Nonexistence of exceptional bundles on $\mathbb{P}^{3}$ with maximal possible ranks

TL;DR

This paper proves the nonexistence of certain exceptional bundles on projective 3-space with specific ranks and degrees, revealing new obstructions beyond known divisibility conditions.

Contribution

It establishes new nonexistence results for exceptional bundles on P^3, identifying obstructions beyond the divisibility condition r|(2d^2+1).

Findings

No exceptional bundles with rank 2d^2+1 and degree d for |d|4.

Existence of no exceptional bundle with rank 27 and degree 11.

New obstructions to the existence of exceptional bundles are identified.

Abstract

We prove that on $\mathbb{P}^{3}$ there is no exceptional bundle with rank $r=2d^{2}+1$ and degree $d$ for every $|d|\geq 4$. In particular, we find a new obstruction for the existence of exceptional bundles other than $r|(2d^{2}+1)$. We also show that there is no exceptional bundle with rank $27$ and degree $11$ to exhibit another different obstruction.

Equations

$$0⟶A⟶B⟶C⟶0$$

$${\mathrm{ext}}^1(B,B)\ge {\mathrm{ext}}^1(A,A)+{\mathrm{ext}}^1(C,C).$$

$$0⟶\underline{Hom}(E,F(-X_y))⟶\underline{Hom}(E,F)⟶\underline{Hom}(E,F{|}_y)\cong {\underline{Hom}}_{X_y}(E_y,F_y)⟶0.$$

$$\mathrm{Hom}(E_U,F_U)\stackrel{f}{⟶}{\mathrm{Hom}}_{X_y}(E_y,F_y)⟶{\mathrm{H}}^1(\underline{Hom}(E_U,F_U(-X_y)))\stackrel{g}{⟶}{\mathrm{H}}^1(\underline{Hom}(E_U,F_U)).$$

$$g:{\mathrm{H}}^0(U,R^1{f}_{\ast }\underline{Hom}(E_U,F_U)\otimes {\mathcal{O}}_Y(-y))\stackrel{y}{⟶}{\mathrm{H}}^0(U,R^1{f}_{\ast }\underline{Hom}(E_U,F_U)).$$

$$L\cap CP=\varnothing .$$

$${\mathrm{End}}_{f^{-1}(s)}(M_{f^{-1}(s)})=\mathbb{C}$$

$${\mathrm{H}}^1(L^{\prime },{\mathcal{G}}_M)={\mathrm{H}}^1(L^{\prime },{\mathrm{GL}}_p)={\mathrm{Bun}}_p(L^{\prime }),$$

$${\mathrm{H}}^0(R^1{p}_{\ast }\underline{Hom}(Q(a_2),Q(a_1)))={\mathrm{H}}^0(R^1{p}_{\ast }\underline{End}(Q)\otimes {\mathcal{O}}_{L^{\prime }}(d))\ne 0.$$

$${\mathrm{H}}^0(R^1{p}_{\ast }{\mathcal{O}}_{\tilde{X}}\otimes {\mathcal{O}}_{L^{\prime }}(d))\ne 0.$$

$$R^1{p}_{\ast }{\mathcal{O}}_{\tilde{X}}=p_{\ast }{\mathcal{O}}_{\tilde{X}}(4H{)}^{\ast }(-1)={\mathcal{O}}_{L^{\prime }}(-1{)}^{\oplus 32}\oplus {\mathcal{O}}_{L^{\prime }}(-2).$$

$${\mathrm{H}}^0(R^1{p}_{\ast }{\mathcal{O}}_{\tilde{X}}\otimes {\mathcal{O}}_{L^{\prime }}(d))={\mathrm{H}}^0({\mathcal{O}}_{L^{\prime }}(d-1{)}^{\oplus 32}\oplus {\mathcal{O}}_{L^{\prime }}(d-2)).$$

$$0⟶E(-4H)⟶E⟶E{|}_{C_s}⟶0,$$

$$\begin{gathered} 0⟶{\mathrm{Hom}}_X(E,E(-4H))⟶{\mathrm{Hom}}_X(E,E)⟶{\mathrm{Hom}}_{C_s}(E_s,E_s) \\ ⟶{\mathrm{Ext}}_X^1(E,E(-4H))⟶{\mathrm{Ext}}_X^1(E,E)=0. \end{gathered}$$

$${\mathrm{hom}}_{C_s}(E_s,E_s)={\mathrm{hom}}_X(E,E)+{\mathrm{ext}}_X^1(E,E(-4H))-{\mathrm{hom}}_X(E,E(-4H))$$

$$0⟶E(-4H)⟶E⟶E{|}_{C_s}⟶0,\forall s\in L^{\prime },$$

$$\mathbb{C}={\mathrm{Hom}}_X(E,E)⟶{\mathrm{Hom}}_X(E,E_{C_s})\cong {\mathrm{Hom}}_{C_s}(E{|}_{C_s},E{|}_{C_s})⟶{\mathrm{Ext}}_X^1(E,E(-4H))=0.$$

$${\mathrm{H}}^1(L^{\prime },{\mathcal{O}}_{L^{\prime }}^{\ast })=\mathrm{Pic}(L^{\prime })=\mathbb{Z}$$

$$0⟶E_{X,d}⟶{\mathcal{O}}_X\otimes {\mathrm{H}}^0({\mathcal{O}}_X(dH))\stackrel{ev}{⟶}{\mathcal{O}}_X(dH)⟶0,$$

$$0⟶F_{C,d}⟶{\mathcal{O}}_C\otimes {\mathrm{H}}^0(C,{\mathcal{O}}_C(dH))⟶{\mathcal{O}}_C(dH)⟶0,$$

$$0⟶{\mathcal{F}}_{\tilde{X},d}⟶p^{\ast }(p_{\ast }{\mathcal{O}}_{\tilde{X}}(dH))\stackrel{ev}{⟶}{\mathcal{O}}_{\tilde{X}}(dH)⟶0.$$

$$C⟶\mathrm{RHom}(A,B)\otimes A\stackrel{ev}{⟶}B.$$

$$0⟶{\mathcal{O}}_X((d-4)H)⟶{\mathcal{O}}_X(dH)⟶{\mathcal{O}}_C(dH)⟶0.$$

$$0⟶{\mathrm{H}}^0({\mathcal{O}}_X((d-4)H))⟶{\mathrm{H}}^0({\mathcal{O}}_X(dH))⟶{\mathrm{H}}^0(C,{\mathcal{O}}_C(dH))⟶{\mathrm{H}}^1({\mathcal{O}}_X((d-4)H))=0.$$

$$\ker$$

$${\mathrm{H}}^0({\mathcal{O}}_X((d-4)H))\otimes {\mathrm{H}}^0({\mathcal{O}}_X(4H))⟶{\mathrm{H}}^0({\mathcal{O}}_X(dH)).$$

$$|4H{|}_X⟶G({\mathrm{h}}^0((d-4)H),{\mathrm{H}}^0({\mathcal{O}}_X(dH))).$$

$$m:L^{\prime }⟶G({\mathrm{h}}^0((d-4)H),{\mathrm{H}}^0({\mathcal{O}}_X(dH))).$$

$$0⟶{\mathrm{H}}^0({\mathcal{O}}_{\tilde{X}}((d-4)H))\otimes {\mathcal{O}}_{\tilde{X}}(-F)⟶{\mathcal{E}}_{\tilde{X},d}⟶{\mathcal{F}}_{\tilde{X},d}⟶0.$$

$$E_{X,d}{|}_C\cong ({\mathrm{H}}^0({\mathcal{O}}_X((d-4)H))\otimes {\mathcal{O}}_C)\oplus F_C.$$

Full text

1. Introduction

1.1. Main Theorem

A vector bundle $E$ on a complex projective variety is called exceptional if $\mathrm{Ext}^{*}(E,E)=\mathbb{C}$. Exceptional vector bundles are very useful in the study of vector bundles and derived categories. For example, they form imporant components of semi-orthogonal decompositions of derived categories [BO95]. They are also closely related to the sharp Bogomolov inequality, the classification of the Chern characters of all stable sheaves on a variety. On $\mathbb{P}^{2}$, the sharp Bogomolov inequality has a precise relation with the exceptional bundles [DLP85]. Exceptional bundles can also form generalized Beilinson spectral sequences. In [CHW17], the authors use this to study the geometry of the moduli spaces of stable sheaves. In [Rud90], various authors studied exceptional vector bundles on surfaces extensively. However, little is known for threefolds.

Unlike on $\mathbb{P}^{2}$ where every exceptional bundle is constructive by mutations [DLP85], classifying exceptional bundles on $\mathbb{P}^{3}$ is a challenging problem [Nog91, Rud95]. The same expectation on constructivity requires proving non-existence of non-constructive exceptional bundles. A necessary condition for an exceptional bundle on $\mathbb{P}^{3}$ with rank $r$ and degree $d$ is that $r|(2d^{2}+1)$ [Kul90]. When this is satisfied, few results are known on the existence of such exceptional bundles. In this paper, we develop techniques to study this question. We give the first non-existence result: For a fixed degree $d$, exceptional bundles of maximal possible rank $r=2d^{2}+1$ either are constructive or do not exist.

Theorem 1.1** (Theorem 5.1).**

On $\mathbb{P}^{3}$, there is no exceptional bundle with rank $2d^{2}+1$ and degree $d$ for every $|d|\geq 4$.

In Section 6, we exhibit another obstruction (Phenomenon 6.2) for the existence of exceptional bundles which is different from that of Theorem 5.1. Using it, we show the following non-existence result that is not covered by Theorem 1.1. This also suggests the feasibility to study the constructivity question further, using the techniques developed in this paper.

Theorem 1.2** (Theorem 6.1).**

On $\mathbb{P}^{3}$, there is no exceptional bundle with rank $27$ and degree $11$.

1.2. Outline of the paper

The strategy to prove Theorem 1.1 is as follows. Suppose $E$ is an exceptional bundle as in Theorem 1.1. We cover $\mathbb{P}^{3}$ by the total family of an appropriate pencil of quartic surfaces (Section 3). On each quartic surface $X$ in this pencil, we construct a bundle $E_{X}$ that must agree with $E|_{X}$ (Section 4). Then we show that all possible bundles on the total family that are swept out by these $E_{X}$ cannot descend to $\mathbb{P}^{3}$ (Section 5).

In Section 6, we prove Theorem 1.2 by showing that the restrictions of $E_{X}$ and $E_{X^{\prime}}$ to the base curve of the pencil are not isomorphic (Phenomenon 6.2).

1.3. Acknowledgements

I want to thank Izzet Coskun, Benjamin Gould and Sixuan Lou for many valuable discussions and conversations. I want to thank Jack Huizenga and Howard Nuer for many useful comments on an early draft of this paper.

2. Preliminaries

In this section we collect facts that relate vector bundles on $\mathbb{P}^{3}$ and quartic surfaces. For more details about exceptional vector bundles, we refer the readers to [Rud90]. For background about $K3$ surfaces and sheaves on them, some good references are [HL97, Huy16].

Definition 2.1**.**

A $K3$ surface is a smooth complex projective surface with $\mathrm{H}^{1}(X,\mathcal{O}_{X})=0$ and $\omega_{X}\cong\mathcal{O}_{X}$.

A smooth quartic surface in $\mathbb{P}^{3}$ is a $K3$ surface [Huy16].

Definition 2.2**.**

Let $X$ be a surface. A vector bundle $E$ on $X$ is rigid if $\mathrm{Ext}^{1}_{X}(E,E)=0$. A vector bundle $F$ is simple if $\mathrm{Hom}_{X}(F,F)=\mathbb{C}$. If $X$ is a $K3$ surface, a vector bundle $G$ is spherical if it is slope stable (see [HL97]) and rigid.

In particular, spherical vector bundles on $K3$ surfaces are simple and rigid. The following lemmas relate exceptional vector bundles on $\mathbb{P}^{3}$ and simple rigid vector bundles on quartic surfaces.

Lemma 2.3** ([Kul90]).**

Let $E$ be an exceptional vector bundle on $\mathbb{P}^{3}$ and $X\subset\mathbb{P}^{3}$ any quartic surface. Then $E|_{X}$ is a simple rigid vector bundle.

Lemma 2.4** ([Zub90]).**

Let $E$ be an exceptional vector bundle on $\mathbb{P}^{3}$ and $X\subset\mathbb{P}^{3}$ a smooth quartic surface with Picard rank 1. Then $E|_{X}$ is a spherical vector bundle.

The following lemma is the crucial ingredient for our proof.

Lemma 2.5**.**

Let $X$ be a $K3$ surface. Then there is at most one spherical vector bundle on $X$ with a given Chern character.

Proof.

Let $E,F$ be spherical vector bundles on $X$ with the same Chern character. By definition, $\mathrm{ext}^{1}(E,E)=0$ and $\mathrm{ext}^{2}(E,E)=\mathrm{hom}(E,E)=1$, we have $\chi(E,E)=\sum_{i\in\mathbb{Z}}\mathrm{ext}^{i}(E,E)=2$. Since $E,F$ have the same Chern character, by Riemann-Roch, we have $\chi(E,F)=\chi(E,E)=2$. Hence either $\mathrm{hom}(E,F)$ or $\mathrm{ext}^{2}(E,F)=\mathrm{hom}(F,E)$ is nonzero, say $\mathrm{hom}(E,F)\neq 0$. Then there is a nonzero map $E\longrightarrow F$. By stability of $E$ and $F$, we have $E\cong F$. ∎

The following fact is known as Mukai’s Lemma.

Lemma 2.6** (Lemma 5.2, [Bri08]).**

Let $\mathcal{A}$ be an abelian category. Suppose

[TABLE]

is a short exact sequence in $\mathcal{A}$ such that $\mathrm{Hom}_{\mathcal{A}}(A,C)=0$. Then we have

[TABLE]

In particular, if $B$ is rigid, then $A$ and $C$ are rigid.

We also need the following lemma, which is a consequence of the cohomology and base change theorems.

Lemma 2.7**.**

Let $Y$ be a smooth curve and $f:X\longrightarrow Y$ be a flat projective morphism. Let $E,F$ be locally free sheaves on $X$. If

1. (1)

For all $y\in Y$, we have $E|_{f^{-1}(y)}\cong F|_{f^{-1}(y)}$, 2. (2)

the dimension of $\mathrm{Ext}^{1}_{f^{-1}(y)}(E|_{f^{-1}(y)},E_{f^{-1}(y)})$ is constant for all $y\in Y$,

then for any $y\in Y$, there exists an open neighborhood $U_{y}\subset Y$ of $y$, such that $E|_{f^{-1}(U)}\cong F|_{f^{-1}(U)}$.

Proof.

For any $y\in Y$ and open subset $U\subset Y$, write $X_{y}:=f^{-1}(y)$ and $X_{U}:=f^{-1}(U)$. For any sheaf $M$ on $X$, write $M_{y}:=M|_{X_{y}}$ and $M|_{U}:=M|_{X_{U}}$.

Let $y\in Y$ be any point. Choose an affine neighborhood $U$ of $y$. Then $\mathrm{Hom}_{X_{U}}(E_{U},F_{U})=\mathrm{H}^{0}(X_{U},\underline{Hom}(E_{U},F_{U}))$. Consider the short exact sequence

[TABLE]

Applying $\mathrm{H}^{0}(-)$, we get

[TABLE]

By the Leray spectral sequence, for any vector bundle $M$ on $X_{U}$, $\mathrm{H}^{1}(X_{U},M)$ admits a filtration whose graded pieces are $\mathrm{H}^{0}(U,R^{1}f_{*}M)$ and $\mathrm{H}^{1}(U,R^{0}f_{*}M)$. Since $U$ is affine, $\mathrm{H}^{1}(U,R^{0}f_{*}M)=0$ for any coherent sheaf $M$. Hence the map $g$ can be identified with the multiplication of the equation of $y$:

[TABLE]

By assumption (2), the dimension of $\mathrm{Ext}^{1}_{X_{y}}(E_{y},F_{y})\cong\mathrm{Ext}^{1}_{X_{y}}(E_{y},E_{y})$ is constant for all $y\in Y$. Since $f$ is projective and $E,F$ are flat over $Y$. By the cohomology and base change theorems, $R^{1}f_{*}\underline{Hom}(E_{U},F_{U})$ is locally free. Then $g$ is injective, $f$ is surjective. By assumption (1), $E_{y}\cong F_{y}$. Hence we may choose a lift $h\in\mathrm{Hom}_{X_{U}}(E_{U},F_{U})$ of $\mathrm{id}\in\mathrm{Hom}_{X_{y}}(E_{y},F_{y})$.

Let $\mathcal{G}^{\prime}=\mathrm{det}(\underline{\mathrm{Hom}}(E_{U},F_{U}))$. The degeneracy locus of $h$ is the vanishing locus of $\mathrm{det}(h)\in\mathrm{H}^{0}(X_{U},\mathcal{G}^{\prime})$. By assumption (2), $\mathcal{G}|_{y^{\prime}}\cong\mathcal{O}_{y^{\prime}}$ for all $y^{\prime}\in Y$. Hence $\mathcal{G}^{\prime}=f^{*}\mathcal{L}$ where $\mathcal{L}=f_{*}\mathcal{G}^{\prime}$ is a line bundle on $Y$. We have the natural identification $\mathrm{H}^{0}(X_{U},\mathcal{G}^{\prime})\cong\mathrm{H}^{0}(U,\mathcal{L})$. Let $Z\subset U$ be the vanishing locus of $\mathrm{det}(h)\in\mathrm{H}^{0}(U,\mathcal{L})$, then $y\notin Z$. Over $U_{s}:=U-Z$, $h:E|_{U_{s}}\longrightarrow F|_{U_{s}}$ is an isomorphism. ∎

3. Pencils of quartic surfaces

In this section we will choose an appropriate pencil of quartic surfaces which sweep out $\mathbb{P}^{3}$ (Proposition 3.4). Let $E$ be an exceptional bundle on $\mathbb{P}^{3}$. On each member $X$ of this pencil, we would like some information about $E|_{X}$. To do this, we cover $X$ by a pencil of curves, such that each curve is the base locus of another pencil of quartic surfaces that contains a smooth member $X^{\prime}$ with Picard rank 1. Then the restrictions of $E$ to these base curves can be understood, since $E|_{X^{\prime}}$ is stable spherical (Lemma 2.4). The requirement for this pencil is Condition 3.1, and we show such pencil exists in Proposition 3.4.

Let $\mathbb{P}^{34}=\mathbb{P}^{34}=\mathbb{P}\mathrm{H}^{0}(\mathcal{O}_{\mathbb{P}^{3}}(4H))$ be the space of quartic surfaces in $\mathbb{P}^{3}$. For a very general line $L\subset\mathbb{P}^{34}$, let $\widetilde{\mathbb{P}^{3}_{L}}=\mathrm{BL}_{C}\mathbb{P}^{3}$, where $C$ is the base locus of $L$. Then $\widetilde{\mathbb{P}^{3}_{L}}$ is the total space of the pencil of quartics parameterized by $L$, it admits morphisms $\pi:\widetilde{\mathbb{P}^{3}_{L}}\longrightarrow L$ and $\nu:\widetilde{\mathbb{P}^{3}_{L}}\longrightarrow\mathbb{P}^{3}$. For any $t\in L$, let $X_{t}=\pi^{-1}(t)$. The requirement for $L$ is as follows.

Condition 3.1**.**

For any $t\in L$, there exists a pencil $L_{t}\subset\mathbb{P}^{34}$ with $t\notin L_{t}$, such that for any $s\in L_{t}$, the pencil $L_{s,t}=\langle X_{s},X_{t}\rangle$ contains a $K3$ surface with Picard rank 1.

To study Condition 3.1, we need to study the Noether-Lefschetz locus for quartic surfaces in $\mathbb{P}^{3}$. In the space of quartic surfaces $\mathbb{P}^{34}$, there is a countable union of hypersurfaces that parametrize smooth quartic surfaces whose Picard rank $>1$ or singular quartic surfaces. We denote these irreducible hypersurfaces by $NL_{i},i\in\mathbb{Z}_{>0}$, and let $NL=\bigcup_{i=1}^{\infty}NL_{i}$. For any $t\in L$, let $\mathrm{cone}_{i}(t)$ be the union of lines $L^{\prime}\subset NL_{i}$ that contains $t$.

Condition 3.1 holds for $t$ if and only if $\mathrm{cone}_{i}(t)$ has codimension $\geq 2$ in $\mathbb{P}^{34}$ for all $i$. To see this, first we assume say $\mathrm{cone}_{1}(t)$ has codimension 1. Then for any pencil $L_{t}$, there exists $s\in L_{t}\cap\mathrm{cone}_{1}(t)$. Then the pencil $L_{s,t}\subset\mathrm{cone}_{1}(t)\subset NL_{1}$ has no member whose Picard rank is 1. Conversely, if $\mathrm{cone}_{i}(t)$ has codimension $\geq 2$ for all $i$, then we may choose $L_{t}$ so that $L_{t}\cap\mathrm{cone}_{i}(t)=\emptyset$ for all $i$. For any $s\in L_{t}$, if the pencil $L_{s,t}$ is contained in $NL$, since $L_{s,t}$ is irreducible and $NL$ is a countable union of $NL_{i}$, there exists some $NL_{i}$, say $NL_{1}$, that contains $L_{s,t}$. By definition $L_{s,t}\subset\mathrm{cone}_{1}(t)$, a contradiction. Hence the pencil $L_{s,t}$ is not contained in $NL$, and Condition 3.1 is satisfied.

Note that if $\mathrm{cone}_{i}(t)$ has codimension 1, then it is a hypersurface, hence $\mathrm{cone}_{i}(t)=NL_{i}$ and $[X_{t}]\in NL_{i}$ is a cone point, in the following sense.

Definition 3.1**.**

For any irreducible subvariety $Y\subset\mathbb{P}^{n}$, a point $p\in Y$ is called a cone point, is for any other point $q\in Y$, the line $\overline{pq}\subset Y$. Denote the set of cone points of $Y$ by $CP(Y)$.

Then we see that $X_{t}$ satisfies Condition 3.1 if and only if $t\in L\subset\mathbb{P}^{34}$ is not a cone point of $NL_{i}$ for any $i$. Let $CP_{i}=CP(NL_{i})$ and $CP=\bigcup_{i=1}^{\infty}CP_{i}$. We may replace the Condition 3.1 for $L$ with the following equivalent condition.

Condition 3.2**.**

[TABLE]

The next lemma shows that cone points are few.

Lemma 3.2**.**

Let $Y\subset\mathbb{P}^{n}$ be an irreducible hypersurface whose degree $d\geq 2$. Then $\emph{dim}CP(Y)<\emph{dim}Y$.

Proof.

Let $L\subset\mathbb{P}^{n}$ be a general line. Then $L$ intersects $Y$ transversely in $d$ distinct points $p_{1},\cdots,p_{d}$. If $d\geq 2$, then $p_{i}$ are not cone points. ∎

In the following we shall prove that $NL_{i}$ is not a hyperplane.

Lemma 3.3**.**

For every $i$, $NL_{i}$is not a hyperplane.

Proof.

Let $v:(\mathbb{P}^{3})^{*}\hookrightarrow\mathbb{P}^{34}$ be the Veronese embedding sending a linear form $f$ to $f^{4}$. Note that the closure $\overline{NL_{i}}$ of $NL_{i}$ is a hypersurface in $\mathbb{P}^{34}$. Then $\overline{NL_{i}}\cap v((\mathbb{P}^{3})^{*})$ contains at least one point $p$. The orbit of $p$ under $\mathbb{P}\mathrm{GL}_{4}$ is $v((\mathbb{P}^{3})^{*})$. Since $\mathbb{P}\mathrm{GL}_{4}$ is connected and Picard rank of quartic surfaces are preserved under $\mathbb{P}\mathrm{GL}_{4}$ action, the orbit of $NL_{i}$ must be $NL_{i}$. Hence $\overline{NL_{i}}$ contains $v((\mathbb{P}^{3})^{*})$. Since the Veronese embedding is non-degenerate, $\overline{NL_{i}}$ cannot be a hyperplane. ∎

By Lemma 3.2 and Lemma 3.3, $CP_{i}$ has codimension $\geq 2$ in $\mathbb{P}^{34}$ for every $i$. Since $CP$ is a contable union of $CP_{i}$, there exists a line $L\subset\mathbb{P}^{34}$ that satisfies Condition 3.2. Furthermore, such $L$ is very general. Therefore we summarize the work of this section in the following proposition.

Proposition 3.4**.**

A very general pencil $L\subset\mathbb{P}^{34}$ satisfies Condition 3.1.

4. Description on quartic surfaces

Let $E$ be an exceptional vector bundle on $\mathbb{P}^{3}$ of rank $2d^{2}+1$ and degree $d$. By Proposition 3.4, we may choose a pencil $L\subset\mathbb{P}^{34}$ that satisfies Condition 3.1. As outlined in the introduction, we will construct a bundle $E_{X}$ on each member $X$ of this pencil, such that the following proposition holds.

Proposition 4.1** (Proposition 4.12).**

On each $X\in L$, there exists a bundle $E_{X}$ such that $E|_{X}\cong E_{X}$.

Proposition 4.1 is the core of this paper, we outline the proof here. We will explictly construct $E_{X}$ first. For any $t\in L$ we cover $X_{t}$ by a pencil $L^{\prime}$ of curves $\{C_{s}\}_{s\in L^{\prime}}$. By the choice of the pencil $L$ (Condition 3.1), $E|_{X}$ and $E_{X}$ can be made isomorphic fiberwise in the family $\{C_{s}\}_{s\in L^{\prime}}$ (Lemma 4.10). We will extend this isomorphism to an open neighborhood (Lemma 2.7), then the bundles on the total family are glued from the local families. Then we will show this glueing is unique by carefully analyzing the relative Harder-Narasimhan filtration.

In this section we fix any $t\in L$, let $X=X_{t}$, $E=E|_{X}$, $L^{\prime}=L_{t}$ and $L^{\prime}_{s}=L_{s,t}$ as in Condition 3.1. Consider the pencil of curves $\{C_{s}=X\cap X_{s}\}_{s\in L^{\prime}}$ on $X$. The base locus of this pencil is 64 points $q_{1},\cdots,q_{64}$. Let $\widetilde{X}=\mathrm{BL}_{q_{1},\cdots,q_{64}}X$ be the total space of this pencil of curves and $\Sigma$ be the exceptional divisor. Then $\widetilde{X}$ admits morphisms $v:\widetilde{X}\longrightarrow X$ and $p:\widetilde{X}\longrightarrow L^{\prime}$.

4.1. Modifications

We define the following notion and discuss its properties when applied to $\widetilde{X}$ in this subsection.

Definition 4.2**.**

Let $f:Z\rightarrow S$ be a flat projective morphism and $M$ be a vector bundle on $Z$. A vector bundle $M^{\prime}$ is called a modification of $M$, if for all $s\in S$, there exists an open neighborhood $U_{s}$, such that $M^{\prime}|_{f^{-1}(U_{s})}\cong M|_{f^{-1}(U_{s})}$. A modification $M^{\prime}$ is called trivial if $M^{\prime}=M\otimes f^{*}\mathcal{L}$ for some line bundle $\mathcal{L}$ on $S$.

Notation 4.3**.**

Under the settings in Definition 4.2, we let $\mathcal{A}_{M}:=f_{*}\underline{End}(M)$ and $\mathcal{G}_{M}\subset\mathcal{A}_{M}$ be the sheaf of groups whose local sections are multiplicative invertible sections of $\mathcal{A}_{M}$.

Suppose $\mathrm{End}_{f^{-1}(s)}(M|_{f^{-1}(s)})$ in Definition 4.2 has constant dimension for all $s\in S$. Then modifications of $M$ are parametrized by the (nonabelian) cohomology set $\mathrm{H}^{1}(S,\mathcal{G}_{M})$. We first note the following two lemmas about modifications.

Lemma 4.4**.**

Using the notations in Definition 4.2, if $M$ is fiberwise simple, i.e.

[TABLE]

for all $s\in S$, then all modifications of $M$ are trivial.

Proof.

Note that $\mathcal{A}_{M}=\mathcal{O}_{S}$ and $\mathcal{G}_{M}=\mathcal{O}_{S}^{*}$, we have $\mathrm{H}^{1}(S,\mathcal{G}_{M})=\mathrm{H}^{1}(S,\mathcal{O}_{S}^{*})=\mathrm{Pic}(S)$. ∎

Lemma 4.5**.**

Let $Q$ be a rigid and fiberwise simple vector bundle on $\widetilde{X}$ and $M=Q^{\oplus p}$ for some $p>0$. Then all rigid modifications of $M$ with respect to $p:\widetilde{X}\longrightarrow L^{\prime}$ are trivial. In particular, for every line bundle $\mathcal{L^{\prime}}$ on $L^{\prime}$, all rigid modifications of $p^{*}\mathcal{L^{\prime}}^{\oplus p}$ are trivial.

Proof.

Recall that $\widetilde{X}=\mathrm{BL}_{q_{1},\cdots,q_{64}}X$ and $\Sigma$ is the exceptional divisor of $v:\widetilde{X}\rightarrow X$. Let $F$ be the fiber class of $p:\widetilde{X}\rightarrow L^{\prime}$.

Since $Q$ is fiberwise simple, $\mathcal{A}_{M}=\mathcal{O}_{L^{\prime}}\otimes\mathrm{Mat}_{p}$ and $\mathcal{G}_{M}=\mathrm{GL}_{p}$, we have

[TABLE]

where $\mathrm{Bun}_{p}(L^{\prime})$ is the set of rank $p$ vector bundles on $L^{\prime}$. Hence any modification $M^{\prime}$ of $M$ is of the form $\oplus_{i=1}^{p}Q(a_{i}F)$. Now assume some $a_{i}\neq a_{j}$, say $d:=a_{1}-a_{2}\geq 1$, we need to show $M^{\prime}$ is not rigid, i.e. $\mathrm{Ext}^{1}_{\widetilde{X}}(M^{\prime},M^{\prime})\neq 0$. It suffices to show $\mathrm{Ext}^{1}_{\widetilde{X}}(p^{*}\mathcal{O}_{L^{\prime}}(a_{2}),p^{*}\mathcal{O}_{L^{\prime}}(a_{1}))\neq 0$, since it is a summand of $\mathrm{Ext}^{1}_{\widetilde{X}}(M^{\prime},M^{\prime})$. By the Leray spectral sequence, it suffices to show

[TABLE]

Since $\mathcal{O}_{\widetilde{X}}$ is a summand of $\underline{End}(Q)$, it suffices to show

[TABLE]

The relative canonical divisor of $p:\widetilde{X}\longrightarrow L^{\prime}$ is $K_{rel}=K_{\widetilde{X}}-p^{*}K_{L^{\prime}}=\Sigma+2F=4H+F$. By relative Serre duality, we have

[TABLE]

Hence

[TABLE]

By assumption, $d\geq 1$. Hence $d-1\geq 0$ and $\mathrm{H}^{0}(R^{1}p_{*}\mathcal{O}_{\widetilde{X}}\otimes\mathcal{O}_{L^{\prime}}(d))\neq 0$. ∎

The next lemma shows rigid modifications can be checked fiberwise.

Lemma 4.6**.**

Let $E,F$ be simple rigid vector bundles on $X$. If $E|_{C_{s}}\cong F|_{C_{s}}$ for all $s\in L^{\prime}$, then $F$ is a modification of $E$.

Proof.

Since $E=v_{*}v^{*}(E),F=v_{*}v^{*}(F)$, the lemma will follow from Lemma 2.7. By assumption, condition (1) in Lemma 2.7 is satisfied, it suffices to check condition (2). By Lemma 2.3, $E$ is a simple rigid vector bundle on $X$. For any $s\in L^{\prime}$, applying $\mathrm{Hom}_{X}(E,-)$ to the short exact sequence

[TABLE]

we have the long exact sequence

[TABLE]

Hence

[TABLE]

is independent of $s\in L^{\prime}$. By Riemann-Roch, $\mathrm{ext}^{1}(E_{s},E_{s})$ is also independent of $s\in L^{\prime}$. ∎

We will need the following version of Theorem 1.2 in [Pol11], which we prove from a different perspective. The relation of these two versions will be explained in Corollary 5.3.

Proposition 4.7**.**

Let $E,F$ be simple vector bundles on $X$ such that $E|_{C_{s}}\cong F|_{C_{s}}$ for all $s\in L^{\prime}$. If $\mathrm{Ext}^{1}_{X}(E,E(-4H))=0$, then $E\cong F$.

Proof.

Using Notation 4.3, the set of modifications of $v^{*}(E)$ is $\mathrm{H}^{1}(L,\mathcal{G}_{v^{*}(E)})$. By Lemma 2.7, $v^{*}(F)\in\mathrm{H}^{1}(L,\mathcal{G}_{v^{*}(E)})$. Applying $\mathrm{Hom}_{X}(E,-)$ to the following short exact sequence

[TABLE]

we get

[TABLE]

Hence $\mathrm{Hom}_{C_{s}}(E|_{C_{s}},E|_{C_{s}})=\mathbb{C}$ for all $s\in L^{\prime}$, $\mathcal{A}_{v^{*}(E)}=\mathcal{O}_{L^{\prime}}$, and $\mathcal{G}_{v^{*}(E)}\cong\mathcal{O}_{L^{\prime}}^{*}$. Therefore

[TABLE]

is in one-to-one correspondence with line bundles on $L^{\prime}$. Let $\Sigma$ be the exceptional divisor of $v$. Note that the restrictions to $\Sigma$ of different twists of $v^{*}(E|_{X})$ by line bundles on $L^{\prime}$ are non-isomorphic. Since $v^{*}(E)|_{\Sigma}$ and $v^{*}(F)|_{\Sigma}$ are both trivial, we have $v^{*}(E|_{X})\cong v^{*}(F|_{X})$. ∎

4.2. The vector bundle $E_{X}$

In this subsection we define the bundle $E_{X}$ on $X$ and discuss its properties. Then we will prove the main result of this section (Proposition 4.12).

In the following we assume $d\geq 0$, the case $d<0$ can be dealt by taking the dual. Let $v_{d}$ be the Mukai vector $(2d^{2}+1,-dH,1)\in H^{*}_{alg}(X)$. We define $E_{X,d}$ on $X$ as the sheaf that fits into the following exact sequence

[TABLE]

and let $\mathcal{E}_{\widetilde{X},d}=v^{*}E_{X,d}$ on $\widetilde{X}$. Let $C\in|4H|_{X}$ be any curve. We also define $F_{C,d}$ as the sheaf on $C$ that fits into the following exact sequence

[TABLE]

and let $\mathcal{F}_{\widetilde{X},d}$ on $\widetilde{X}$ be the sheaf swept out by $F_{C,d}$. In other words, $\mathcal{F}_{\widetilde{X},d}$ fits into the following exact sequence

[TABLE]

The first observation is that $E_{X,d}$ is simple and rigid. This is a consequence of a more general proposition.

Proposition 4.8**.**

Let $Y$ be a $K3$ surface and $A,B\in\mathcal{D}^{b}(Y)$ be simple rigid objects. Then the mapping cone of $ev:\mathrm{RHom}(A,B)\otimes A\longrightarrow B$ is simple and rigid.

Proof.

Let $C$ be the mapping cone. Let $S=\mathbb{C}\oplus\mathbb{C}[-2]\in\mathcal{D}^{b}(Y)$ and $p_{i}:S\rightarrow\mathcal{H}^{i}(S)[-i]$ be the projection map. We need to show $\mathrm{RHom}(C,C)\cong S$. Applying $\mathrm{RHom}(-,-)$ to the following exact triangle in both arguments:

[TABLE]

Note that $\mathrm{RHom}(A,A)\cong\mathrm{RHom}(B,B)\cong S$ by assumption, we get a commutative diagram whose rows and columns are all exact triangles:

{\mathrm{RHom}(C,C)}$${\mathrm{RHom}(C,A)\otimes\mathrm{RHom}(A,B)}$${\mathrm{RHom}(C,B)}$${\mathrm{RHom}(A,B)^{*}\otimes\mathrm{RHom}(A,C)}$${\mathrm{RHom}(A,B)^{*}\otimes S\otimes\mathrm{RHom}(A,B)}$${\mathrm{RHom}(A,B)^{*}\otimes\mathrm{RHom}(A,B)}$${\mathrm{RHom}(B,C)}$${\mathrm{RHom}(B,A)\otimes\mathrm{RHom}(A,B)}$${\mathrm{RHom}(B,B)\cong S.}$$\scriptstyle{p_{0}}$$\scriptstyle{p_{0}^{*}}

For any $M,N\in\mathcal{D}^{b}(Y)$, by Serre Duality we have $\mathrm{RHom}(M,N)^{*}\cong\mathrm{RHom}(N,M)[2]$. Hence applying dual will flip the diagram about the anti-diagonal and shift its degree by 2. Therefore $\mathrm{RHom}(C,C)\cong\mathrm{RHom}(B,B)\cong S$.

∎

Corollary 4.9**.**

The sheaf $E_{X,d}$ is simple and rigid for $d>0$. In particular, it is locally free.

Proof.

By the Kodaira Vanishing Theorem, we have $\mathrm{H}^{1}(\mathcal{O}_{X}(dH))=0$. Hence we have $\mathrm{RHom}(\mathcal{O}_{X},\mathcal{O}_{X}(dH))=\mathrm{H}^{0}(\mathcal{O}_{X}(dH))$ for $d>0$. The claim follows from Proposition 4.8. ∎

Consider the following exact sequence

[TABLE]

Taking cohomology, we have

[TABLE]

Hence $\mathrm{h}^{0}(C,\mathcal{O}_{C}(dH))$ is independent of the choice of $C\in|4H|_{X}$. Furthermore, when $d\geq 4$, we see that $\mathrm{H}^{0}(C,\mathcal{O}_{C}(dH))$ is a strict quotient of $\mathrm{H}^{0}(\mathcal{O}_{X}(dH))$. Hence $F_{C,d}$ is a strict quotient of $E_{X,d}|_{C}=\mathcal{E}_{d}|_{C}$. We have the following commutative diagram

[TABLE]

On each $C\in|4H|_{X}$, $K_{\widetilde{X},d}|_{C}\cong\mathcal{O}_{C}\otimes\mathrm{H}^{0}(\mathcal{O}_{X}(-C))$. However, we will see that $K_{\widetilde{X},d}$ is not a constant family. Consider the multiplication map

[TABLE]

This induces a morphism

[TABLE]

Precomposing this map with the inclusion $L^{\prime}\subset|4H|_{X}$, we have a morphism

[TABLE]

Let $S$ be the tautological subbundle of $G(\mathrm{h}^{0}((d-4)H),\mathrm{H}^{0}(\mathcal{O}_{X}(dH)))$, then by construction $K_{\widetilde{X},d}=m^{*}S$. Explicitly, $K_{\widetilde{X},d}\cong\mathrm{H}^{0}(\mathcal{O}_{X}((d-4)H))\otimes\mathcal{O}_{\widetilde{X}}(-F)$, where $F$ is the fiber class of $p$. Hence we have the following exact sequence

[TABLE]

We will also need the following observations.

Lemma 4.10**.**

The sequence (4) splits on each fiber. More generally, for every $C\in|4H|_{X}$ we have

[TABLE]

Proof.

On each $C\in|4H|_{X}$, the evaluation map $ev:\mathrm{H}^{0}(\mathcal{O}_{X}(dH))\longrightarrow\mathcal{O}_{X}(dH)$ vanishes on the subbundle

[TABLE]

Hence the sequence (4) splits on $C$. ∎

Lemma 4.11**.**

The sequence (4) is the relative Harder-Narasimhan filtration of $\mathcal{E}_{\widetilde{X},d}$, in the sense that the restriction of (4) to every smooth fiber is the Harder-Narasimhan filtration of the restriction of $\mathcal{E}_{\widetilde{X},d}$.

Proof.

Note that for $d\geq 4$, $F_{C_{s},d}$ is stable on every smooth fiber $C_{s}$ by [But94]. Since $\mathrm{H}^{0}(\mathcal{O}_{X}((d-4)H))\otimes\mathcal{O}_{C_{s}}$ is semistable and $\mu(F_{C_{s},d})<\mu(\mathcal{O}_{C_{s}})$, the lemma follows from the uniqueness of the Harder-Narasimhan filtration. ∎

Now we show the main result of this section.

Proposition 4.12**.**

Suppose $E$ is an exceptional bundle on $\mathbb{P}^{3}$ with rank $2d^{2}+1$ and degree $-d$ for some $d>0$. Then for any smooth quartic $X$ in a pencil that satisfies Condition 3.1, we have $E|_{X}\cong E_{X,d}$ (defined in (1)).

Proof.

The proof will be separated into several steps. First we will see that $v^{*}(E|_{X})$ is a rigid modification of $v^{*}(E_{X,d})$ (Definition 4.2). Then we find an invariant for such modifications. Next, we show that the invariants of $v^{*}(E|_{X})$ and $v^{*}(E_{X,d})$ are equal. Finally, this implies that $v^{*}(E|_{X})\cong v^{*}(E_{X,d})$. Since when $d\leq 3$ the relative Harder-Narasimhan filtration of $v^{*}(E|_{X})$ is different from the case $d\geq 4$, we deal with this case first.

Let $X$ be a quartic surface in a pencil that satisfies Condition 3.1. The Mukai vector of $E|_{X}$ is of the form $v_{d}=(2d^{2}+1,-dH,a)$. Since $E|_{X}$ is simple and rigid, $v_{d}^{2}=-2$, hence $a=1$.

Step 1: Prove the proposition for $0<d\leq 3$.

In this case, there exists a constructive exceptional bundle $E$ on $\mathbb{P}^{3}$ that fits into the exact sequence

[TABLE]

Since $d\leq 3$, the restriction map $\mathrm{H}^{0}(\mathcal{O}_{\mathbb{P}^{3}}(dH))\rightarrow\mathrm{H}^{0}(\mathcal{O}_{X}(dH))$ is an isomorphism, and $E|_{X}$ fits into the following exact sequence

[TABLE]

Since $E$ is constructive, by Theorem 2.3 of [BP93], $\mathrm{Hom}_{\mathbb{P}^{3}}(E,E(4H))=0$. By Proposition 4.7, any exceptional bundle $F$ with the same rank and degree as $E$ has $F\cong E$, hence Proposition 4.12 is proved for $d\leq 3$. In the following we assume $d\geq 4$.

Step 2: Show that $v^{*}(E|_{X})$ is a rigid modification of $v^{*}(E_{X,d})$.

If $X$ is smooth with Picard rank 1, then by Lemma 2.4, $E|_{X}\in M_{X,H}(v_{d})$. By [BM14] and Corollary 4.13 of [Liu22] (or a direct computation), $E_{X,d}$ is stable for any $d>0$. Hence $E|_{X}\cong E_{X,d}$ when $\mathrm{Pic}(X)=\mathbb{Z}H$.

For singular $X$ or $X$ with higher Picard rank, let $L^{\prime}\subset|4H|_{X}$ be as in the beginning of the section and $\widetilde{X}$ be the universal family of $L^{\prime}$. Since $L^{\prime}$ satisfies Condition 3.1, for any $s\in L^{\prime}$, there exists $u\in L^{\prime}_{s}=\langle X,X_{s}\rangle$ such that $X_{u}$ is smooth with Picard rank 1. In particular $X_{u}\neq X$, hence $C_{s}=X\cap X_{u}$. By Lemma 4.10, we have

[TABLE]

Hence $v^{*}(E|_{X})$ is a modification of $\mathcal{E}_{X,d}$ (Definition 4.2). By Lemma 2.3, it is rigid.

Step 3: Find the invariant.

The relative Harder-Narasimhan factors of $v^{*}(E|_{X})$ are modifications of the relative Harder-Narasimhan factors of $\mathcal{E}_{X,d}$. By Lemma 4.11, (4) is the relative Harder-Narasimhan filtration of $\mathcal{E}_{X,d}$. Hence the relative Harder-Narasimhan filtration of $v^{*}(E|_{X})$ is of the following form

[TABLE]

where $G_{1}$ is a modification of $\mathrm{H}^{0}(\mathcal{O}_{\widetilde{X}}((d-4)H))\otimes\mathcal{O}_{\widetilde{X}}(-F)$ and $G_{2}$ is a modification of $\mathcal{F}_{\widetilde{X},d}$. By Lemma 4.4, $G_{2}\cong\mathcal{F}_{\widetilde{X},d}\otimes\mathcal{O}_{\widetilde{X}}(bF)$ for some $b\in\mathbb{Z}$. Since $G_{1},G_{2}$ are relative Harder-Narasimhan factors by Lemma 4.11, we have $p_{*}\underline{Hom}(G_{1},G_{2})=0$, hence

[TABLE]

Since $v^{*}(E|_{X})$ is rigid, by Mukai’s Lemma (Lemma 2.6) $G_{1}$ is also rigid. By Lemma 4.5, we have $G_{1}\cong\mathrm{H}^{0}(\mathcal{O}_{\widetilde{X}}((d-4)H))\otimes\mathcal{O}_{\widetilde{X}}(-F+aF)$ for some $a\in\mathbb{Z}$. Let $p:=\mathrm{h}^{0}(\mathcal{O}_{X}((d-4)H))$, $c:=b-a$, and

[TABLE]

Then by the Leray spectral sequence, $\mathrm{Ext}^{1}(\mathcal{F}_{\widetilde{X},d},\mathcal{O}_{\widetilde{X}}(-F))$ admits a filtrations whose factors are $\mathrm{H}^{0}(R^{1}p_{*}\mathcal{G})$ and $\mathrm{H}^{1}(p_{*}\mathcal{G})$. The invariant that we will use is $\mathrm{h}^{1}(R^{1}p_{*}\mathcal{G})$. We will show that the invariants of $v^{*}(E|_{X})$ and $v^{*}(E_{X,d})$ are equal. Explicitly, the goal is to show

[TABLE]

Step 4: Prove (6).

First note that $\mathrm{ext}^{1}(\mathcal{F}_{\widetilde{X},d},\mathcal{O}_{\widetilde{X}}(-F))=p$ for the following reason. Since $\mathcal{E}_{\mathcal{X},d}$ is rigid, any general classes $\delta_{1},\delta_{2}\in\mathrm{Ext}^{1}(\mathcal{F}_{\widetilde{X},d},\mathcal{O}_{\widetilde{X}}(-F)^{\oplus p})$ have the middle term isomorphic to $\mathcal{E}_{\widetilde{X},d}$. Since $\mathrm{Hom}(\mathcal{F}_{\widetilde{X},d},\mathcal{O}_{\widetilde{X}}(-F))=0$, we have that $\delta_{1}=g(\delta_{2})$ for some $g\in\mathrm{GL}_{p}$ under the natural action of $\mathrm{GL}_{p}$ on $\mathrm{Ext}^{1}(\mathcal{F}_{\widetilde{X},d},\mathcal{O}_{\widetilde{X}}(-F)^{\oplus p})$ (say, by Lemma 6.3 in [CH18]). Hence we have

[TABLE]

namely $\mathrm{ext}^{1}(\mathcal{F}_{\widetilde{X},d},\mathcal{O}_{\widetilde{X}}(-F))\leq p$. However, if $\mathrm{ext}^{1}(\mathcal{F}_{\widetilde{X},d},\mathcal{O}_{\widetilde{X}}(-F))<p$, then the sequence (4) partially splits, i.e. there exists some summand $\mathcal{O}_{\widetilde{X}}(-F)\subset\mathcal{O}_{\widetilde{X}}(-F)^{\oplus p}$, such that the restriction of (4) to it admits a section. In particular, $\mathcal{O}_{\widetilde{X}}(-F)$ is a summand of $\mathcal{E}_{\widetilde{X},d}$. Hence $\mathcal{E}_{\widetilde{X},d}$ is not simple, this contradicts Corollary 4.9.

Next, we claim that $\mathrm{H}^{0}(R^{1}p_{*}\mathcal{G})=0$. Suppose not, then there exists a nonzero class $\delta\in\mathrm{H}^{0}(R^{1}p_{*}\mathcal{G}^{\oplus p})$ whose middle term is denoted by $E^{\prime}$. By construction, there exists some fiber $C_{s}$, such that the restriction of $\delta$ to $C_{s}$ is nonsplit. However, since the sequence (4) in $\mathrm{Ext}^{1}(\mathcal{F}_{\widetilde{X},d},\mathcal{O}_{\widetilde{X}}(-F)^{\oplus p})$, $E^{\prime}$ is a deformation of $E_{\widetilde{X},d}$. Since $E_{\widetilde{X},d}$ is rigid, $E^{\prime}\cong E_{\widetilde{X},d}$. By Lemma 4.10, $E^{\prime}|_{C_{s}}$ splits, a contradiction. Hence $\mathrm{H}^{0}(R^{1}p_{*}\mathcal{G})=0$, and $\mathrm{h}^{1}(p_{*}\mathcal{G})=p\neq 0$.

Note that $v^{*}(E|_{X})$ is a modification of $E_{X,d}$. By Lemma 4.10, $E|_{C_{s}}\cong\mathcal{O}_{C_{s}}^{\oplus p}\oplus F_{C_{s},d}$. By the same reason applied to sequence (5),

[TABLE]

must vanish. Hence $\mathrm{h}^{0}(R^{1}p_{*}\mathcal{G}(c))=0$ and $\mathrm{h}^{1}(p_{*}\mathcal{G}(c))=p$.

Step 5: Show that $v^{*}(E|_{X})\cong v^{*}(E_{X,d})$.

Since $\mathrm{h}^{1}(p_{*}\mathcal{G})=p\neq 0$ and $p_{*}\mathcal{G}$ is a sheaf on $\mathbb{P}^{1}$, we must have $c=0$. Hence $a=b$. Since the restrictions of $v^{*}(E|_{X})$ and $\mathcal{E}_{\widetilde{X},d}$ to $\Sigma$ are both trivial, we have $a=b=0$. Hence $v^{*}(E|_{X})$ and $\mathcal{E}_{\widetilde{X},d}$ are both rigid objects in $\mathrm{Ext}^{1}(\mathcal{F}_{\widetilde{X},d},\mathcal{O}_{\widetilde{X}}(-F)^{\oplus p})$, we must have $v^{*}(E|_{X})\cong\mathcal{E}_{\widetilde{X},d}$. Taking pushforward along $v$, we get

[TABLE]

∎

5. Main results

Recall that we have chosen a pencil of quartic surfaces $L\subset\mathbb{P}^{34}$ that satisfies Condition 3.1, and $\widetilde{\mathbb{P}^{3}}=\widetilde{\mathbb{P}^{3}_{L}}$ is the total space of the pencil of quartics parametrized by $L$. In this section we prove the following main theorem of this paper. The proof follows a similar strategy as that of Proposition 4.12.

Theorem 5.1**.**

On $\mathbb{P}^{3}$, there exists no exceptional bundles of rank $2d^{2}+1$ and degree $d$ for every $|d|\geq 4$.

We first make the following observation.

Lemma 5.2**.**

Let $E$ be an exceptional vector bundle on $\widetilde{\mathbb{P}^{3}}$ and $F$ be a vector bundle such that $E|_{X_{t}}\cong F|_{X_{t}}$ for all $t\in L$. Then $F$ is a modification of $E$ in the sense of Definition 4.2.

Proof.

The lemma will follow from Lemma 2.7. By assumption, condition (1) in Lemma 2.7 is satisfied. Hence it suffices to check condition (2) in Lemma 2.7. Since $\widetilde{\mathbb{P}^{3}}=\mathrm{BL}_{C}\mathbb{P}^{3}$, we have

[TABLE]

Since $E$ is exceptional, we have $\mathrm{Ext}^{2}(E,E(-X_{t}))\cong\mathrm{Ext}^{1}(E,E)^{*}=0$ by Serre Duality. Hence the following exact sequence

[TABLE]

implies that $\mathrm{Ext}^{1}_{X_{t}}(E_{t},E_{t})=0$ for all $t\in L$, condition (2) in Lemma 2.7 is satisfied. ∎

As an application, we prove Theorem 1.2 in [Pol11] from a different perspective.

Corollary 5.3** (Theorem 1.2, [Pol11]).**

Let $E,F$ be exceptional bundles on $\mathbb{P}^{3}$ with the same rank and degree. If $\mathrm{Ext}^{1}_{\mathbb{P}^{3}}(E,E(4H))=0$, then $E\cong F$.

Proof.

By assumption, the following exact sequence

[TABLE]

implies that $\mathrm{Ext}^{1}_{X}(E|_{X},E|_{X}(-4H))=\mathrm{Ext}^{1}_{X}(E|_{X},E|_{X}(4H))^{*}=0$. By Proposition 4.7, we have $E|_{X}\cong F|_{X}$ for all $X\in L$. By Lemma 5.2, $\nu^{*}F$ is a modification of $\nu^{*}(E)$. By Lemma 2.3, they are fiberwise simple. By Lemma 4.4, the modification is trivial. ∎

We construct a sheaf $\mathcal{E}_{d}$ on $\widetilde{\mathbb{P}^{3}}$ that is swept out by $E_{X,d}$ when $X$ varies in the pencil $L$, i.e. $\mathcal{E}_{d}|_{X}\cong E_{X,d}$ for all $[X]\in L$. One way to do this is to let $\mathcal{E}_{d}$ fit into the following exact sequence

[TABLE]

Recall that $C$ is the base curve of $L\subset\mathbb{P}^{34}$ and $\widetilde{\mathbb{P}^{3}}=\mathrm{BL}_{C}\mathbb{P}^{3}$. Let $Y$ be the exceptional divisor of $\nu:\widetilde{\mathbb{P}^{3}}\longrightarrow\mathbb{P}^{3}$. Let $W_{n}:=\mathrm{H}^{0}(C,\mathcal{O}_{C}(n))$ for all $n\in\mathbb{Z}$, $w_{n}=\mathrm{dim}W_{n}$, and $F$ be the fiber class of $\pi$. Write $d=4k+e$ for $k\in\mathbb{Z}$ and $0\leq e<4$ uniquely.

Lemma 5.4**.**

For $d\geq 0$, we have

[TABLE]

where $F$ is the fiber class of $\pi$, and $F_{C,d}$ is defined in (2).

Proof.

Since $C$ is a $(4,4)$-complete intersection, we have $Y\cong C\times L$. In $\mathrm{Pic}(\widetilde{\mathbb{P}^{3}})$, we have $Y+F=4H$. To compute $\pi_{*}\mathcal{O}_{\widetilde{\mathbb{P}^{3}}}(dH)$, we will apply $\pi_{*}$ to the following exact sequence on $\widetilde{\mathbb{P}^{3}}$:

[TABLE]

Let $X$ be any fiber of $\pi:\widetilde{\mathbb{P}^{3}}\rightarrow L$. Then $X$ has $\mathrm{H}^{1}(X,\mathcal{O}_{X})=0$ and $\omega_{X}\cong\mathcal{O}_{X}$. By the Kodaira Vanishing Theorem and Serre duality, we have $\mathrm{H}^{1}(X,\mathcal{O}_{X}(nH))=0$ for all $n\in\mathbb{Z}$. Hence we have

[TABLE]

Then we have

[TABLE]

Then we claim that for $d\geq 0$, we have

[TABLE]

We use induction on $k$. By the projection formula, we have

[TABLE]

When $k=0$, namely $0\leq d<4$, we have $\mathrm{H}^{0}(X,\mathcal{O}_{X}((d-4)H))=0$ for every fiber $X$ of $\pi$. Hence $\pi_{*}\mathcal{O}_{\widetilde{\mathbb{P}^{3}}}(dH-Y)=0$. Therefore $\pi_{*}\mathcal{O}_{\widetilde{\mathbb{P}^{3}}}(dH)\cong\pi_{*}\mathcal{O}_{Y}(dH)\cong\mathcal{O}_{L}\otimes W_{d}$, the claim holds in this case. Now suppose the claim holds for $(k-1)$. Then

[TABLE]

Hence $\pi_{*}\mathcal{O}_{\widetilde{\mathbb{P}^{3}}}(dH)$ fits into the following exact sequence on $L$:

[TABLE]

This sequence splits, since

[TABLE]

Hence the claim is proved.

Restricting the sequence (7) to $Y\cong C\times L$, we get

[TABLE]

Recall that $F_{C,d}$ fits into the following exact sequence by construction (2):

[TABLE]

On the summand $\mathcal{O}_{C\times L}\otimes W_{d}$, the map $ev|_{C\times L}$ is the pullback of $ev_{C}$ along $\nu$. On other summands, $ev|_{C\times L}$ vanishes. Hence we have

[TABLE]

∎

Now we prove Theorem 5.1.

Proof of Theorem 5.1.

Suppose $E$ is an exceptional bundle on $\mathbb{P}^{3}$ with rank $2d^{2}+1$ and degree $-d$. We assume $d\geq 0$, the case where $d<0$ can be proved by taking dual. By Proposition 4.12,

[TABLE]

Since $E$ is exceptional on $\mathbb{P}^{3}$, $\nu^{*}(E)$ is exceptional on $\widetilde{\mathbb{P}^{3}}$. By Lemma 5.2, $\nu^{*}(E)$ is a modification of $\mathcal{E}_{d}$ (Definition 4.2). By Corollary 4.9, $\mathcal{E}_{d}|_{X}$ is simple and rigid for all fibers $X$ of $\pi$. By Lemma 4.4, $\nu^{*}E$ is a trivial modification of $\mathcal{E}_{d}$. In particular, since $\nu^{*}(E)|_{Y}$ is trivial, we must have $\mathcal{E}_{d}|_{Y}\cong\mathcal{O}_{Y}(mF)^{\oplus q}$ for some $m,q\in\mathbb{Z}$. However, by Lemma 5.4, we have

[TABLE]

Recall that $W_{n}=\mathrm{H}^{0}(C,\mathcal{O}_{C}(nH))$. When $d\geq 4$, $W_{d-4j}\neq 0$ for $1\leq j\leq k$. Hence $\mathcal{E}_{d}$ is not of the form $\mathcal{O}_{Y}(mF)^{\oplus q}$, a contradiction.

∎

6. Another obstruction

In this section we discuss a different phenomenon from that of Theorem 5.1 which obstructs the existence of exceptional bundles. The following theorem is a consequence of this different obstruction.

Theorem 6.1**.**

On $\mathbb{P}^{3}$, there exists no exceptional bundles of rank $27$ and degree $11$.

In this section, for any smooth quartic surface $X\subset\mathbb{P}^{3}$ with Picard rank 1, we fix the Mukai vector $v=(27,11H,9)\in\mathrm{H}^{*}_{alg}(X)$ and let $E_{X}\in M_{X,H}(v)$. Suppose $E$ is an exceptional bundle with rank $27$ and degree $11$. Then for such an $X$ we have $E|_{X}\cong E_{X}$ [Zub90]. Let $\{X_{t}\subset\mathbb{P}^{3}\}_{t\in L},L\subset\mathbb{P}^{34}$ be a general pencil of quartic surfaces and $C\subset\mathbb{P}^{3}$ be its base curve. If such an exceptional bundle $E$ exists, then

[TABLE]

for any $t,t^{\prime}\in L$. However, in fact this is not true: the following phenomenon obstructs the existence of exceptional bundles of rank $27$ and degree $11$.

Phenomenon 6.2** (Corollary 6.11).**

For general $t,t^{\prime}\in L$, $E_{X_{t}}|_{C}\not\cong E_{X_{t^{\prime}}}|_{C}$.

Note that Phenomenon 6.2 is different from the the reason for Theorem 5.1. In Theorem 5.1, $E_{X_{t}}|_{C}\cong E_{X_{t^{\prime}}}|_{C}$ for any $t,t^{\prime}\in L$. The difficulty there is to rule out modifications (Definition 4.2) of $E_{X_{t}}|_{C}$, since they are not simple. However, in the case of rank $27$ and degree $11$, we have $E_{X_{t}}|_{C}$ is simple for general $t\in L$. The difficulty here is to show $E_{X_{t}}|_{C}$ and $E_{X_{t^{\prime}}}|_{C}$ are non-isomorphic for general $t,t^{\prime}\in L$. This also shows that the condition $\mathrm{Ext}^{1}_{X}(E,E(4H))=0$ in [Pol11] is not a sufficient condition for the existence of exceptional bundles.

We first describe $E_{X}$ for a smooth quartic surface $X$ with Picard rank 1. By the algorithm in [Liu22], $E_{X}$ can be constructed as follows. Let $M=T_{\mathbb{P}^{3}}(-H)$. Let $N_{X}\in\mathcal{D}^{b}(X)$ fits into the following exact triangle

[TABLE]

where the extension is described below. Note that $\mathrm{Ext}_{X}^{1}(\mathcal{O}_{X}(H),\mathcal{O}_{X}[1])\cong\mathrm{H}^{0}(X,\mathcal{O}_{X}(H))^{*}$. Hence we may take the coevaluation map $\mathcal{O}_{X}(H)\overset{coev}{\longrightarrow}\mathcal{O}_{X}[2]\otimes\mathrm{H}^{0}(X,\mathcal{O}_{X}(H))$ and let $N_{X}$ be the cone. Then $E_{X}$ can be constructed as the canonical extension that fits into the following exact triangle

[TABLE]

We have a canonical identification $\mathrm{Ext}^{1}(N_{X},M|_{X})^{*}\cong\mathrm{H}^{0}(X,\mathcal{O}_{X}(2H))$. For simplicity, we set the following notation.

Notation 6.3**.**

For any quartic surface $X\subset\mathbb{P}^{3}$ and $d\in\mathbb{Z}$, let $V_{X,d}:=\mathrm{H}^{0}(X,\mathcal{O}_{X}(dH))$. For $d\leq 3$, they are fixed quotients of $\mathrm{H}^{0}(\mathbb{P}^{3},\mathcal{O}_{\mathbb{P}^{3}}(dH))$, we denote them by $V_{d}$ in this case.

Using Notation 6.3, the long exact sequence of cohomology sheaves of the sequence (9) is

[TABLE]

The map $f$ is given by the composition of the following maps

[TABLE]

where $m:V_{1}\otimes V_{1}\longrightarrow V_{2}$ is the multiplication map. Note that using the same description, $f$ can be defined on $\mathbb{P}^{3}$. We let $F$ be the cokernel:

[TABLE]

Note that $\mathrm{Ext}^{1}_{\mathbb{P}^{3}}(\mathcal{O}_{\mathbb{P}^{3}}(H),F)=0$, hence the same construction cannot produce a simple bundle on $\mathbb{P}^{3}$. However, on each quartic $X$ there are non-trivial extensions.

Lemma 6.4**.**

We have $\mathrm{Ext}_{X}^{1}(\mathcal{O}_{X}(H),F|_{X})=\mathbb{C}$ for all $X\in L$.

Proof.

Applying $\mathrm{Hom}_{X}(\mathcal{O}_{X}(H),-)$ to sequence (11), we get the following exact sequence

[TABLE]

Computing each term, we have

[TABLE]

The map $g$ can be explicitly described as follows. Let $g_{1}:V_{1}^{*}\longrightarrow(\wedge^{2}V_{1})^{*}\otimes V_{1}$ be the adjoint map of the canonical projection $V_{1}^{*}\otimes V_{1}^{*}\longrightarrow(\wedge^{2}V_{1})^{*}$. Then $g$ is the composition of the following maps:

[TABLE]

Let $x_{0},\cdots,x_{3}$ be coordinates on $\mathbb{P}^{3}$, then by explicit computations we have

[TABLE]

∎

We define the vector bundle $E_{X}$ on any $X\in L$ to be the unique non-split extension

[TABLE]

The author wonders but does not know whether $F|_{C}$ is stable. However the following proposition is sufficient for our purpose.

Proposition 6.5**.**

The bundle $F|_{C}$ is simple.

We will show Proposition 6.5 by proving the following two lemmas.

Lemma 6.6**.**

The bundle $F$ is simple on $\mathbb{P}^{3}$.

Proof.

Applying $\mathrm{Hom}_{\mathbb{P}^{3}}(F,-)$ to (11), we have

[TABLE]

Hence $\mathrm{Hom}_{\mathbb{P}^{3}}(F,F)=\mathrm{ker}(r)$. By applying $\mathrm{Hom}_{\mathbb{P}^{3}}(-,\mathcal{O}_{\mathbb{P}^{3}})$ and $\mathrm{Hom}_{\mathbb{P}^{3}}(-,M)$ to (11), we see that

[TABLE]

The map $r$ is the composition of the following maps

[TABLE]

where $\mathrm{tr}^{*}$ is the dual of the trace map $V_{1}^{*}\otimes V_{1}\rightarrow\mathbb{C}$. By explicit coordinates computations, we see that

[TABLE]

∎

Lemma 6.7**.**

The bundle $F|_{X}$ is simple on $X$.

Proof.

Applying $\mathrm{Hom}_{\mathbb{P}^{3}}(F,-)$ to the exact sequence

[TABLE]

we have

[TABLE]

By Lemma 6.6, $\mathrm{Hom}_{\mathbb{P}^{3}}(F,F)=\mathbb{C}$, hence it suffices to show $\mathrm{Ext}_{\mathbb{P}^{3}}^{1}(F,F(-4H))=0$. Applying $\mathrm{Hom}_{\mathbb{P}^{3}}(F,-$) to (11) twisted by $\mathcal{O}_{\mathbb{P}^{3}}(-4H)$, we have

[TABLE]

Applying $\mathrm{Hom}_{\mathbb{P}^{3}}(-,M(-4H))$ to (11), we have

[TABLE]

Hence $\mathrm{Ext}_{\mathbb{P}^{3}}^{1}(F,M(-4H))=\mathrm{Ext}_{\mathbb{P}^{3}}^{1}(F,F(-4H))=0$, and $F|_{X}$ is simple for all $X$.

∎

Proof of Proposition 6.5.

Applying $\mathrm{Hom}_{X}(F|_{X},-$) to the exact sequence

[TABLE]

we have

[TABLE]

By Lemma 6.7, it suffices to show $\mathrm{Ext}_{X}^{1}(F|_{X},F|_{X}(-4H))=0$. Applying $\mathrm{Hom}_{X}(F|_{X},-)$ to the restriction of (11) to $X$, we have

[TABLE]

Applying $\mathrm{Hom}_{X}(-,M|_{X}(-4H))$ to the restriction of (11) to $X$, we have

[TABLE]

Hence $\mathrm{Ext}_{X}^{1}(F|_{X},M|_{X}(-4H))=0$. To show $\mathrm{Ext}_{X}^{1}(F|_{X},F|_{X}(-4H))=0$, it suffices to show the map $u$ is injective, equivalently, to show $u^{*}$ is surjective. Consider the following commutative diagram

[TABLE]

where the columns are exact. Hence it suffices to show the surjectivity of the map

[TABLE]

which is described as follows. Applying $\mathrm{Hom}_{X}(-,M|_{X})$ to the Euler sequence

[TABLE]

we get $ev^{*}:\mathrm{Hom}_{X}(M|_{X}(-4H),M|_{X})\longrightarrow\mathrm{Hom}_{X}(\mathcal{O}_{X}(-4H),M|_{X})\otimes V_{1}$. Let $e_{2}^{*}:V_{1}\otimes V_{2}^{*}\rightarrow V_{1}^{*}$ be the adjoint map of the comultiplication map $V_{2}^{*}\rightarrow V_{1}^{*}\otimes V_{1}^{*}$. Then $v$ is the composition of the following maps

[TABLE]

By an explicit computation in coordinates, $v$ is surjective. ∎

We also need the following fact.

Proposition 6.8**.**

For any $X\in L$, we have $\mathrm{Hom}_{C}(F|_{C},E_{X}|_{C})=\mathbb{C}$.

Proof.

Applying $\mathrm{Hom}_{C}(-,E_{X}|_{C})$ to the sequence (12), we get

[TABLE]

We compute each term in the following.

To compute $\mathrm{Hom}_{C}(\mathcal{O}_{C},E_{X}|_{C})$, we use the sequence (9). Applying $\mathrm{H}^{0}(X,-)$ to (9), we get

[TABLE]

We have nutural identification $\mathrm{H}^{0}(X,M|_{X})=V_{1}^{*}$, and the map $h_{1}$ is the adjoint of the multiplication map $m:V_{1}\otimes V_{1}\longrightarrow V_{2}$. Hence $\mathrm{H}^{0}(X,E_{X})=\mathrm{coker}(h_{1})$. Now we apply $\mathrm{H}^{0}(X,-)$ to the sequnce

[TABLE]

we get

[TABLE]

We will show $\mathrm{H}^{1}(X,E_{X}(-4H))=0$ by applying $\mathrm{H}^{0}(X,-)$ to

[TABLE]

Let $K_{1}$ be the cokernel of the comultiplication $V_{X,5}^{*}\longrightarrow V_{X,4}^{*}\otimes V_{1}^{*}$. Then $\mathrm{H}^{2}(X,M|_{X}(-4H))=K_{1}$. We also have $\mathrm{H}^{1}(X,M|_{X}(-4H))=0$ and $\mathrm{H}^{1}(X,N_{X})=V_{X,4}^{*}\otimes V_{1}$. Hence $\mathrm{H}^{1}(X,E_{X}(-4H))$ can be idetified with the kernel of the following composition of maps

[TABLE]

where $e_{2}$ is the adjoint of the multiplication map $V_{1}\otimes V_{1}\longrightarrow V_{2}$. By an explicit computation in coordinates, we see that $h_{2}$ is injective. Hence $\mathrm{H}^{1}(X,E_{X}(-4H))=0$ and we have $\mathrm{H}^{0}(C,E_{X}|_{C})=\mathrm{coker}(h_{1})$.

To compute $\mathrm{Hom}_{C}(M|_{C},E_{X}|_{C})$, we first compute $\mathrm{Hom}_{X}(M|_{X},E_{X})$. Applying $\mathrm{Hom}_{X}(M,-)$ to (9), we have

[TABLE]

Hence $\mathrm{Hom}_{X}(M,E_{X})=V_{2}$. Applying $\mathrm{Hom}|_{X}(M|_{X},-)$ to the sequence

[TABLE]

we have

[TABLE]

We will show $\mathrm{Ext}_{X}^{1}(M|_{X},E_{X}(-4H))=0$ by applying $\mathrm{Hom}_{X}(M|_{X},-)$ to the sequence (9) twisted by $\mathcal{O}_{X}(-4H)$. We get

[TABLE]

Note that $e_{3}$ is exactly the map $v^{*}$, where $v$ is defined in the proof of Lemma 6.5. Hence $e_{3}$ is injective. We have $\mathrm{Ext}_{X}^{1}(M|_{X},E_{X}(-4H))=0$ and $\mathrm{Hom}_{C}(M|_{C},E_{X}|_{C})=V_{2}$ for any $X$.

Getting back to the sequence (13), the map $h$ is the composition of the following maps

[TABLE]

where $m^{*}$ is the comultiplication map $V_{2}^{*}\longrightarrow V_{1}^{*}\otimes V_{1}^{*}$. By an explicit computation in coordinates, the kernel of $h$ is the identity element in $V_{2}^{*}\otimes V_{2}$. Hence $\mathrm{Hom}_{C}(F|_{C},E_{X}|_{C})=\mathbb{C}$.

∎

The following lemma allows us to prove certain bundles are non-isomorphic.

Lemma 6.9**.**

On $C$, let

[TABLE]

be extension classes such that $\mathrm{Hom}_{C}(F|_{C},E_{i})=\mathbb{C},i=1,2$. If $E_{1}\cong E_{2}$, then $\delta_{1}=\lambda\delta_{2}$ for some scalar $\lambda\in\mathbb{C}^{*}$.

Proof.

Let $f:E_{1}\rightarrow E_{2}$ be an isomorphism. Then the composition

[TABLE]

is non-zero. Since $\mathrm{Hom}(F|_{C},E_{2})=\mathbb{C}$, we have the following commutative diagram

[TABLE]

By Proposition 6.5, $F|_{C}$ and $\mathcal{O}_{C}(H)$ are both simple. Hence $f_{1},f_{2}$ are non-zero scalar multiplications. ∎

The following observation plays a central role in the proof of Theorem 6.1.

Proposition 6.10**.**

For distinct $t_{i}\in L,i=1,2$, the restriction maps

[TABLE]

have distinct images, as subspaces of the fixed space $\mathrm{Ext}_{C}^{1}(\mathcal{O}_{C}(H),F_{C})$.

Proof.

Recall that $\widetilde{\mathbb{P}^{3}}=\mathrm{BL}_{C}\mathbb{P}^{3}$, $\nu:\widetilde{\mathbb{P}^{3}}\rightarrow\mathbb{P}^{3}$ is the projection and $\pi:\widetilde{\mathbb{P}^{3}}\rightarrow L$ is the $K3$ fibration. Let $Y\cong C\times L$ be the exceptional divisor of $\nu$. The pushforward $\pi_{*}$ on the restriction map $\nu^{*}F\longrightarrow\nu^{*}F|_{Y}$ induces the map

[TABLE]

The fiber of $\mathrm{res}$ over $t\in L$ is the map

[TABLE]

Relativize the proof of Lemma 6.4 by using the relative Serre Duality, we have

[TABLE]

Hence the restrictions of $\mathrm{res}:\mathcal{O}_{L}(-1)\longrightarrow\mathrm{Ext}_{C}^{1}(\mathcal{O}_{C}(H),F_{C})\otimes\mathcal{O}_{L}$ to fibers either have pairwise distinct image, or vanish at some $t\in L$.

By the algorithm in [Liu22] or a direct computation, we have $\mathrm{Ext}_{X_{t}}^{1}(E_{X_{t}},E_{X_{t}}(-4H))=0$ for any smooth quartic surface $X_{t}$ with Picard rank 1. Hence $E_{X_{t}}|_{C}$ is simple, the map $\mathrm{res}_{t}$ is non-zero, hence the map $\mathrm{res}$ is also non-zero. If $\mathrm{res}$ vanishes at some $t\in L$, then the sheaf $R^{2}\pi_{*}\underline{Hom}_{\widetilde{\mathbb{P}^{3}}}(\mathcal{O}_{\widetilde{\mathbb{P}^{3}}}(H),\nu^{*}F(-Y))$ must have torsion. We exclude this by showing

[TABLE]

has constant dimension for $t\in L$. Applying $\mathrm{Hom}_{\mathbb{P}^{3}}(F(-4H),-)$ to the exact sequence

[TABLE]

we get

[TABLE]

Hence $\mathrm{Ext}_{X_{t}}^{2}(\mathcal{O}_{X_{t}}(H),F|_{X_{t}}(-4H))$ has constant dimension, we must have that the map $\mathrm{res}:\mathcal{O}_{L}(-1)\longrightarrow\mathrm{Ext}_{C}^{1}(\mathcal{O}_{C}(H),F_{C})\otimes\mathcal{O}_{L}$ has non-constant image. ∎

Corollary 6.11**.**

For distinct $t,t^{\prime}\in L$, we have $E_{X_{t}}|_{C}\not\cong E_{X_{t^{\prime}}}|_{C}$.

Proof.

By Proposition 6.8, $E_{X}$ satisfies $\mathrm{Hom}(F|_{C},E_{X})=\mathbb{C}$ for any $X\in L$. If $E_{X_{t}}|_{C}\cong E_{X_{t^{\prime}}}|_{C}$ for any $t\neq t^{\prime}$, by Lemma 6.9, the extension classes given by (12) are scalar multiples. By Proposition 6.10, this is not true. ∎

Combining all preparations, we now prove Theorem 6.1.

Proof of Theorem 6.1.

Suppose $E$ is an exceptional bundle on $\mathbb{P}^{3}$ of rank $27$ and degree $11$. Take distinct $t_{1},t_{2}\in L$ so that $\mathrm{Pic}(X_{t_{i}})\cong\mathbb{Z}H,i=1,2$. By [Zub90], we have

[TABLE]

Restricting to $C$, we have $E_{X_{t_{1}}}|_{C}\cong E|_{C}\cong E_{X_{t_{2}}}|_{C}$. By Corollary 6.11, this is not true. ∎