12, where $a,b,c$ are non -
negative integers and $a>0$. On the other hand, we have

$(Z_{h}^{2})=(Z-aR_{2}-bH_{1}-cF_{1})^{2}=-1+2a-2a^{2}-2b^{2}-2c^{2}\leqslant-1+2a-2a^{2},$

which gives $(Z_{h}^{2})=-1$ because $K_{Y}\cdot Z_{h}=1$ and
$\deg\omega_{\scriptscriptstyle Z_{h}}\geqslant-2$. But then
$\deg\omega_{\scriptscriptstyle Z_{h}}=0$ by definition, i. e. normalization
of $Z_{h}$ is either rational or elliptic, which contradicts Proposition 2.13.

The proof that $H^{0}(Y,A_{2}^{*})=0$ under the assumption $Z_{h}\neq Z$ is
complete. $\Box$

It follows from Lemma 2.18 that for $i=2$ the curve $Z$ must be irreducible —
in contradiction with Corollary 2.17. Together with Lemma 2.11 we have thus
proven the vanishing of
$\operatorname{Ext}^{{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}}(A_{2},A_{1})$.

Suppose next that $i=3$. Then again for $Z\sim A_{3}^{*}$ one gets: $Z\cdot
R_{2}=-1$, $Z\cdot R_{8}=2$, $Z\cdot H_{1}=Z\cdot F_{1}=0$ and the
intersections of $Z$ with all other fiber components of $p_{\scriptscriptstyle
Y}$ are positive. Now the arguments from the case of $i=2$ apply literally
here to show that
$\operatorname{Ext}^{{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}}(A_{3},A_{1})=0$.

######  Lemma 2.19.

Let $Z_{h}\neq Z$. Then $i\neq 4$.

###### Proof.

Assume the contrary (so that $Z\sim A_{4}^{*}$). One checks via Proposition
2.6 that $Z\cdot R_{2}=Z\cdot R_{5}=Z\cdot H_{1}=Z\cdot F_{1}=0$ and the
intersections of $Z$ with all other fiber components of $p_{\scriptscriptstyle
Y}$ are positive. Note also that $Z\cdot R_{8}=2$. Then all ‘‘ positive ’’
components of $Z_{v}$ are excluded exactly as in the proof of Lemma 2.18.

Thus we obtain

$Z=Z_{h}+aR_{2}+bR_{5}+cH_{1}+dF_{1}$

by construction and Lemma 2.12, where $a,b,c,d\in\mathbb{Z}$ are non -
negative, $a+b+c+d\neq 0$. On the other hand, we have

$(Z_{h}^{2})=(Z-aR_{2}-bR_{5}-cH_{1}-dF_{1})^{2}=-1+2ab-2a^{2}-2b^{2}-2c^{2}-2d^{2},$

which gives $(Z_{h}^{2})=-3$ and $Z_{h}\simeq\mathbb{P}^{1}$, contradicting
Proposition 2.13.

Hence $H^{0}(Y,A_{4}^{*})=0$ when $Z_{h}\neq Z$. $\Box$

It follows from Lemma 2.19 that for $i=4$ the curve $Z$ must be irreducible —
in contradiction with Corollary 2.17. Together with Lemma 2.11 this proves the
vanishing of
$\operatorname{Ext}^{{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}}(A_{4},A_{1})$.

######  Lemma 2.20.

Let $Z_{h}\neq Z$. Then $i\neq 5$.

###### Proof.

Assume the contrary (so that $Z\sim A_{5}^{*}$). One checks via Proposition
2.6 that $Z\cdot R_{2}=Z\cdot F_{1}=0$ and the intersections of $Z$ with all
other fiber components of $p_{\scriptscriptstyle Y}$ are positive. Note also
that $Z\cdot R_{8}=1$. Then all ‘‘ positive ’’ components of $Z_{v}$ are
excluded similarly as in the proof of Lemma 2.18 (cf. the case of $i=6$
below).

Thus we obtain

$Z=Z_{h}+aR_{2}+bF_{1}$

by construction and Lemma 2.12, where $a,b\in\mathbb{Z}$ are non - negative,
$a+b\neq 0$. On the other hand, we have

$(Z_{h}^{2})=(Z-aR_{2}-bF_{1})^{2}=-1-2a^{2}-2b^{2},$

which gives $(Z_{h}^{2})=-3$ and $Z_{h}\simeq\mathbb{P}^{1}$, contradicting
Proposition 2.13.

Hence $H^{0}(Y,A_{5}^{*})=0$ when $Z_{h}\neq Z$. $\Box$

######  Lemma 2.21.

Let $Z_{h}\neq Z$. Then $i\neq 6$.

###### Proof.

Assume the contrary (so that $Z\sim A_{6}^{*}$). One checks via Proposition
2.6 that $Z\cdot R_{2}=Z\cdot F_{1}=-1$ and the intersections of $Z$ with all
other fiber components of $p_{\scriptscriptstyle Y}$ are positive. Note also
that $Z\cdot R_{8}=3$. Then the curve $Z_{1}:=Z-F_{1}$ satisfies $Z_{1}\cdot
R_{8}=1$. Furthermore, $Z_{1}$ can not contain the curve
$\Phi_{1}-R_{2}-R_{5}$, as well as the curves from the fibers $\Phi_{i}$,
$i>1$ . Indeed, if $H_{1}\subset Z_{1}$, say, then from $(Z_{1}-H_{1})\cdot
R_{8}=-1$ we deduce that $Z_{1}=R_{8}+H_{1}+mR_{2}$ for some $m\in\mathbb{Z}$,
which gives $Z_{1}\cdot R_{2}=-2m^{2}$ — in contradiction with $Z\cdot
R_{2}=-1$.

Thus one gets only two possibilities for $Z_{1}$ (and $Z$): $Z_{h}+mR_{2}$ or
$Z_{h}+R_{5}+mR_{2}$. In particular, we obtain
$(Z_{h}^{2})\leqslant-1+2m-2m^{2}\leqslant-1$ for $Z\cdot R_{5}\geqslant 0$,
contradicting Lemma 2.12 and Proposition 2.13.

Hence $H^{0}(Y,A_{6}^{*})=0$ when $Z_{h}\neq Z$. $\Box$

Lemmas 2.20 and 2.21 lead to contradiction with Corollary 2.17 exactly as
above

Further, if $i>6$ is not equal $11$ (cf. Lemma 2.11), then we have $Z\cdot
R_{8}=1$, $Z\cdot R_{2}\geqslant-1$, $Z\cdot F_{1}\geqslant 0$ and the
intersections of $Z$ with all other fiber components of $p_{\scriptscriptstyle
Y}$ are positive. Now the arguments, similar to those in the proofs of Lemmas
2.20 and 2.21, lead to contradiction. We have thus proven that
$\operatorname{Ext}^{{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}}(A_{i},A_{1})=0$
for all $i>1$ not equal to $11$.

Let us finally show that
$\operatorname{Ext}^{{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}}(A_{11},A_{1})=0$
as well. According to Lemma 2.11 it suffices to prove that
$H^{2}(Y,A_{11}^{*})=0$. Assume on the contrary that $K_{Y}+A_{11}\sim\
[\text{some curve}\ Z]$. One checks via Proposition 2.6 that $Z\cdot
R_{2}=Z\cdot H_{1}=Z\cdot F_{1}=Z\cdot F^{\prime}=0$ for
$F^{\prime}:=\Phi_{3}-F_{1}-F_{2}$ and the intersections of $Z$ with all other
fiber components of $p_{\scriptscriptstyle Y}$ are positive. Note also that
$Z\cdot R_{8}=4$. Let us represent again $Z$ as the sum of horizontal $Z_{h}$
and vertical $Z_{v}$ curves with respect to the morphism
$p_{\scriptscriptstyle Y}$

######  Lemma 2.22.

The curve $Z\sim K_{Y}+A_{11}$ is irreducible.

###### Proof.

Assume the contrary. Let us show that only $R_{2},H_{1},F_{1}$ and
$F^{\prime}$ can belong to $Z_{v}$ (cf. the proof of Lemma 2.18).

Indeed, if $Z$ has $H_{2}$, say, as its irreducible component, so that
$Z-H_{2}$ is again a curve, then we get

$(Z-H_{2})\cdot R_{8}=2=Z_{h}\cdot R_{8}+(Z_{v}-H_{2})\cdot R_{8}.$

This yields the following options for $Z_{v}-H_{2}$: (i) $R_{5}+\Phi+mR_{2}$,
$3R_{5}+mR_{2}$ (when $Z_{h}=R_{8}$) or (ii) $R_{5}+mR_{2}$, $2R_{5}+mR_{2}$,
$\Phi+mR_{2}$, $mR_{2}$ (when $Z_{h}\cdot R_{8}\geqslant 0$) for some
(positive) $m\in\mathbb{Z}$ and an irreducible curve $\Phi\subset\Phi_{i}$,
$i>1$. It follows from $Z\cdot H_{1}=0$ and $R_{8}\cdot H_{i}=2$ that only
case (ii) is possible. This implies that

$(Z_{h}^{2})=(Z-Z_{v})^{2}=-1-2Z\cdot Z_{v}+(Z_{v})^{2}<-3,$

since $Z\cdot Z_{v}>0$ by construction, which contradicts Lemma 2.12.

Further, if $R^{\prime}=\Phi_{1}-R_{2}-R_{5}$ is an irreducible component of
$Z$, then from $(Z-R^{\prime})\cdot R_{8}=-1$ we deduce that
$Z=R_{8}+R^{\prime}+mR_{2}$, which contradicts $Z\cdot R_{2}=0$. Finally, if
$R_{5}\subset Z$, then from $(Z-R_{5})\cdot R_{8}=3$ and the preceding results
we deduce the following possibilities for $Z$:

$R_{8}+R_{5}+2H_{1}+mR_{2},\qquad R_{8}+R_{5}+2F_{1}+mR_{2},\qquad
R_{8}+R_{5}+2F^{\prime}+mR_{2},$ $R_{8}+R_{5}+H_{1}+F_{1}+mR_{2},\qquad
R_{8}+R_{5}+H_{1}+F^{\prime}+mR_{2},\qquad
R_{8}+R_{5}+F_{1}+F^{\prime}+mR_{2},$ $R_{8}+3R_{5}+H_{1}+mR_{2},\qquad
R_{8}+3R_{5}+F_{1}+mR_{2},\qquad R_{8}+3R_{5}+F^{\prime}+mR_{2},\qquad
R_{8}+5R_{5}+mR_{2}$

(when $Z_{h}=R_{8}$) or

$Z_{h}+R_{5}+mR_{2},\qquad Z_{h}+2R_{5}+mR_{2},\qquad
Z_{h}+3R_{5}+mR_{2},\qquad Z_{h}+4R_{5}+mR_{2}$
$Z_{h}+R_{5}+H_{1}+mR_{2},\qquad Z_{h}+R_{5}+F_{1}+mR_{2},\qquad
Z_{h}+R_{5}+F^{\prime}+mR_{2},$ $Z_{h}+2R_{5}+H_{1}+mR_{2},\qquad
Z_{h}+2R_{5}+F_{1}+mR_{2},\qquad Z_{h}+2R_{5}+F^{\prime}+mR_{2}$

(when $Z_{h}\cdot R_{8}\geqslant 0$). In all cases, we either get
contradiction with $Z\cdot R_{2}=Z\cdot H_{1}=Z\cdot F_{1}=Z\cdot
F^{\prime}=0$ or obtain $(Z_{h}^{2})<-3$, contradicting Lemma 2.12.

Hence we have

$Z=Z_{h}+aR_{2}+bH_{1}+cF_{1}+dF^{\prime},$

where $a,b,c,d\in\mathbb{Z}$ are non - negative, $a+b+c+d\neq 0$. Now exactly
as in the proof of Lemma 2.19 one gets contradiction with Proposition 2.13.

The proof that $H^{2}(Y,A_{11}^{*})=0$ under the assumption $Z_{h}\neq Z$ is
complete. $\Box$

It follows from Lemma 2.22 that $Z=Z_{h}$. But then from $(Z^{2})=-1$ and
$\deg\omega_{\scriptscriptstyle Z}=0$ we get contradiction exactly as in the
proof of Proposition 2.13. This concludes the proof that
$\operatorname{Ext}^{{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}}(A_{11},A_{1})=0$.

We now turn to the case of $j>1$.

######  Lemma 2.23.

$\operatorname{Ext}^{{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}}(A_{12},A_{11})=\operatorname{Ext}^{{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}}(A_{11},A_{j})=0$
for all $1<j<11$ not equal $8$.

###### Proof.

Let us consider the case of
$\operatorname{Ext}^{{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}}(A_{12},A_{11})$
first. Again, since

$\chi(A_{12},A_{11}):=\chi(Y,A_{12}^{*}\otimes A_{11})=0$

via Proposition 2.6, it suffices to prove that $H^{0}(Y,A_{12}^{*}\otimes
A_{11})=H^{2}(Y,A_{12}^{*}\otimes A_{11})=0$. Moreover, since
$(K_{Y}+A_{12}-A_{11})\cdot K_{Y}=-2$, we immediately get $H^{2}=0$. Thus it
remains to show that $H^{0}=0$.

Assume on the contrary that $A_{11}-A_{12}\sim\ [\text{some curve}\ Z]$. We
have $(Z^{2})=0$ and $Z\cdot K_{Y}=2$. Let us represent $Z$ as the sum of
horizontal $Z_{h}$ and vertical $Z_{v}$ curves with respect to the morphism
$p_{\scriptscriptstyle Y}$ (cf. the above case of $j=1$).

Note that $Z\cdot R_{2}=-1$, i. e. $R_{2}\subseteq Z_{v}$, and one computes
the curve $Z-R_{2}$ intersects any possible irreducible component of $Z_{v}$
_non - negatively_. In particular, if $Z_{v}\neq R_{2}$, then we obtain
$(Z_{h}^{2})\leqslant-2$ (cf. the proof of Lemma 2.12). This together with
$Z_{h}\cdot K_{Y}=2$ shows that $\deg\omega_{\scriptscriptstyle
Z_{h}}\leqslant 0$. Hence the arithmetic genus of (the underlying reduced
scheme of) $Z_{h}$ is either $0$ or $1$. Now the result (and the proof) of
Proposition 2.13 apply literally here to provide contradiction.

Thus we get $Z=Z_{h}+R_{2}$. Note that $(Z_{h}^{2})=(Z^{2})=0$ in this case.
Then for any irreducible component $\widetilde{Z}_{h}\subsetneq Z_{h}$ we have
$(\widetilde{Z}_{h}^{2})<0$ and $\widetilde{Z}_{h}\cdot K_{Y}=1$. In
particular, we obtain that
$\deg\omega_{\scriptscriptstyle\widetilde{Z}_{h}}\leqslant 0$, which gives
contradiction via Proposition 2.13 as above. Hence the curve $Z_{h}$ is
_irreducible_. Then the linear system $|Z_{h}|$ either consists of only
$Z_{h}$ or else it is basepoint - free. But in the latter case, since
$c_{2}(X)=12$ and $\chi_{\text{top}}(Z_{h})=-2$ (for generic $Z_{h}$), the
morphism given by $|Z_{h}|$ will have a _singular_ fiber of arithmetic genus
$2$, which once again contradicts Proposition 2.13. Thus we get
$H^{0}(Y,\mathcal{O}_{Y}(Z_{h}))=H^{0}(Y,\mathcal{O}_{Y}(Z))=\mathbb{C}$. This
yields a _unique non - split_ rank $2$ vector bundle $\mathcal{E}$ on Y
corresponding to a non - trivial class in
$\text{Ext}^{1}(\mathcal{O}_{Y},\mathcal{O}_{Y}(Z))\simeq\mathbb{C}$.

Choose an affine open subset $U\subset\mathbb{P}^{1}$ and refer to the surface
$p_{\scriptscriptstyle Y}^{-1}(U)$ as an elliptic curve $E$ over the function
field $\mathbb{C}(U)$. Then $\mathcal{O}_{Y}$ is identified with the structure
sheaf of $E$ and $\mathcal{O}_{Y}(Z)$ corresponds to some $0$ \- cycle $\xi$
on $E$ (of length $12$). In turn, the restriction
$\mathcal{E}\big{|}_{p_{\scriptscriptstyle Y}^{-1}(U)}$ can be regarded as an
extension of $\mathcal{O}_{E}$ by $\mathcal{O}_{E}(\xi)$, which is thus
trivial due to
$\text{Ext}^{1}(\mathcal{O}_{E},\mathcal{O}_{E}(\xi))=H^{1}(E,\mathcal{O}_{E}(\xi))=0$.
On the other hand, the $\text{Gal}(\overline{\mathbb{C}(U)}/\mathbb{C}(U))$ \-
action on $E(\overline{\mathbb{C}(U)})$,5)5)5) For any field k, we denote by
$\text{Gal}(\bar{{\bf k}}/{\bf k})$ the Galois group of an algebraic closure
$\bar{{\bf k}}\supseteq{\bf k}$, whereas $E({\bf k})$ denotes the set of ${\bf
k}$ \- points on the curve $E$. with varying $U$, generates the action of the
global monodromy of $p_{\scriptscriptstyle Y}$ (cf. 3.5 below). In particular,
since the bundle $\mathcal{E}$ is unique, it is invariant under the
monodromy.6)6)6) Indeed, any element $\tau\in\text{Gal}$ acts on the group
$\text{Ext}^{1}(\mathcal{O}_{Y},\mathcal{O}_{Y}(Z))$ in a natural way, so that
$\tau^{*}\mathcal{E}=\mathcal{E}$. Then various trivializations
$\mathcal{E}\big{|}_{p_{\scriptscriptstyle
Y}^{-1}(U)}\simeq\mathcal{O}_{E}\oplus\mathcal{O}_{E}(\xi)$ agree for
different choices of $U$, hence they extend over the whole base
$\mathbb{P}^{1}$ of $p_{\scriptscriptstyle Y}$ to show that $E$ actually
_splits_ , a contradiction. Thus we get $H^{0}(Y,A_{12}^{*}\otimes A_{11})=0$.

Further, turning to the case of
$\operatorname{Ext}^{{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}}(A_{11},A_{j})$,
where $1<j<11$ is not equal $8$, we have

$\chi(A_{11},A_{j}):=\chi(Y,A_{11}^{*}\otimes A_{j})=0$

via Proposition 2.6. Then $H^{0}(Y,A_{11}^{*}\otimes A_{j})=0$ because of
$(A_{j}-A_{11})\cdot K_{Y}=-2$. Now, assuming that $H^{2}(Y,A_{11}^{*}\otimes
A_{j})\neq 0$ for some $j$, let us fix a curve $Z\sim K_{Y}+A_{11}-A_{j}$.
Note that again $(Z^{2})=0$ and $Z\cdot K_{Y}=2$.

Consider $j\in\\{2,3,7,10\\}$. One computes $Z\cdot R_{2}=-1$ for these $j$,
i. e. $R_{2}\subset Z$, and the curve $Z-R_{2}$ intersects any fiber component
of $Z$ non - negatively. Then running the preceding arguments verbatim we
obtain contradiction. Similarly, if $j=6$, then we have $Z\cdot R_{2}=Z\cdot
F_{1}=-1$ and the curve $Z-R_{2}-F_{1}$ intersects any fiber component of $Z$
non - negatively, whence a contradiction as earlier. Finally, if
$j\in\\{4,5,9\\}$, then already $Z$ intersects all of its fiber components non
- negatively, so that $Z$ is horizontal and irreducible as above, which is
again impossible.

The proof of lemma is complete. $\Box$

######  Lemma 2.24.

$\operatorname{Ext}^{{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}}(A_{i},A_{8})=\operatorname{Ext}^{{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}}(A_{8},A_{j})=0$
for all $i>8$ and $1<j<8$.

###### Proof.

Similarly as above, since $\chi(A_{i},A_{8})=\chi(A_{8},A_{j})=0$ for all
$i>8$ and $j<8$, it suffices to prove that $H^{0}(Y,A_{i}^{*}\otimes
A_{8})=H^{2}(Y,A_{i}^{*}\otimes A_{8})=H^{2}(Y,A_{j}\otimes A_{8}^{*})=0$.
Note that $H^{0}(Y,A_{j}\otimes A_{8}^{*})=0$ because $(A_{j}-A_{8})\cdot
K_{Y}=-1$. We also have $H^{2}(Y,A_{i}^{*}\otimes A_{8})=0$, $i\neq 11$, and
$H^{0}(Y,A_{11}^{*}\otimes A_{8})=0$ be the same reason.

Suppose first that $i\neq 11$ and there exists a curve $Z\sim A_{8}-A_{i}$.
One computes $(Z^{2})=-1$, $Z\cdot K_{Y}=1$, $Z\cdot R_{8}=3$, $Z\cdot
R_{2}\geqslant-1$ and $Z$ intersects all other fiber components of the
morphism $p_{\scriptscriptstyle Y}$ non - negatively. Now similarly as in the
proofs of Lemmas 2.21, 2.22, we first exclude those irreducible fiber
components that intersect $Z$ positively, and then derive contradiction with
Lemma 2.12 and Proposition 2.13.

In turn, equality $H^{2}(Y,A_{11}^{*}\otimes A_{8})=0$ follows as previously,
by observing that $(K_{Y}+A_{11}-A_{8})\cdot R_{8}=2$ and $K_{Y}+A_{11}-A_{8}$
intersects all the fiber components of $p_{\scriptscriptstyle Y}$ non -
negatively.

Now assume that $K_{Y}+A_{8}-A_{j}\sim[\text{some curve}\ Z]$. Again we have
$(Z^{2})=-1$ and $Z\cdot K_{Y}=1$. Firstly, if $j\leqslant 4$, then $Z\cdot
R_{8}=5$. At the same time, we have $Z\cdot H_{1}=-1$, which implies that
$Z_{1}:=Z-H_{1}$ is a curve. Again $(Z_{1}^{2})=(Z^{2})=-1$, $Z_{1}\cdot
R_{8}=3$, $Z_{1}\cdot R_{2}=-1$ and $Z_{1}$ intersects all other fiber
components of $p_{\scriptscriptstyle Y}$ non - negatively. Similarly as in the
proof of Lemmas 2.21, we first exclude those irreducible fiber components that
intersect $Z$ positively, and then derive contradiction with Lemma 2.12 and
Proposition 2.13.

Secondly, if $j=6$, then $Z\cdot R_{8}=6$. At the same time, we have $Z\cdot
R_{2}=Z\cdot F_{1}=-1$, which implies that $Z_{1}:=Z-R_{2}-F_{1}$ is a curve.
Note that $(Z_{1}^{2})=(Z^{2})=-1$, $Z_{1}\cdot R_{8}=4$ and $Z_{1}$
intersects all other fiber components of $p_{\scriptscriptstyle Y}$ non -
negatively. Similarly as in the proof of Lemma 2.22, we first exclude those
irreducible fiber components that intersect $Z$ positively, and then derive
contradiction with Lemma 2.12 and Proposition 2.13.

Finally, if $j=5$ or $7$, then $Z\cdot R_{8}=4$, $Z\cdot R_{2}\geqslant-1$ and
$Z$, $Z-R_{2}$ intersect all other fiber components of $p_{\scriptscriptstyle
Y}$ non - negatively. Again the preceding considerations give contradiction.

The proof of lemma is complete. $\Box$

For all other $i>j>1$, not considered previously (cf. Lemmas 2.23 and 2.24),
we have $K_{Y}\cdot(A_{j}-A_{i})=0$ and again $\chi(A_{i},A_{j})=0$. In
particular, once $H^{0}(Y,A_{i}^{*}\otimes A_{j})\neq 0$ (resp.
$H^{2}(Y,A_{i}^{*}\otimes A_{j})\neq 0$), there exists a curve $Z\sim
A_{j}-A_{i}$ (resp. $Z\sim K_{Y}+A_{i}-A_{j}$) contained in the fibers of
$p_{\scriptscriptstyle Y}$. In other words, we have $Z=Z_{v}$ in the previous
notation, which implies that $A_{j}-A_{i}$ (resp. $K_{Y}+A_{i}-A_{j}$) is an
_integral_ combination of $K_{Y}$ and the classes of
$R_{l},H_{i},F_{h},G_{k}$. This situation is only possible for the following
values of $(i,j)$:

$(10,2),\qquad(4,3),\qquad(9,5),\qquad(7,6),\qquad(12,6),\qquad(12,7).$

The corresponding divisors $A_{j}-A_{i}$ are

$-3K_{Y}+H_{2},\qquad-K_{Y}+R_{5},\qquad 2K_{Y}-G_{2},\qquad-
F_{1},\qquad-4K_{Y}+F_{2},\qquad-4K_{Y}+F_{1}+F_{2}.$

It is now immediate that $H^{0}(Y,A_{i}^{*}\otimes
A_{j})=H^{2}(Y,A_{i}^{*}\otimes A_{j})=0$. For example, if $(i,j)=(10,2)$ and
$-3K_{Y}+H_{2}\sim\ [\text{some curve}\ Z]$, then $Z-H_{2}\sim-3K_{Y}$ is also
a curve for $Z\cdot H_{2}=-2$, a contradiction. In turn, if
$H^{0}(Y,4K_{Y}-H_{2})\neq 0$, then from 2.4 we deduce that $2F$ contains
$H_{2}$, which is also impossible. The remaining cases of $(i,j)$ are treated
similarly and employ in addition the fact that $C+F\sim 5K_{Y}$ does not
contain rational curves.

Theorem 2.10 is completely proved. $\Box$

##  3\. A phantom

###  3.1. Description of $\operatorname{Jac}\,Y$

We retain the notation of Section 2.

Recall that the _Hesse pencil_ is a one - dimensional linear system of cubics
on $\mathbb{P}^{2}$ of the form

(3.2) $t_{0}(x^{3}+y^{3}+z^{3})+t_{1}xyz=0.$

Here $x,y,z$ are projective coordinates on $\mathbb{P}^{2}$ and
$[t_{0}:t_{1}]\in\mathbb{P}^{1}$ parameterizes the members of the pencil (we
refer to [4] for an overview of the many properties of (3.2)).

Blowing up the base points one resolves the indeterminacies of the rational
map $\mathbb{P}^{2}\dashrightarrow\mathbb{P}^{1}$ given by the Hesse pencil.
This yields a minimal elliptic surface $p_{\scriptscriptstyle
S}:S\longrightarrow\mathbb{P}^{1}$ whose Mordell – Weil group is generated by
$9$ sections of $p_{\scriptscriptstyle S}$ and is isomorphic to
$(\mathbb{Z}/3\mathbb{Z})^{2}$.

######  Proposition 3.3.

We have $S=\operatorname{Jac}\,Y$. In particular, there is a natural inclusion
$\mu_{3}\subset(\mathbb{Z}/3\mathbb{Z})^{2}\subset\operatorname{Aut}\,S$,
where $\mu_{3}$ is the Galois group of the covering $\psi$ (see the end of
2.1)

###### Proof.

Recall that fibration $p_{\scriptscriptstyle S}$ has exactly $4$ fibers
$\Phi_{i}$ of type $I_{3}$ and no other singular fibers (the images of
$\Phi_{i}$ on $\mathbb{P}^{2}$ form the _Hesse configuration_ of $12$ lines
passing through $9$ inflection points common to all smooth curves (3.2)).

In turn, the surface $Y$ is obtained from $\operatorname{Jac}\,Y$ via the
logarithmic transformations along the fibers corresponding to $C$ and $F$,
respectively. This implies that the only singular fibers of the elliptic
fibration $\operatorname{Jac}\,Y\longrightarrow\mathbb{P}^{1}$ are $4$ curves
of type $I_{3}$ (cf. Figure 3 and the beginning of 2.4). Now the equality
$S=\operatorname{Jac}\,Y$ follows from [8]. The last assertion of lemma is due
to the fact that $\mu_{3}$ preserves the fibers of $p_{\scriptscriptstyle Y}$
(by construction) and acts naturally on $\operatorname{Jac}\,Y$ (cf. [4,
Section 4]). $\Box$

######  Notation.

All elliptic surfaces we consider in the text may be identified in a natural
way with genus $1$ curves over the field $\mathbb{C}(\mathbb{P}^{1})$. Then
for notation’s simplicity we will not distinguish in what follows between any
such surface (over $\mathbb{C}$) and the corresponding curve (over
$\mathbb{C}(\mathbb{P}^{1})$). Similarly, all non - fiber curves (a. k. a.
_multi - sections_) on a given elliptic surface are identified with
$\overline{\mathbb{C}(\mathbb{P}^{1})}$ \- points on the corresponding
elliptic curve, and vice versa.

######  Remark 3.4.

Let $R\subset Y$ be a multi - section of some degree $n$ (i. e. the induced
morphism $p_{\scriptscriptstyle Y}\big{|}_{R}$ has degree $n$). Then the genus
$1$ curve $Y/{\bf k}$ acquires a point over a degree $n$ extension ${\bf
k}\supset\mathbb{C}(\mathbb{P}^{1})=:{\bf k}_{0}$. This yields a degree $n$
isogeny $Y\longrightarrow\operatorname{Jac}\,Y$ of ${\bf k}_{0}$ \- curves.
Indeed, by definition we have an isomorphism
$\theta:\operatorname{Jac}^{n}\,Y\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{Jac}\,Y$,
as well as the Poincaré line bundle $\Theta$ on
$Y\times\operatorname{Jac}^{n}\,Y$ (all over ${\bf k}_{0}$), where
$\operatorname{Jac}^{n}\,Y$ is the moduli space of degree $n$ line bundles on
$Y$. Now the morphism $Y\longrightarrow\operatorname{Jac}\,Y$, given by
$y\mapsto y^{*}\Theta\big{|}_{Y\times 0}$ (shifts by all $y\in\bar{{\bf
k}}_{0}$) and the subsequent composition with $\theta$, is
$\text{Gal}(\bar{{\bf k}}_{0}/{\bf k}_{0})$ \- equivariant and maps $R\in
Y({\bf k})$ to $0\in\operatorname{Jac}\,Y$. This is exactly the claimed ${\bf
k}_{0}$ \- isogeny $Y\longrightarrow\operatorname{Jac}\,Y$.7)7)7) More
precisely, as follows from the construction the graph of
$Y\longrightarrow\operatorname{Jac}\,Y$ is invariant under the action of
$\text{Gal}({\bf k}/{\bf k}_{0})$, hence also under the action of
$\text{Gal}(\bar{{\bf k}}_{0}/{\bf k}_{0})$, which implies that it is defined
over ${\bf k}_{0}$. In particular, letting $R:=R_{8}$ (see Proposition 2.6) we
get $n=6$, and it follows from Proposition 3.3 that the group
$\mu_{3}\subseteq\operatorname{Aut}\,Y$ acts on $Y$ via fiberwise shifts by
some $c\in\operatorname{Jac}^{6}\,Y$.

###  3.5. The monodromy of $Y$

Put $B:=p_{\scriptscriptstyle
Y}(Y\setminus\left\\{C,F,\Phi_{1},\ldots,\Phi_{4}\right\\})$ (notation is as
earlier). Then all fibers of the elliptic fibration $p_{\scriptscriptstyle
Y}:Y\longrightarrow\mathbb{P}^{1}$ restricted to $B$ are smooth. This yields a
homomorphism $\pi_{1}(B)\to\operatorname{SL}_{2}(\mathbb{Z})$, defined up to
the conjugation, and its image $\Gamma_{Y}$ is called the _(global) monodromy
group_ (of $p_{\scriptscriptstyle Y}$ or $Y$). Note that $\Gamma_{Y}$ is
generated by $5$ elements $g_{F},g_{1},\ldots,g_{4}$ (for
$\pi_{1}(\mathbb{P}^{1}\setminus\\{p_{\scriptscriptstyle Y}(C)\\})=\\{1\\}$),
which satisfy the relation

$g_{F}\cdot g_{1}\cdot\ldots\cdot g_{4}=1,$

where $g_{F}\in\operatorname{SL}_{2}(\mathbb{Z})$ is the image of a loop on
$B$ around $p_{\scriptscriptstyle Y}(F)$, etc. (see e. g. [11, Lemma 2.1]).

Let $\mu_{3}$ be as in Proposition 3.3. Choose some generator
$\sigma\in\mu_{3}$. It follows from the construction of $Y$ (via a ‘‘ cubic
root of $F$ ’’) that $\sigma\in\Gamma_{Y}$. Recall also that the monodromy of
$S=\operatorname{Jac}\,Y$ is the _modular group_

$\Gamma(3):=\text{Ker}\big{[}\operatorname{SL}_{2}(\mathbb{Z})\twoheadrightarrow\operatorname{SL}_{2}(\mathbb{F}_{3})\big{]}.$

By construction there is a natural (outer) action of $\sigma$ on $\Gamma(3)$.
This allows one to identify $\Gamma_{Y}$ with an extension of
$\mu_{3}\ni\sigma$ by $\Gamma(3)$ (cf. 4.1 below). We will see in the next
subsection that $\sigma$ ‘‘ detects ’’ phantoms in $D^{b}(Y)$.

###  3.6. A phantom in $D^{b}(Y)$

Consider the length $12$ exceptional collection ${\bf
EC}:=\left\\{A_{1},\ldots,A_{12}\right\\}$ from Theorem 2.10. We will assume
that it is full and bring this to contradiction.

Let $F_{\zeta}$ be the general fiber of $p_{\scriptscriptstyle Y}$ treated as
a genus $1$ curve over the field $\mathbb{C}(\mathbb{P}^{1})$.

######  Lemma 3.7.

${\bf EC}$ generates the category $D^{b}(F_{\zeta})$. In other words, the
sheaves $\mathcal{O}_{F_{\zeta}}$, $\mathcal{O}_{F_{\zeta}}(\pm R_{8})$
generate $D^{b}(F_{\zeta})$.

###### Proof.

Given any element $A\in\operatorname{Pic}F_{\zeta}$, we regard it as a formal
difference of two multi - sections of $p_{\scriptscriptstyle Y}$, which yields
an element $A_{Y}\in\operatorname{Pic}Y$ naturally associated with $A$. In
this way, to every complex of direct sums of line bundles on $F_{\zeta}$ one
associates a complex of direct sums of line bundles on $Y$, since
$\operatorname{Hom}\,(A_{1},A_{2}):=H^{0}(F_{\zeta},A_{1}^{*}\otimes A_{2})$
for any $A_{i}\in\operatorname{Pic}F_{\zeta}$. Moreover, for each _exact_ such
complex on $F_{\zeta}$ one gets an exact complex on $Y$, which defines a
functor (cf. 2.8)

$\Phi:\mathcal{D}:=\big{[}\text{full triangulated subcategory of}\
D^{b}(F_{\zeta})\ \text{generated by various}\
A\in\operatorname{Pic}F_{\zeta}\big{]}\longrightarrow D^{b}(Y).$

It is immediate from the construction that two categories $\mathcal{D}$ and
$\Phi(\mathcal{D})$ are equivalent. Further, all vector bundles on $F_{\zeta}$
can be naturally identified via $\Theta$ from Remark 3.4 with
$\text{Gal}(\bar{{\bf k}}_{0}/{\bf k}_{0})$ \- invariant $0$ \- cycles (a. k.
a. sky - scraper sheaves) on $\operatorname{Jac}\,Y$, which in turn are
identified with $0$ \- cycles on $F_{\zeta}$.8)8)8) This follows basically
from [5, Theorem 7], where it is proved that every indecomposable vector
bundle on an elliptic curve is uniquely determined by its rank and
determinant, considered as an element in $\operatorname{Pic}$. Note also that
given any vector bundle $V$ on $F_{\zeta}$, with the associated $0$ \- cycle
$\xi_{V}$, one has a canonical identification
$\operatorname{Hom}\,(V,\star)=\operatorname{Hom}\,(\xi_{V},\star)$.

Thus all vector bundles on $F_{\zeta}$ belong to $\mathcal{D}$ and we get
$\Phi(\mathcal{D})=\mathcal{D}=D^{b}(F_{\zeta})\subset D^{b}(Y)$. The last
statement of lemma follows from the fact that
$D^{b}(Y)=\left<A_{1},\ldots,A_{12}\right>$ by assumption and that
$A_{i}\big{|}_{F_{\zeta}}$ is either $\mathcal{O}_{F_{\zeta}}$ or
$\mathcal{O}_{F_{\zeta}}(\pm R_{8})$ for any $1\leqslant i\leqslant 12$.
$\Box$

The automorphism $\sigma$ (see 3.5) induces an autoequivalence on $D^{b}(Y)$.
Then it follows from Lemma 3.7 that
$\sigma^{*}\mathcal{O}_{F_{\zeta}},\sigma^{*}\mathcal{O}_{F_{\zeta}}(\pm
R_{8})$ also generate $D^{b}(F_{\zeta})$. Moreover, since
$\mathcal{O}_{\varepsilon}(\pm R_{8})=\mathcal{O}_{\varepsilon}$ for any
effective $0$ \- cycle $\varepsilon$ on $F_{\zeta}$, one may assume that
$\sigma^{*}\mathcal{O}_{F_{\zeta}},\sigma^{*}\mathcal{O}_{F_{\zeta}}(R_{8})$
generate $D^{b}(F_{\zeta})$ for
$\mathcal{O}_{\varepsilon}=\text{Cone}(\mathcal{O}_{F_{\zeta}}\to\mathcal{O}_{F_{\zeta}}(R_{8}))$
and $\mathcal{O}_{F_{\zeta}}(-R_{8})=\mathcal{O}_{\varepsilon}[1]$ (here
$\varepsilon$ corresponds to the multi - section $R_{8}$).

######  Lemma 3.8.

There exists a semiorthogonal decomposition
$D^{b}(F_{\zeta})=\left<\sigma^{*}\mathcal{O}_{F_{\zeta}},\sigma^{*}\mathcal{O}_{F_{\zeta}}(R_{8})\right>$.

###### Proof.

Recall that $\sigma^{*}\mathcal{O}_{F_{\zeta}}=\mathcal{O}_{F_{\zeta}}(c)$ for
some effective $0$ \- cycle $c$ on $F_{\zeta}$ of degree $6$ (see Remark 3.4).
Now, if
$\mathcal{O}_{F_{\zeta}}(c)\neq\sigma^{*}\mathcal{O}_{F_{\zeta}}(R_{8})$, then
for $\sigma^{*}\mathcal{O}_{F_{\zeta}}(R_{8})=\mathcal{O}_{F_{\zeta}}(R_{8})$
and $R_{8}\cdot F_{\zeta}=6$ (by construction of $R_{8}$) we get
$\operatorname{Ext}^{{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}}(\sigma^{*}\mathcal{O}_{F_{\zeta}},\sigma^{*}\mathcal{O}_{F_{\zeta}}(R_{8}))=0$.
The latter yields a SOD
$D^{b}(F_{\zeta})=\left<\sigma^{*}\mathcal{O}_{F_{\zeta}},\sigma^{*}\mathcal{O}_{F_{\zeta}}(R_{8})\right>$.

Finally, if
$\mathcal{O}_{F_{\zeta}}(c)=\sigma^{*}\mathcal{O}_{F_{\zeta}}(R_{8})$, then
$\mathcal{O}_{F_{\zeta}}$ and $\mathcal{O}_{F_{\zeta}}(R_{8})$ are quasi -
isomorphic, which is impossible as the function
$\chi(F_{\zeta},\star):D^{b}(F_{\zeta})\longrightarrow\mathbb{Z}$ attains
different values on them. $\Box$

Lemma 3.8 contradicts [28, Corollary 1.3]. Hence ${\bf EC}$ is not full and
there exists a phantom
$\mathcal{A}=\left<A_{1},\ldots,A_{12}\right>^{\bot}\neq 0$ (cf. the beginning
of 2.8). 9)9)9) It would be interesting to distinguish the (derived) class of
the curve $F_{\zeta}$ (resp. describe $\mathcal{A}$ in terms of $F_{\zeta}$)
using the results and technique of the paper [3].

######  Remark 3.9.

Since $q(Y)=p_{g}(Y)=0$ and the class $K_{Y}$ is deformation - invariant, one
can easily prove that any (sufficiently general) algebraic surface
$Y^{\prime}$ in the same deformation class as $Y$ is again a $(2,3)$ \-
Dolgachev, carrying an exceptional collection
$A^{\prime}_{1},\ldots,A^{\prime}_{12}$ such that $A^{\prime}_{i}$ deforms to
$A_{i}$ for all $i$. Furthermore, both fibrations $p_{\scriptscriptstyle
Y^{\prime}}:Y^{\prime}\longrightarrow\mathbb{P}^{1}$ and
$p_{\scriptscriptstyle Y}:Y\longrightarrow\mathbb{P}^{1}$ are relatively
diffeomorphic, with any such diffeomorphism
$s:Y/\mathbb{P}^{1}\stackrel{{\scriptstyle\sim}}{{\to}}Y^{\prime}/\mathbb{P}^{1}$
satisfying $s(A_{i})=A^{\prime}_{i},1\leqslant i\leqslant 12$, and
$s(\Gamma_{Y})=\Gamma_{Y^{\prime}}$ (cf. 3.5). This gives a phantom
$\mathcal{A^{\prime}}\subset D^{b}(Y^{\prime})$ exactly as in 3.6
above.10)10)10) We will develop more general (and conceptual) arguments to
construct exceptional collections and phantoms in Section 4 below.

##  4\. Log - transfer

###  4.1. Setup

We are going to extend the result of Theorem 2.10 (cf. Remark 3.9) to the case
of an arbitrary $(2,3)$ \- Dolgachev surface $\mathfrak{S}$ with
$\operatorname{Jac}\,\mathfrak{S}=S$ (notation is as in 3.1). For technical
reasons, though, we had to impose an Assumption at the end of this subsection.

Fix two distinct points $p$ and $q$ on $\mathbb{P}^{1}$ such that the fibers
$F_{p}:=p_{\scriptscriptstyle S}^{-1}(p)$ and $F_{q}:=p_{\scriptscriptstyle
S}^{-1}(q)$ are smooth. Recall that $\mathfrak{S}$ is obtained by _logarithmic
transformations_ along $F_{p}$ and $F_{q}$ with multiplicities $2$ and $3$,
respectively, as follows (see [7, V.13] for details). Take $p$ for instance
and a small disk $\Delta_{p}\subset\mathbb{P}^{1}$ around it so that all
fibers of $p_{\scriptscriptstyle S}$ are smooth over $\Delta_{p}$. Choose a
trivialization of the family of lattices $H^{1}(p_{\scriptscriptstyle
S}^{-1}(t),\mathbb{Z})\subset\mathbb{C}$, where $t\in\Delta_{p}$ is a
parameter with $p=\\{t=0\\}$, and identify

$p_{\scriptscriptstyle S}^{-1}(\Delta_{p})=\\{(z,t)\ |\
z\in\mathbb{C}/H^{1}(p_{\scriptscriptstyle S}^{-1}(t),\mathbb{Z}),\
t\in\Delta_{p}\\}.$

Then there is a $\mu_{2}$ \- action on $p_{\scriptscriptstyle
S}^{-1}(\Delta_{p})$ as follows:

(4.2) $(z,t)\mapsto(z+\frac{1}{2},-t).$

The quotient $p_{\scriptscriptstyle S}^{-1}(\Delta_{p})/\mu_{2}$ is a smooth
elliptic surface over $\Delta_{p}$ with unique singular fiber over $p=0$ that
is multiple of multiplicity $2$. The surfaces $p_{\scriptscriptstyle
S}^{-1}(\Delta_{p})/\mu_{2}$ and $S\setminus F_{p}$ can be glued into a smooth
compact elliptic surface. Applying further the same construction with $q$ and
$\mu_{3}$ one obtains $\mathfrak{S}$.

Denote by
$p_{\scriptscriptstyle\mathfrak{S}}:\mathfrak{S}\longrightarrow\mathbb{P}^{1}$
for what follows the corresponding elliptic fibration. We will also denote the
fibers of $p_{\scriptscriptstyle\mathfrak{S}}$ over $p$ and $q$ by the same
symbols as above. The surface $\mathfrak{S}$ thus admits an open cover by
analytic subsets $U_{p}:=p_{\scriptscriptstyle\mathfrak{S}}^{-1}(\Delta_{p})$,
$U_{q}:=p_{\scriptscriptstyle\mathfrak{S}}^{-1}(\Delta_{q})$ and
$U_{pq}:=\mathfrak{S}\setminus\\{F_{p}\cup F_{q}\\}$, with
$\mathfrak{S}\setminus\\{U_{p}\cup U_{q}\\}\simeq
S\setminus\\{p_{\scriptscriptstyle S}^{-1}(\Delta_{p})\cup
p_{\scriptscriptstyle S}^{-1}(\Delta_{q})\\}$ as _complex manifolds_. The
principal goal of this section is to understand when a given triple

(4.3)
$(L_{p}\in\operatorname{Pic}U_{p},L_{q}\in\operatorname{Pic}U_{q},L_{pq}\in\operatorname{Pic}U_{pq})$

of line bundles is induced by restriction from a line bundle on $\mathfrak{S}$
(this will ultimately allow one relate exceptional collections in $D^{b}(S)$
and $D^{b}(\mathfrak{S})$). The key role here will be played by the monodromy
group $\Gamma_{\mathfrak{S}}$ of
$p_{\scriptscriptstyle\mathfrak{S}}:\mathfrak{S}\longrightarrow\mathbb{P}^{1}$
for which we make the following additional

######  Assumption.

$\Gamma_{\mathfrak{S}}$ fits into an exact sequence

$1\to\Gamma(3)\to\Gamma_{\mathfrak{S}}\to\mu_{3}\to 1.$

Here $\mu_{3}$ gives the monodromy action around $F_{q}$ and $\Gamma(3)$ is
induced by the monodromy of $p_{\scriptscriptstyle S}$.

######  Remark 4.4.

The Assumption is crucial for the coming constructions (see e. g. the end of
4.6). Note that it is met for one particular, hence generic, surface
$\mathfrak{S}$ (see 3.5 and Remark 3.9). It is not at all clear however
whether the extension structure of $\Gamma_{\mathfrak{S}}$ takes place for
_any_ given $\mathfrak{S}$ (compare with [12, Section 3]). We will give an
outline towards the evidence for such a structure later in the Appendix
below.11)11)11) This Appendix aims to put the present technical results in a
broader perspective (yet the complete arguments will appear elsewhere).

Recall that $\mathfrak{S}$ is simply - connected. Furthermore, it is glued out
of $U_{p}$, $U_{q}$ and $U_{pq}$, so that for the triple (4.3) to come from a
line bundle it is necessary _and sufficient_ the components of this triple to
be $\Gamma_{\mathfrak{S}}$ \- invariant. Furthermore, it follows from the
Assumption that
$p_{\scriptscriptstyle\mathfrak{S}}\big{|}_{U_{pq}}=p_{\scriptscriptstyle
S}\big{|}_{U_{pq}}$, since by construction the natural $\mu_{3}$ \- action on
the family of lattices $H^{1}(p_{\scriptscriptstyle S}^{-1}(t),\mathbb{Z})$
preserves the action of $\Gamma(3)$. Next proposition is an immediate
consequence of these considerations.

######  Proposition - definition 4.5.

Let $\widetilde{L}\in\operatorname{Pic}S$ be such that
$\widetilde{L}\big{|}_{p_{\scriptscriptstyle S}^{-1}(\Delta_{q})}$ is
$\mu_{3}$ \- invariant. Then one defines the triple (4.3) by descending
$\widetilde{L}\big{|}_{p_{\scriptscriptstyle S}^{-1}(\Delta_{q})}$ to $U_{q}$,
restricting $\widetilde{L}$ to $S\setminus\\{p_{\scriptscriptstyle
S}^{-1}(\Delta_{p})\cup p_{\scriptscriptstyle
S}^{-1}(\Delta_{q})\\}=\mathfrak{S}\setminus\\{U_{p}\cup U_{q}\\}$ and
extending over $U_{p}$ by $\Gamma_{\mathfrak{S}}$ \- invariance, so that this
triple coincides with
$(\mathfrak{L}\big{|}_{U_{p}},\mathfrak{L}\big{|}_{U_{q}},\mathfrak{L}\big{|}_{U_{pq}})$
for some $\mathfrak{L}\in\operatorname{Pic}\mathfrak{S}$.

###  4.6. Log - transfer construction I

Take some line bundle $L\in\operatorname{Pic}S$ and let
$\mathcal{E}:=p_{\scriptscriptstyle S*}L$ be the pushforward of $L$ under
$p_{\scriptscriptstyle S}$.

######  Lemma 4.7.

The sheaf $\mathcal{E}$ is either trivial (zero)
$\mathcal{O}_{\mathbb{P}^{1}}$ \- module or it is a line bundle on
$\mathbb{P}^{1}$.12)12)12) Compare with [7, III.11, Corollary 11.2].

###### Proof.

The claim is obvious when $L=\mathcal{O}_{S}$. Thus we may assume that $L$ is
a $(-1)$ \- curve because these generate $\operatorname{Pic}S$ and
$p_{\scriptscriptstyle S*}M_{1}\otimes p_{\scriptscriptstyle
S*}M_{2}=p_{\scriptscriptstyle S*}(M_{1}\otimes M_{2})$ for any
$\mathcal{O}_{S}$ \- modules $M_{i}$ such that both $p_{\scriptscriptstyle
S*}M_{i}$ are non - trivial. Let $F$ be a fiber of $p_{\scriptscriptstyle S}$.
We have $h^{0}(F,L\big{|}_{F})=1$ when $F$ is smooth. Now suppose that
$F=F_{i}$ is a ‘‘ triangle ’’ (cf. 3.1). We have
$h^{0}(F_{i},L\big{|}_{F_{i}})\geqslant 1$ because $L$ is a section of
$p_{\scriptscriptstyle S}$. Let $F_{ij}\simeq\mathbb{P}^{1}$, $1\leqslant
j\leqslant 3$, be the irreducible components of $F_{i}$. Then
$L\big{|}_{Fi1}=\mathcal{O}_{\mathbb{P}^{1}}(1)$, say, and
$L\big{|}_{Fij}=\mathcal{O}_{\mathbb{P}^{1}}$ for $j>1$. In particular, any
global section from $H^{0}(F_{i},L\big{|}_{F_{i}})$ induces one of
$\mathcal{O}_{\mathbb{P}^{1}}(1)$ on $F_{i1}$, whereas it is (the same)
constant on the other $F_{ij}$. Now, if $s_{1},s_{2}\in
H^{0}(F_{i},L\big{|}_{F_{i}})$ are two different sections, then their linear
combination gives a linear form on $F_{i1}=\mathbb{P}^{1}$ with two _distinct_
values at $F_{i1}\cap F_{i2}$ and $F_{i1}\cap F_{i3}$, respectively, a
contradiction. We thus obtain a family over $\mathbb{P}^{1}$ of one -
dimensional linear spaces $H^{0}(F,L\big{|}_{F})$, which are precisely the
fibres of $\mathcal{E}$, i. e. $\mathcal{E}$ is a line bundle as claimed.
$\Box$

######  Remark 4.8.

Interestingly enough, Lemma 4.7 indicates that with any line bundle on $S$ is
associated a _meromorphic projective structure_ on the base $\mathbb{P}^{1}$,
which brings the theory of integrable systems into play. We will outline in
A.1 of the Appendix how our present geometric constructions can be put into a
‘‘ physical ’’ perspective in this way.

Consider $\widetilde{L}:=p_{\scriptscriptstyle
S}^{*}\mathcal{E}\in\operatorname{Pic}S$ (see Lemma 4.7). Let $t$ be the
projective coordinate on $\mathbb{P}^{1}$, for which $p=0$, $q=\infty$ and the
transition function of $\mathcal{E}$ is $t^{-d}$, some
$d\in\mathbb{Z}\cup\\{\infty\\}$, so that either
$\mathcal{E}\simeq\mathcal{O}_{\mathbb{P}^{1}}(d)$ or $\mathcal{E}=0$ when
$d=\infty$. Then, since the restriction
$\widetilde{L}\big{|}_{p_{\scriptscriptstyle S}^{-1}(\Delta_{p})}$ is trivial
on $F_{p}$, it is invariant under the action (4.2). The same applies to $q$.
It follows that $\widetilde{L}$ may be identified with a line bundle
$\mathfrak{L}$ as in Proposition 4.5. The following property is evident
(although important):

######  Lemma 4.9.

$p_{\scriptscriptstyle
S*}\widetilde{L}=p_{\scriptscriptstyle\mathfrak{S}*}\mathfrak{L}=\mathcal{O}_{\mathbb{P}^{1}}(d)$.

So far the specifics of situation in 4.1 was used in an essential way. We
proceed with relating $D^{b}(S)$ and $D^{b}(\mathfrak{S})$.

###  4.10. Log - transfer construction II

Let $L$ and $\widetilde{L}$ be as in 4.6. Denote by
$\gamma:\widetilde{L}\longrightarrow L$ the natural (forgetful) morphism of
sheaves. Choose an affine open cover $\mathfrak{S}=\bigcup U_{i}$ such that
$L\big{|}_{U_{i}}\simeq\widetilde{L}\big{|}_{U_{i}}\simeq\mathcal{O}_{U_{i}}$
all $i$. Then $\gamma\big{|}_{U_{i}}$ becomes multiplication by some _regular_
function $f_{i}\in\mathcal{O}_{U_{i}}\setminus\\{0\\}$. Moreover, the
collection $\left\\{(f_{i},U_{i})\right\\}$ defines an _effective_ divisor
$C_{L}\subset\mathfrak{S}$, on which the kernel of $\gamma$ is supported.
Obviously, this procedure can be reversed, so that one reconstructs $L$ from
the pair $(\widetilde{L},C_{L})$.13)13)13) This correspondence also admits an
IS interpretation — to be discussed in the Appendix (cf. Remark 4.8). It
follows that $L$ meets the conditions of Proposition 4.5 _if and only if_ the
divisor $C_{L}$ meets these conditions. A priori it need not, however, and
below we will associate with $C_{L}$, _canonically_ , an algebraic cycle
$C_{L}^{\text{log}}$ on $S$, to which Proposition 4.5 does apply.

Namely, replacing the preceding $U_{i}$ with analytic subsets (GAGA) and
employing the identification $\widetilde{L}=\mathfrak{L}$ at the end of 4.6,
we get a natural $\mu_{3}$ \- action on the line bundle $\widetilde{L}$. Take
generator $\sigma\in\mu_{3}$ and denote its action on $\widetilde{L}$ by
$\sigma^{*}$. Then, in particular, $\sigma^{*}\ker\,\gamma$ is a coherent
subsheaf of $\widetilde{L}$, isomorphic to $\ker\,\gamma$. We thus obtain
another effective divisor $C_{L}^{\sigma}$ being the support of
$\sigma^{*}\ker\,\gamma$. This yields the following:

######  Proposition - definition 4.11.

For the line bundle on $S$, associated with the divisor
$C_{L}^{\mathrm{log}}:=C_{L}+C_{L}^{\sigma}+C_{L}^{\sigma^{2}}$, the
conditions of Proposition - definition 4.5 hold.

Now we define the line bundle $L^{\text{log}}$ via the natural morphism
$\widetilde{\gamma}:\widetilde{L}\longrightarrow L^{\text{log}}$ (‘‘
multiplication - by - local - equation ’’) with the kernel supported on
$C_{L}^{\text{log}}$. Then according to Propositions 4.5 and 4.11 the objects
$L^{\text{log}}$, $\widetilde{\gamma}$, $C_{L}^{\text{log}}$ (and
$\widetilde{L}$ of course) can be considered as the corresponding gadgets on
the surface $\mathfrak{S}$.

###  4.12. Exceptionality

Now we are going to show that the above correspondence $L\mapsto
L^{\text{log}}$ defines an equivalence between $D^{b}(S)$ and a (full)
subcategory of $D^{b}(\mathfrak{S})$ in such a way that an exceptional
collection of line bundles on $S$ will remain exceptional after log. Here is a
key statement:

######  Proposition 4.13.

We have
$\operatorname{Ext}^{{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}}(L_{1}^{\mathrm{log}},L_{2}^{\mathrm{log}})=0$
for any two line bundles $L_{i}\in\operatorname{Pic}S$ satisfying
$\operatorname{Ext}^{{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}}(L_{1},L_{2})=0$.

###### Proof.

Let us associate with each $L_{i}$ the pair $(\widetilde{L_{i}},C_{L_{i}})$ as
in 4.10. Then $\widetilde{L_{i}}=p_{\scriptscriptstyle
S}^{*}\mathcal{O}_{\mathbb{P}^{1}}(d_{i})$ for some
$d_{i}\in\mathbb{Z}\cup\\{\infty\\}$ by construction. Note also that the
cycles $C_{L_{i}}$ belong to the fibers of $p_{\scriptscriptstyle S}$ and
hence $p_{\scriptscriptstyle S*}L_{i}=p_{\scriptscriptstyle
S*}L_{i}^{\text{log}}$ for all $i$.

Recall next that
$\operatorname{Ext}^{{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}}(L_{1},L_{2})=H^{{\>\raisebox{3.0pt}{\text{\circle*{1.5}}}}}(S,L_{1}^{*}\otimes
L_{2})\ (=0)$ and similarly for $L_{i}^{\text{log}}$ (cf. the proof of Theorem
2.10). Then the vanishing of
$\operatorname{Ext}^{0}(L_{1}^{\text{log}},L_{2}^{\text{log}})$ follows right
away because

$H^{0}(\mathfrak{S},L_{1}^{*\text{log}}\otimes
L_{2}^{\text{log}})=H^{0}(\mathfrak{S},\widetilde{L^{*}_{1}}\otimes\widetilde{L_{2}})=H^{0}(\mathbb{P}^{1},\mathcal{O}_{\mathbb{P}^{1}}(d_{2}-d_{1}))=H^{0}(S,\widetilde{L^{*}_{1}}\otimes\widetilde{L_{2}})=H^{0}(S,L_{1}^{*}\otimes
L_{2})=0$

by Lemma 4.9 and Leray spectral sequence (to simplify the notation we use the
same symbol $\widetilde{L_{i}}$ for the corresponding line bundle on both $S$
and $\mathfrak{S}$).

Further, since

$H^{0}(S,L_{1}^{*}\otimes
L_{2})=0=\operatorname{Ext}^{2}(L_{1},L_{2})=H^{0}(S,\omega_{\scriptscriptstyle
S}\otimes L_{1}\otimes L^{*}_{2})$

by assumption and Serre duality, and $p_{\scriptscriptstyle
S*}\omega_{\scriptscriptstyle S}=\mathcal{O}_{\mathbb{P}^{1}}(-1)$ (see the
construction of $S$ in 3.1), it follows that $|d_{1}-d_{2}|=\infty$ (i. e.
$p_{\scriptscriptstyle S*}(L_{1}\otimes L^{*}_{2})=p_{\scriptscriptstyle
S*}(L^{*}_{1}\otimes L_{2})=0$ in the setting at the end of 4.6). Then,
similarly as earlier, we obtain

$\operatorname{Ext}^{2}(L_{1}^{\text{log}},L_{2}^{\text{log}})=H^{0}(\mathfrak{S},\omega_{\scriptscriptstyle\mathfrak{S}}\otimes
L_{1}^{\text{log}}\otimes L_{2}^{*\text{log}})=H^{0}(\mathfrak{S},0)=0$

because
$p_{\scriptscriptstyle\mathfrak{S}*}\omega_{\scriptscriptstyle\mathfrak{S}}=0$
by (2.5) and definition of $p_{\scriptscriptstyle\mathfrak{S}*}$ (cf. the
proof of Lemma 4.7).

It thus remains to prove the vanishing of
$\operatorname{Ext}^{1}(L_{1}^{\text{log}},L_{2}^{\text{log}})=H^{1}(\mathfrak{S},L_{1}^{*\text{log}}\otimes
L_{2}^{\text{log}})$. Note that

$H^{1}(\mathfrak{S},L_{1}^{*\text{log}}\otimes
L_{2}^{\text{log}})=H^{0}(\mathbb{P}^{1},R^{1}p_{\scriptscriptstyle\mathfrak{S}*}(L_{1}^{*\text{log}}\otimes
L_{2}^{\text{log}}))$

by Leray and equality $|d_{1}-d_{2}|=\infty$. Using the latter condition once
again, by relative Serre duality [7, III.12] we get

$H^{0}(\mathbb{P}^{1},R^{1}p_{\scriptscriptstyle\mathfrak{S}*}(L_{1}^{*\text{log}}\otimes
L_{2}^{\text{log}}))=H^{1}(\mathbb{P}^{1},\omega_{\scriptscriptstyle\mathfrak{S}/\mathbb{P}^{1}}\otimes
p_{\scriptscriptstyle\mathfrak{S}*}(L_{1}^{\text{log}}\otimes
L_{2}^{*\text{log}}))=0,$

where
$\omega_{\scriptscriptstyle\mathfrak{S}/\mathbb{P}^{1}}:=\omega_{\scriptscriptstyle\mathfrak{S}}\otimes
p_{\scriptscriptstyle\mathfrak{S}}^{*}\mathcal{O}_{\mathbb{P}^{1}}(2)$ is the
_dualizing sheaf_ of $p_{\scriptscriptstyle\mathfrak{S}}$. This completes the
proof. $\Box$

######  Remark 4.14.

The proof of Proposition 4.13 actually shows that
$\operatorname{Ext}^{1}(L_{1},L_{2})=0$ for any $L_{i}\in\operatorname{Pic}S$
satisfying
$\operatorname{Ext}^{0}(L_{1},L_{2})=\operatorname{Ext}^{2}(L_{1},L_{2})=0$.
Of course, this is due to the special geometric situation dictated by the
surface $S$, and it need not hold for arbitrary rational surfaces. One may
call a collection of exceptional objects $A_{1},\ldots,A_{N}$ _quasi -
exceptional_ if the conditions, similar to those in 2.8, are satisfied, but
the vanishing of $\operatorname{Ext}^{1}(A_{i},A_{j})$ is not required. It
would be interesting to clarify the categorical meaning of quasi - exceptional
collections.

From Proposition 4.13 we deduce the main result of this section:

######  Corollary 4.15.

Let $A_{1},\ldots,A_{12}$ be an exceptional collection of maximal length of
line bundles on $S$ (cf. Remark 2.9 and [42]). Then
$A_{1}^{\mathrm{log}},\ldots,A_{12}^{\mathrm{log}}$ is an exceptional
collection of maximal length on $\mathfrak{S}$.

###### Proof.

Fix some $i$ and put $L:=A_{i}$. Then, in the notations of 4.6, it suffices to
prove that $\widetilde{L}\neq 0$ (aka $d\neq\infty$). This is evident,
however, by definition of $\gamma:\widetilde{L}\longrightarrow L$ from 4.10.
$\Box$

## Appendix

###  A.1. Local systems

We suggest the reader to consult e. g. [2] and [43] for the notions and facts
used in this subsection (the literature on the subject is quite vast and we
had to limit ourselves while referring).

Recall that the fibration $p_{\scriptscriptstyle
S}:S\longrightarrow\mathbb{P}^{1}=:\mathfrak{B}$ is _modular_ , with
$\mathfrak{B}$ being the $\Gamma(3)$ \- curve, so that there is a natural
(quotient) morphism $\mathfrak{B}\longrightarrow\mathbb{P}^{1}$ of degree $12$
ramified _only_ at $\\{0,1,\infty\\}$. This yields a triangulation
$\displaystyle\bigcup_{i=1}^{24}\overline{\Delta_{i}}=\mathfrak{B}$, symmetric
with respect to the antipodal involution, and a _meromorphic projective
structure_ subject to this triangulation, given by the sheaf
$p_{\scriptscriptstyle S*}\mathbb{Z}_{S}$. On the open locus
$\displaystyle\bigcup_{i=1}^{24}\Delta_{i}$, over which $p_{\scriptscriptstyle
S}$ is smooth, the latter is given by trivializations of the family
$H^{1}(p_{\scriptscriptstyle S}^{-1}(t),\mathbb{Z})$ from 4.1 over each open
triangle $\Delta_{i}$, compatibly with the action of $\Gamma(3)$. In addition,
the so - obtained trivial rank $2$ vector bundle over each $\Delta_{i}$ is
supplied with a flat (Gauss – Manin) connection, defined _up to gauge_ , which
can be equivalently described by the (Schrödinger) operator
$P_{i}:=d/dz^{2}+\varphi(z)\star$ for some holomorphic quadratic differential
$\varphi(z)dz^{\otimes 2}$ and complex variable $z\in\Delta_{i}$.

Further, the choice of a line bundle $L$ on $S$ yields a basis
$\\{g_{1},g_{2}\\}$ in the space of solutions to $P_{i}(g)=0$, $1\leqslant
i\leqslant 24$. Namely, trivialization $L\simeq\mathbb{C}\times\Delta_{i}$ for
each $i$, given by some global section $s_{i}(z)$ of $L\big{|}_{\Delta_{i}}$,
yields a holomorphic family of vectors $g_{1}:=s_{i}+\bar{s_{i}}$,
$g_{2}:=\frac{1}{\sqrt{-1}}(s_{i}-\bar{s_{i}})\in H^{1}(p_{\scriptscriptstyle
S}^{-1}(t),\mathbb{Z})\otimes\mathbb{R}$, which are parallel with respect to
the connection corresponding to $P_{i}$.

One interprets the global sections of $L$ as follows. Consider the Lie algebra
$sl_{2}(z,z^{-1})$ over the Laurent field $\mathbb{C}((z))$. It admits a non -
trivial (Gelfand – Fuchs) $2$ \- cycle, which gives a central extension
$\mathcal{V}$ of $sl_{2}(z,z^{-1})$, the _Virasoro algebra_. The space of
$P_{i}$ above is identified with the coadjoint representation of
$\mathcal{V}$. Then, for each $1\leqslant i\leqslant 24$, a given $s\in
H^{0}(S,L)$ determines the coadjoint orbit $O_{i}:=\mathcal{V}\cdot P_{i}$ in
such a way that

  * •

for any $i\neq j$, there is $g_{ij}\in\Gamma(3)$ with
$O_{i}=g_{ij}(O_{j}):=\mathcal{V}\cdot g_{ij}^{*}P_{j}$,

  * •

$g_{ii}=1$ for all $i$,

  * •

for any $i$, $j$, $k$, we have $g_{ij}g_{jk}g_{ki}=1$.

This follows from the fact that the $\mathcal{V}$ \- orbits of $P_{i}$ are
determined by monodromy of the corresponding second order differential
equations (note once again that $P_{i}$ a priori depend on $L$ — aka on
$s_{i}$ above). The collection $\\{O_{i},g_{ij}\\}$ defines another
meromorphic projective structure on $\mathfrak{B}$ (with connection in the
same gauge class as the initial one).

######  Example A.2.

Let $j:\mathfrak{B}\longrightarrow\mathbb{P}^{1}$ be the $j$ \- map of
$p_{\scriptscriptstyle S}$ of degree $12$. Then the corresponding projective
structure is given by
$\varphi:=\displaystyle\big{(}\frac{j^{\prime\prime}}{j^{\prime}}\big{)}^{\prime}+\displaystyle\frac{1}{2}\big{(}\frac{j^{\prime\prime}}{j^{\prime}}\big{)}^{2}$,
the _Schwartz derivative_ of $j$, with $L$ being one of the sections of
$p_{\scriptscriptstyle S}$. This projective structure is nothing but the one
associated with the homological invariant of the elliptic fibration on $S$
(cf. [7, III.11] or the proof of Lemma 3.9 in [12]).14)14)14) All this
probably goes back to the elliptic functions approach to the geometry of
icosahedron in [30].

######  Remark A.3.

The preceding arguments intimately relate our algebro - geometric
considerations with ‘‘ Lax dynamics ’’ $\dot{X}=[M,X]$, $M\in\mathcal{V}$, in
vector bundles on $\mathbb{P}^{1}$. It would be interesting to obtain some
integrable systems in this way (cf. [18], [40], [52], [45]). Furthermore, one
may interpret the pair $(\widetilde{L},C_{L})$ in 4.10 as a ‘‘ spectral data
’’ associated with $L$, that is a (spectral) curve plus a line bundle on it.
This brings in a relation of our present setting with Nahm, KdV and KZ
equations, on the one hand (see [25], [38], [51]), and with Painlevé equations
on the other (see [27]). Namely, the change of projective structure on
$\mathbb{P}^{1}$ given by $L$ and $s$ as above should correspond, dually, to
evolution of $(\widetilde{L},C_{L})$ in the ‘‘ phase space ’’ $S$. In this
regard one may identify generic fiber $F_{\zeta}$ of $p_{\scriptscriptstyle
S}$ with (normalization of) $C_{L}$ and represent $S$ in terms of a pair of
commuting differential operators (cf. [16]). It would be also interesting to
relate this picture with the _quantum Calogero – Painlevé correspondence_
between Painlevé and non - stationary Schrödinger equations (see e. g. [46,
Section 7]).

###  A.4. Representation theory

Constructions in 4.6 and 4.10 are related to the following modification of the
projective structure on $\mathfrak{B}$ associated with a given
$L\in\operatorname{Pic}S$. Namely, let $p(z)$ be a complex polynomial, with
$\deg p=3$ and $\operatorname{mult}_{0}p=2$. Consider the corresponding
morphism $p:\mathbb{P}^{1}\longrightarrow\mathfrak{B}$ and new projective
structure on $\mathfrak{B}$, given by the pullbacks $p^{-1}(\Delta_{i})$,
$p^{*}P_{i}$ for all $i$ (cf. A.1). By construction, the monodromy $\Gamma$ of
this new structure is a non - trivial extension of $\mu_{3}$ by $\Gamma(3)$,
as in 4.1. It is then tempting to propose the following:

######  Conjecture A.5.

In the previous setting, projective structure on $\mathbb{P}^{1}$ associated
with $L$ and $p$ coincides with the one given by $L^{\mathrm{log}}$ on
$\mathfrak{S}$, so that $\Gamma=\Gamma_{\mathfrak{S}}$.

Our point is that co - adjoint representations of $\Gamma(3)$ (resp.
$\Gamma$), together with a triangulation of $\mathbb{P}^{1}$, should determine
$D^{b}(S)$ (resp. $D^{b}(\mathfrak{S})$). More precisely, the ‘‘ fusion
matrices ’’ $g_{ij}$ in A.1 define the ‘‘ conformal blocks ’’ $s$, from which
one reconstructs $L$. The picture here is quite similar to TQFT (cf. [51]). In
turn, applying $p^{*}$ as above may be regarded as a ‘‘ t’Hooft transform ’’
(or a ‘‘ change of magnetic charge ’’) for initial integrable system
associated with $L$ (cf. Remark A.3). This corresponds to the change of a
$\mathcal{V}$ \- $\Gamma(3)$ \- bimodule structure to a $\mathcal{V}$ \-
$\Gamma$ \- bimodule structure on the space of quadratic differentials
$\varphi(z)$. Conjecture A.5 then claims that the latter bimodule admits a
_geometric_ realization as $L^{\mathrm{log}}$ on the surface $\mathfrak{S}$.
One could also verify the Assumption from 4.1 in this way.

######  Remark A.6.

The previous considerations may be compared with (geometric) construction in
[26] of certain Enriques surfaces. Namely, one starts with _quadratic_ $p$ and
the base change as above, which yields a K3 surface
$\widetilde{S}\longrightarrow S$ (in our case $\widetilde{S}$ will be some
elliptic surface of Kodaira dimension $\geqslant 0$). The corresponding
Enriques surface is then obtained as the quotient of $\widetilde{S}$ by an
involution that comes from the Galois one composed with the fiberwise shift by
some $2$ \- torsion section (such a group action is missing in our situation).

Two comparisons are in order. First, one may interpret $S$ (resp.
$\mathfrak{S}$) as a moduli space of quiver representations (cf. [1], [36]),
taking values in an abelian category of $\mathcal{V}$ \- $\Gamma(3)$ \-
bimodules (resp. of $\mathcal{V}$ \- $\Gamma$ \- bimodules). The quiver here
is the dual graph of the triangulation on $\mathfrak{B}$ from A.1. In turn,
with each $L$ is associated the bimodule $\widetilde{L}$ (‘‘ representation
’’), for which the curve $C_{L}$ plays the role of ‘‘ ramification divisor ’’
of the Virasoro algebra $\mathcal{V}$. Similar picture is observed in the case
of _AS regular quadratic_ (Clifford) algebras (see [10], [44]). Namely, any
such $\mathcal{A}$ is given by relations, corresponding to a net of conics in
$\mathbb{P}^{2}$. The latter has the _discriminant curve_
$C_{\mathcal{A}}\subset\mathbb{P}^{2}$, $\deg C_{\mathcal{A}}=3$, which
together with a pair of line bundles on it determines $\mathcal{A}$. Let us
also indicate that $\mathcal{A}$ yields a projective structure on
$\mathfrak{B}$ corresponding to a degree four map
$\mathbb{P}^{1}\longrightarrow\mathbb{P}^{1}$ given by some pencil of binary
quartics. We will revisit this $\mathcal{V}$ vs. $\mathcal{A}$ correspondence
from the categorical point of view in A.7 below.

Finally, an arbitrary exceptional collection of $L$ on $S$ (resp. on
$\mathfrak{S}$) may be regarded as a collection of ‘‘ conformal blocks ’’ (cf.
A.7), so that one could formulate the semiorthogonality of them in terms of
certain ‘‘ Ind \- Res (Frobenius) type ’’ relation between the corresponding
$\mathcal{V}$ \- $\Gamma(3)$ \- bimodules (resp. $\mathcal{V}$ \- $\Gamma$ \-
bimodules).

###  A.7. Categorification

Let us now return to the considerations of Section 4 and provide some evidence
for why
$\mathcal{A}=\left<A_{1}^{\mathrm{log}},\ldots,A_{12}^{\mathrm{log}}\right>^{\bot}\neq
0$. To reach contradiction with $\mathcal{A}=0$ we interpret both $K$ \-
groups for $S$ and $\mathfrak{S}$ in terms of the Grothendieck groups of the
bimodules considered in A.4 — at the end of the day the latter two groups will
not be isomorphic (we will write $\overline{K}_{0}$ to distinguish the $K$ \-
group of modules from that of sheaves). The idea is that once $\mathcal{A}$ is
trivial we get $D^{b}(S)\simeq D^{b}(\mathfrak{S})$ and so
$\overline{K}_{0}(S)=\overline{K}_{0}(\mathfrak{S})$ (cf. Proposition 4.13 and
Remark 2.9). More precisely, we argue that $\widetilde{L}$, corresponding to
the $3$ \- multiple fibre $F_{q}$ of $p_{\scriptscriptstyle\mathfrak{S}}$,
gives a non - trivial torsion element in $\overline{K}_{0}(\mathfrak{S})$,
whereas $\overline{K}_{0}(S)\simeq\mathbb{Z}^{12}$.15)15)15) This should be
compared with invariants constructed in [48] in order to distinguish rational
cyclic group quotients from irrational ones (all over $\mathbb{Q}$). See also
[33] for further general outlook.

To $\widetilde{L}$ there corresponds the character $\psi$ on
$\mathcal{V}\times\Gamma$ of Virasoro - monodromy representation associated
with the local system on $\mathbb{P}^{1}$ defined by $\widetilde{L}$. We view
$\psi$ as an element in the group ring
$\mathbb{C}\left[\overline{K}_{0}(\mathfrak{S})\right]$. Then the condition
$p_{\scriptscriptstyle\mathfrak{S}*}\widetilde{L}^{\otimes
3}=\mathcal{O}_{\mathbb{P}^{1}}$ translates into $\psi^{3}=1$. At the same
time, due to a non - trivial monodromy around $F_{q}$, we have $\psi\neq 1$,
whence a torsion in $\overline{K}_{0}(\mathfrak{S})$. This is our sketch of
why $D^{b}(\mathfrak{S})$ possesses phantoms in general.

We conclude this section by revisiting the present approach from the point of
view of _non - commutative geometry_ in order to explain the distinguished
role played by fibers of an elliptic fibration in the description of $D^{b}$
(cf. the above curves $C_{L}$). Our exposition will follow the ‘‘
semicommutative ’’ approach as discussed in the wonderful work [44] — this
aims to put our direct geometric reasoning on the general grounds of _sheaves
of categories_ (see below). Recall that one starts with an elliptic curve $E$
and a pair of degree two morphisms
$p_{0},p_{1}:E\longrightarrow\mathbb{P}^{1}$. Then out of the line bundles
$\mathcal{L}_{i}:=p_{i}^{*}\mathcal{O}_{\mathbb{P}^{1}}(1)$ are constructed
two algebras $S_{i}$, their degree two extension $R\supset S_{i}$ and an $R$
\- algebra $\overline{H}$ generated by $\text{End}_{S_{i}}(R)$, which is a
certain quadratic $\mathbb{Z}$ \- algebra (‘‘ twisted dihedral group algebra
’’). This $\overline{H}$ turns out to be an Azumaya algebra with a maximal
order $\overline{S}$. The algebra $\overline{S}$ is also quadratic and
$\mathbb{Z}$ \- graded.

Now, coming back to our elliptic surface $S$ (or, more generally, to
$\mathfrak{S}$), it turns out that it is encoded by the center of an
appropriate $\overline{S}$ as above and that $D^{b}(S)$ is equivalent to the
bounded derived category of finite - dimensional $\overline{S}$ \- modules.
The generating (simple) modules are also _$\overline{H}$ \- bimodules_ and can
be extracted from the graded pieces of $\overline{S}$. This suggests a
relation between the present constructions and those from A.4 in the following
way. First, arguing as in [24, Section 3], one interprets $D^{b}(S)$ as a
category of global sections of the sheaf of categories associated with afore -
mentioned quiver (same for $D^{b}(\mathfrak{S})$). Recall that the
corresponding representations take values in the $\mathcal{V}$ \- $\Gamma(3)$
\- bimodules (resp. $\mathcal{V}$ \- $\Gamma$ \- bimodules). Second, using the
gluing and geometric realization technique of e. g. [41, Section 5], one
should be able to reinterpret $D^{b}(S)$ (resp. $D^{b}(\mathfrak{S})$) in
terms of the _$\overline{H}$ \- bimodules_ for an appropriate $\overline{H}$
as above. This should also give a fully faithful embedding
$D^{b}(S)\hookrightarrow D^{b}(\mathfrak{S})$ (compare with Corollary 4.15).
Finally, the fact that both categories are actually distinct can be proved by
an explicit computation of the $K$ \- groups of respective modules, similarly
as we did at the beginning of this subsection.16)16)16) Alternatively, one may
apply [15, Lemma 4.3], which also gives that $D^{b}(S)\neq
D^{b}(\mathfrak{S})$.

######  Remark A.8.

More analogies with the preceding discussion are in order. First, it is a
construction of trigonometric solutions to the Yang – Baxter equation, the ‘‘
classical ’’ ($R$ \- matrix) one in [9] and an ‘‘ $A_{\infty}$ ’’ (Fukaya –
Seidel) one in [35]. Note also a similarity between the preceding ‘‘ Virasoro
- Clifford ’’ correspondence and the ‘‘ Helix - Sklyanin ’’ one (cf. [39, Ch.
8]) in the context of non - commutative deformations of $\mathbb{P}^{n}$ (see
[24, Section 7] for instance).17)17)17) A feasible reasoning should probably
come from the interpretation in [29] of Hochschield cohomology as a Lie
algebra structure on the _(graded) Picard group(s)_. Further, various surfaces
$X_{q}$ in [44, Section 5] (see also [44, Section 8]), corresponding to the
above triples $(E,p_{0},p_{1})$ and associated with a given rational elliptic
surface $X$ (i. e. $D^{b}(X_{q})$ is obtained from $D^{b}(X)$ via certain
gluing bi - module), can be regarded as deformations of the $t$ \- structure
on $D^{b}(X)$ (compare with [24, 7.4]). This motivates a similar relation to
hold between $D^{b}(\mathfrak{S})$ and $D^{b}(S)$ (cf. A.9 below). Finally, we
note that in the case of _generic Enriques surface_ $\Sigma$ there is a
similar interpretation for $D^{b}(\Sigma)$ via
$\text{Perf}(\mathcal{P},\mathcal{R})$ (see [34, Lemma 6.14]), where
$\mathcal{P}$ is a stacky projective plane and $\mathcal{R}$ is an Azumaya
sheaf on it. The argument in [34] is based on the construction of $\Sigma$ as
a codimension $3$ linear section of the join of two $\mathbb{P}^{2}$s (cf.
[34, Lemma 6.10]). This suggests yet another outlook on categorical properties
of Dolgachev surfaces and their explicit construction (compare with Remark
A.6).

###  A.9. A - side

This final subsection discusses some analogies with the Seiberg – Witten
theory on $4$ \- manifolds. Here _moduli spaces of instantons_ on $S$ will
play the role of geometric counterparts for $D^{b}(S)$. Let us recall the
relevant notions.

First of all, if $M$ is the $4$ \- manifold underlying $S$, then $K_{S}$
corresponds to a $\text{spin}^{c}$ \- structure on $M$ (same also applies to
$\mathfrak{S}$ being homeomorphic to $M$). This yields a decomposition of the
complexified tangent bundle $T_{S}\otimes\mathbb{C}=(W^{-})^{*}\otimes W^{+}$
and the (gauge class of) Dirac operator $D_{S}:H^{0}(S,W^{+})\to
H^{0}(S,W^{-})$. Here $W^{\pm}$ are rank two vector bundles satisfying
$\bigwedge^{2}W^{\pm}=-K_{S}$. Fixing a rank $2$ vector bundle $\mathcal{V}$
on $S$ with Chern classes $c_{i}(\mathcal{V})$ one obtains a moduli space of
twisted Dirac operators $D_{S}^{a}:H^{0}(S,\mathcal{V}\otimes W^{+})\to
H^{0}(S,\mathcal{V}\otimes W^{-})$ for various Hermitian connections $a$ on
$\mathcal{V}$. Imposing further restrictions on the solutions of the equations
$D_{S}^{a}\star=0$, as well as on the curvature of $a$, yields various moduli
spaces of _topological instantons_ whose geometric characteristics provide
Donaldson, SW, etc. invariants of $M$, specific to the smooth structure
induced by $S$.

######  Example A.10 (cf. [50, Ch. 1, §2]).

Consider a collection of vectors $e_{i}\in\text{Pic}\,S$ satisfying: (i)
$e_{i}\equiv K_{S}\mod 2$; (ii) $K_{S}^{2}-4c_{2}(S)\leqslant
e_{i}^{2}\leqslant 0$; (iii) $e_{i}\cdot K_{S}=0$. The hyperplanes (‘‘ walls
’’) $e_{i}^{\bot}$ divide $\operatorname{Pic}S\otimes\mathbb{R}$ into a system
of chambers $\mathcal{C}_{k}$. Then we can form the moduli space $J_{k}^{l}$
of Dirac operators, corresponding to vector bundles $\mathcal{V}$ with
$c_{1}(\mathcal{V})=c_{1}(S)$ and $c_{2}(\mathcal{V})=c_{2}(S)+k$ as above,
which are also semistable with respect to polarization belonging to the
$k^{\text{th}}$ chamber $\mathcal{C}_{k}$. Imposing further a _quasiparabolic
structure_ on $\mathcal{V}$ with respect to some curve $C\subset S$, one gets
a family of moduli spaces $J_{k,\alpha}^{l}$ of $\alpha$ \- parabolic vector
bundles, $\alpha\in[0,1/2]$. These $J_{k,\alpha}^{l}$ also turn out to be
birationally isomorphic.

Going back to the categorical picture let us say first that the choice of a
spin structure and topological data as above parallels that of a numerically
exceptional collection (cf. Remark 2.9). Further, the exceptionality condition
for collections of objects in $D^{b}$ may be considered as an analog of Dirac
or Seiberg – Witten equations, which brings in a similarity between $D^{b}$
and various ‘‘ Jacobians ’’ $J_{k}^{l}$ (see Example A.10). We expect the
properties of the latter might guide the properties of the former. Of most
interest to us is the behavior under the logarithmic transformations (cf.
Section 4).

Namely, we propose that the ‘‘ core ’’ $D^{b}(S)$ is preserved under these
transformations (cf. the discussion at the end of (A.7)), which is reminiscent
of the above birationality of $J_{k,\alpha}^{l}$.

######  Remark A.11.

Representing one log transform on a minimal elliptic surface as a _rational
blow - down_ one finds that the SW invariants are preserved (see e. g. [19,
Lecture 4]). On the other hand, performing such transform on $S$ yields an
_algebraic_ rational elliptic surface $\Sigma$, with $D^{b}(\Sigma)$ being a
deformation of $D^{b}(S)$ (see [6]). This should give yet another hint for the
behavior of $D^{b}$ under the logarithmic transformations of elliptic surfaces
in general. Namely, we expect that the canonical class $K_{S}$ defines a
_stability condition_ on $D^{b}(S)$ w. r. t.
$H^{2}(S,\mathbb{Z})=H^{2}(\Sigma,\mathbb{Z})$ (cf. [37, Section 5]), which
deforms to another stability condition given by $K_{\Sigma}$. This deformation
procedure should in turn preserve (essential) parts of $D^{b}(S)$ and
$D^{b}(\Sigma)$.

Acknowledgments. The work of the first author was performed at the Center of
Pure Mathematics, MIPT, with financial support of the project FSMG - 2023 -
0013. The second author was supported by the Basic Research Program of the
National Research University Higher School of Economics.

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