Non-emptiness of Brill-Noether Loci over very general quintic hypersurface

TL;DR

This paper investigates the non-emptiness of Brill-Noether loci on very general quintic hypersurfaces in projective 3-space, introducing a Petri map analogy to produce components of expected dimension.

Contribution

It establishes the non-emptiness of specific Brill-Noether loci on quintic hypersurfaces and develops a Petri map framework analogous to the curve case.

Findings

Confirmed non-emptiness of certain Brill-Noether loci

Constructed components of expected dimension using Petri map

Extended Brill-Noether theory to surfaces in projective space

Abstract

In this article we study Brill-Noether loci of moduli space of stable bundles over smooth surfaces. We define Petri map as an analogy with the case of curves. We show the non-emptiness of certain Brill-Noether loci over very general quintic hypersurface in $\mathbb{P}^3$, and use the Petri map to produce components of expected dimension.

Equations

$$\mu (G)\le \mu (F).$$

$$0\to \mathcal{E}\to \mathcal{E}(\mathcal{D})\to \mathcal{E}(\mathcal{D})/\mathcal{E}\to 0$$

$$0\to {\varphi }_{{}_{\ast }}(\mathcal{E})\to {\varphi }_{{}_{\ast }}(\mathcal{E}(\mathcal{D}))\stackrel{\gamma }{\to }{\varphi }_{{}_{\ast }}(\mathcal{E}(\mathcal{D})/\mathcal{E})\to R^1{\varphi }_{{}_{\ast }}(\mathcal{E})\to 0.$$

$$\mathrm{Support}(B_{r,X,H}^k)=\{E\in \mathcal{M}:h^0(X,E)\ge k+1\}.$$

$$\text{Support}(B_{r,X,H}^k)=\{E\in \mathcal{M}:h^0(X,E)\ge k+1\}.$$

$$\text{dim}(\mathcal{M})-(k+1)(k+1-\chi (r;c_1,\cdots ,c_s))$$

$$B_{r,X,H}^{k+1}\subset \text{Sing}(B_{r,X,H}^k)$$

$${\rho }_{r,X,H}^k:=\text{dim}(\mathcal{M})-(k+1)(k+1-\chi (r;c_1,\cdots ,c_s))$$

$$0\to E\to E_{ϵ}\to E\to 0.$$

$$\left[\begin{array}{cc}\text{Id}& 0\\ {\varphi }_{ij}& \text{Id}\end{array}\right]$$

$$\left[\begin{array}{cc}\text{Id}& 0\\ {\varphi }_{ij}& \text{Id}\end{array}\right]\left[\begin{array}{c}s{|}_{U_i}\\ s_i^{\prime }\end{array}\right]=\left[\begin{array}{c}s{|}_{U_j}\\ s_j^{\prime }\end{array}\right].$$

$$H^1(X,E\otimes E^{\ast })⟶H^1(X,E),({\nu }_{ij})\mapsto ({\nu }_{ij}(s)).$$

$$E_{\epsilon}\in\text{Ker}\big{(}H^{1}(X,E\otimes E^{*})\longrightarrow H^{1}(X,E),\hskip 8.5359pt(\nu_{ij})\mapsto(\nu_{ij}(s))\big{)}.$$

$$\alpha :H^0(X,E)\otimes H^1(X,E\otimes E^{\ast })⟶H^1(X,E).$$

$$\mu :H^0(X,E)\otimes H^{n-1}(X,K_X\otimes E^{\ast })⟶H^{n-1}(X,K_X\otimes E\otimes E^{\ast }).$$

$$\mu :H^0(X,E)\otimes H^1(X,K_X\otimes E^{\ast })⟶H^1(X,K_X\otimes E\otimes E^{\ast }).$$

$$\text{dim}(T)=\text{dim}(\mathcal{M})-h^0(X,E)h^1(X,E)+\text{dim Ker}(\mu ).$$

$$0\to F\to H^0(C,A)\otimes {\mathcal{O}}_X\to A\to 0.$$

$$0\to H^0(C,A{)}^{\ast }\otimes {\mathcal{O}}_X\to F^{\ast }\to {\mathcal{O}}_C(C)\otimes A^{\ast }\to 0.$$

$$0\to {\mathcal{O}}_X(-2{)}^{\oplus 2}\to F^{\ast }\otimes {\mathcal{O}}_X(-2)\to {\mathcal{O}}_C(H)\otimes A^{\ast }\to 0.$$

$$0\to {\mathcal{O}}_X^{\oplus 2}\to E\to {\mathcal{O}}_C(C)\otimes A^{\ast }\to 0$$

$$0\to (E^{\ast }\otimes K_X{)}^{\oplus 2}\to \mathcal{E}nd(E)\otimes K_X\to {\mathcal{O}}_C(C)\otimes A^{\ast }\otimes (E^{\ast }\otimes K_X){|}_{{}_C}\to 0.$$

$$\begin{array}{rl}0& \to H^0(X,K_X)\to H^0(C,{\mathcal{O}}_C(C)\otimes A^{\ast }\otimes (E^{\ast }\otimes K_X){|}_{{}_C})\\ & \to H^1(X,(E^{\ast }\otimes K_X{)}^{\oplus 2})\stackrel{\eta }{\to }{H}^1(X,\mathcal{E}nd(E)\otimes K_X)\to \dots \end{array}$$

$$0\to {\mathcal{O}}_C(-C)\otimes A\to E^{\ast }{|}_{{}_C}\to {\mathcal{O}}_C^{\oplus 2}\to A\to 0$$

$$0\to {\mathcal{O}}_C(-C)\otimes A\to E^{\ast }{|}_{{}_C}\to A^{\ast }\to 0.$$

$$0$$

$$\to H^0(C,{\mathcal{O}}_C(C)\otimes A^{\ast \otimes 2}\otimes K_X{|}_{{}_C})\to \dots$$

$$0\to {\mathcal{O}}_X\to E\to {\mathcal{I}}_Z\otimes {\mathcal{O}}_X(3)\to 0$$

Full text

1. Introduction

Let $X$ be a smooth, irreducible, projective variety of dimension $n$ over $\mathbb{C}$, $H$ be an ample divisor on $X$, and let $\mathcal{M}:=\mathcal{M}_{X,H}(r;c_{1},\cdots,c_{s})$ be the moduli space of rank $r$, $H$-stable vector bundles $E$ over $X$ with Chern classes $c_{i}(E)=c_{i},$ where $s:=\text{min}\{r,n\}$. A Brill-Noether locus $B_{r,X,H}^{k}$ is a closed subscheme of $\mathcal{M}$ whose support consists of points $E\in\mathcal{M}$ such that $h^{0}(X,E)\geq k+1$. Göttsche et al ([6]) and M. He ([7]) studied the Brill-Noether loci of stable bundles over $\mathbb{P}^{2}$, and Yoshioka ([15], [16]), Markman ([4]), Leyenson ([10], [11]) studied it for $K3$ surfaces. In the case of smooth, projective, irreducible curves $C$ over $\mathbb{C}$, the Brill-Noether loci of the moduli space, $\text{Pic}^{d}(C)$, of degree $d$ line bundles on $C$ is well-studied. The questions like non-emptiness, connectedness, irreducibility, singular locus etc of Brill-Noether loci are known when $C$ is a general curve in the sense of moduli (see e.g. [1]). This concept was generalized for vector bundles over curves by Newstead, Teixidor and others. For an account of the results and history in this case see [5] and the references therein.

Recently, in [2], authors have constructed a Brill-Noether loci over higher dimensional varieties under the additional cohomology vanishing assumptions: $H^{i}(X,E)=0,\forall i\geq 2$ and for all $E\in\mathcal{M}$. This is a natural generalization of the Brill-Noether loci over the curves for higher dimensional varieties. In [2], [3], authors gave several examples of non-empty Brill-Noether locus, and examples of Brill-Noether locus where “expected dimension” is not same as the exact dimension. In all these examples, the surfaces and/or varieties chosen have the canonical bundle has no non-zero sections. In this article, we defined “Petri map” over a smooth projective variety with canonical bundle ample, as an analogue of that for curves. Similar to the case of curves, the injectivity of the “Petri map” implies the existence of smooth points in Brill-Noether loci. We then use this fact to prove the existence of a smooth point and hence a component of expected dimension in the Brill-Noether loci over a very general quintic hypersurface in $\mathbb{P}^{3}$ where the canonical bundle is ample and globally generated.

Notation: We work throughout over the field $\mathbb{C}$ of complex numbers. If $X$ is a smooth, projective variety, we denote by $K_{X}$ the canonical bundle on $X$. For a coherent sheaf $\mathcal{F}$ on $X$, we denote by $H^{i}(X,\mathcal{F})$ the $i$-th cohomology group of $\mathcal{F}$ and by $h^{i}(X,\mathcal{F})$ its (complex) dimension. If $V$ is a vector bundle on $X$, we denote by $V^{*}$ the dual of $V$.

2. Brill-Noether Loci

In this section we will briefly recall the construction of Brill-Noether loci over higher dimensional varieties, following [2].

Let $X$ be an irreducible, smooth, projective variety of dimension $n$, and let $H$ be an ample divisor on $X$. For a torsion free sheaf $F$ over $X$, let $c_{i}(F)$ denotes the $i$-th Chern class of $F$. Set $\mu(F)=\mu_{H}(F):=\frac{c_{1}(F)\cdot H^{n-1}}{\text{rank}(F)}$.

Definition 2.1**.**

A torsion-free sheaf $F$ over $X$ of rank $r$ is called $H$-semistable if for all non-zero subsheaf $G$ of $F$ with ${\rm{rank}}(G)<{\rm{rank}}(F)$, we have

[TABLE]

We say $F$ is $H$-stable if the above inequality is strict.

Let $\mathcal{M}:=\mathcal{M}_{{}_{X,H}}(r;c_{1},\cdots,c_{s})$ be the moduli space of rank $r$, $H$-stable vector bundles $E$ over $X$ with Chern classes $c_{i}(E)=c_{i},$ where $s:=\text{min}\{r,n\}$. Assume that $\mathcal{M}$ is a fine moduli space, and let $\mathcal{E}\to\mathcal{M}\times X$ be an universal family such that for any $t\in\mathcal{M},E_{t}:=\mathcal{E}|_{{}_{t\times X}}$ is a rank $r,H$-stable bundle over $X$ with Chern classes $c_{i}(E_{t})=c_{i}$. Choose an effective divisor $D$ on $X$ such that $H^{i}(X,E_{t}(D))=0,\forall i\geq 1$ and $\forall t\in M$. Let $\mathcal{D}:=\mathcal{M}\times D$ be the product divisor, and $\phi:\mathcal{M}\times X\to\mathcal{M}$ be the projection map. From the exact sequence

[TABLE]

on $\mathcal{M}\times X$, we get an exact sequence on $\mathcal{M}$:

[TABLE]

Note that, $\gamma$ is a map between two locally free sheaves of ranks $\chi(E_{t}(D))$ and $\chi(E_{t}(D))-\chi(E_{t})$ respectively, on $\mathcal{M}$. For an integer $k\geq-1$, let $B_{r,X,H}^{k}\subset\mathcal{M}$ be the $(\chi(E_{t}(D))-(k+1))$-th determinantal variety associated to the map $\gamma$. Now assume $H^{i}(X,E_{t})=0,\forall i\geq 2$ and $\forall t\in\mathcal{M}$. Then we have

[TABLE]

When $\mathcal{M}$ is not a fine moduli space, it is possible to carry out this construction locally and then can be glued together to get a global algebraic object. We summarize the above construction as

Theorem 2.2**.**

([2], Theorem 2.3) Let $X$ be a smooth, irreducible, projective variety of dimension $n$, $H$ be a fixed ample divisor on $X$, and $\mathcal{M}:=\mathcal{M}_{{}_{X,H}}(r;c_{1},\cdots,c_{s})$ be a moduli space of rank $r,H$-stable vector bundles $E$ on $X$ with fixed Chern classes $c_{i}(E)=c_{i},s=\text{min}\{r,n\}.$ Assume that for any $E\in\mathcal{M},H^{i}(X,E)=0$ for $i\geq 2.$ Then for any $k\geq-1,$ there exists a determinantal variety $B_{r,X,H}^{k}\subset\mathcal{M}$ such that

[TABLE]

Moreover, each non-empty irreducible component of $B_{r,X,H}^{k}$ has dimension at least

[TABLE]

where $\chi(r;c_{1},\cdots,c_{s}):=\chi(E_{t})$ for any $t\in\mathcal{M}$ and

[TABLE]

whenever $B_{r,X,H}^{k}\neq\mathcal{M}.$

Definition 2.3**.**

The variety $B_{r,X,H}^{k}$ is called the $k$-th Brill-Noether locus of the moduli space $\mathcal{M}$ and the number

[TABLE]

is called the generalized Brill-Noether number.

By the above theorem, the dimension of $B_{r,X,H}^{k}$ is at least $\rho_{r,X,H}^{k}$. We call $\rho_{r,X,H}^{k}$, the expected dimension of the Brill-Noether locus $B_{r,X,H}^{k}$. When there is no confusion about $X$ and $H$, we will simply denote these by $B_{r}^{k}$ and $\rho_{r}^{k}$.

3. Petri Map

In this section we will define “Petri Map” for higher dimensional varieties, as an analogue of the one defined for curves. We note that the description of Petri map over curves as given in [5] works for higher dimensional varieties also. For convenience, we recall this description.

Let $X$ be an irreducible, smooth, projective variety of dimension $n$, $H$ be an ample divisor on $X$, and let $\mathcal{M}:=\mathcal{M}_{{}_{X,H}}(r;c_{1},\cdots,c_{s})$ be the moduli space of rank $r$, $H$-stable vector bundles $E$ over $X$ with Chern classes $c_{i}(E)=c_{i},$ where $s:=\text{min}\{r,n\}$. The tangent space to $\mathcal{M}$ at a point $E\in\mathcal{M}$ is given by $H^{1}(X,E\otimes E^{*})$, where $E^{*}$ is the dual of $E$. A tangent vector to $\mathcal{M}$ at $E$ can be identified with a vector bundle $E_{\epsilon}$ on $X_{\epsilon}:=X\times\text{Spec}(k[\epsilon]/\epsilon^{2})$ whose restriction on $X$ is $E$ and it fits into the exact sequence

[TABLE]

We call it a first order deformation of $E$.

One can give an explicit description of the bundle $E_{\epsilon}$ as follows: Let $\{U_{i}\}$ be an open cover of $X$ such that $E_{i}:=E|_{U_{i}}$ is the trivial bundle. Set $U_{ij}:=U_{i}\cap U_{j}$, and let $\phi_{ij}\in H^{0}(U_{ij},E\otimes E^{*})$ be the co-boundary map corresponding to $\phi\in H^{1}(X,E\otimes E^{*})$. Consider the trivial extension of $E_{i}$ to $U_{i}\times\text{Spec}(k[\epsilon]/\epsilon^{2})$ given by $E_{i}\oplus\epsilon E_{i}$. Then the matrix

[TABLE]

will give the gluing data for the bundle $E_{\epsilon}$.

Assume that a section $s$ of $E$ can be extended to a section of $E_{\epsilon}$. Then we have local sections $(s_{i}^{\prime})\in H^{0}(U_{i},E_{i})$ such that $(s|_{U_{i}},s_{i}^{\prime})$ defines a section of $E_{\epsilon}$. If this is the case, then we have

[TABLE]

This gives two conditions: $(s|_{U_{i}})|_{U_{ij}}=(s|_{U_{j}})|_{U_{ij}}$ and $\phi_{ij}(s)=s_{j}^{\prime}-s_{i}^{\prime}$. The first condition is automatically satisfied, since $s$ is a global section and from the second condition we see that, in this case, $(\phi_{ij}(s))$ satisfies the co-cycle condition. In other words, $(\phi_{ij}(s))$ is in the kernel of the map

[TABLE]

Let $E\in B_{r}^{k},k\geq 0$ and let $s\in H^{0}(X,E)$. Then the first order deformation of $E$, as an element of $B_{r}^{k}$, is the subset $\{E_{\epsilon}:$ the section $s$ can be extended to a section of $E_{\epsilon}\}$ of $H^{1}(X,E\otimes E^{*})$, i.e.

[TABLE]

Now assume $E\in B_{r}^{k}-B_{r}^{k+1}$ and let $T$ be the tangent space to $B_{r}^{k}$ at the point $E$. From the discussion above, we have a map

[TABLE]

This induces the map

[TABLE]

Note that, $T$ can be identified with $(\text{Im}(\mu))^{\bot}$. We call the map $\mu$, the Petri map.

Remark 3.1**.**

Let $X$ be an irreducible, smooth, projective surface and $H$ be an ample divisor on $X$. In this case, Petri map, as defined above, is the cup product map

[TABLE]

If $E$ is a smooth point in the moduli space $\mathcal{M}$, we have

[TABLE]

Thus, if the Petri map is injective, then $E$ is a smooth point of $B_{r}^{k}$ and and the component of $B^{k}_{r}$ through $E$ has the expected dimension.

Remark 3.2**.**

The Petri map can also be derived from [7]. Indeed, by taking $\Lambda=\Lambda^{\prime}=(H^{0}(X,E),Id,E)$ in [7, Corollary 1.6], we see that the above Petri map is dual of the map $Ext^{1}(E,E)\to Hom(H^{0}(X,E),H^{1}(X,E))$ in the given exact sequence.

4. Brill-Noether loci over quintic hypersurface

Let $X$ be a very general quintic hypersurface in $\mathbb{P}^{3}$. Then we have $\text{Pic}(X)\simeq\text{Pic}(\mathbb{P}^{3})\simeq\mathbb{Z}$, $K_{X}\simeq\mathcal{O}_{X}(1)$, $\chi(X,\mathcal{O}_{X})=5$. Let $H$ be a hyperplane class, and $\mathcal{M}(c_{2}):=\mathcal{M}_{X,H}(2;3H,c_{2})$ be the moduli space of rank two $H$-stable bundles on $X$ with first Chern class $3H$, and second Chern class $c_{2}$. It is known ([13]) that $\mathcal{M}(c_{2})$ is irreducible for $c_{2}\geq 14$, generically smooth for $c_{2}\geq 21$, and has the expected dimension $4c_{2}-60$. Also note that $H^{2}(X,E)=0,\forall E\in\mathcal{M}$. Thus the hypothesis of the Theorem 2.2 is satisfied.

Let $C$ be a smooth, irreducible, projective curve in the complete linear system $|3H|$. Then the genus of $C,g:=31$.

Proposition 4.1**.**

With the notations as above, assume that there is a base point free line bundle of degree $\geq 16$ on $C$ with exactly two sections, then $B^{1}_{2}\subset\mathcal{M}(c_{2})$ is non-empty.

Proof.

Let $A$ be a base point free line bundle on $C$ with $h^{0}(C,A)=2$. Consider the elementary transformation

[TABLE]

Then $F$ is a rank two vector bundle on $X$ with $c_{1}(F)=-3H$, $c_{2}(F)=\text{deg}(A)$, $h^{0}(X,F)=0=h^{1}(X,F)$ ([8, Chapter 5, Proposition 5.2.2]). Dualizing the above exact sequence we get

[TABLE]

Thus $h^{0}(X,F^{*})=2+h^{0}(C,\mathcal{O}_{C}(C)\otimes A^{*})\geq 2$.

Claim: $F^{*}\in B^{1}_{2}$.

It is sufficient to show that $F$ is $H$-stable. Let $\mathcal{O}_{X}(m)$ destabilizes $F$. Then $m\geq-1$. On the other hand, from (2), we have $m\leq 0$. Since $h^{0}(X,F)=0$, $m\neq 0$. Thus we are reduced to show that $h^{0}(X,F\otimes\mathcal{O}_{X}(1))=0$. Note that $F\otimes\mathcal{O}_{X}(1)\simeq F^{*}\otimes\mathcal{O}_{X}(-2)$. Tensoring the exact sequence (3) by $\mathcal{O}_{X}(-2)$ we get

[TABLE]

Since $\text{deg}(\mathcal{O}_{C}(H)\otimes A^{*})<0$, $h^{0}(C,\mathcal{O}_{C}(H)\otimes A^{*})=0$ and consequently $h^{0}(X,F^{*}\otimes\mathcal{O}_{X}(-2))=0.$ ∎

Since $C\in|3H|$, $C$ is a smooth complete intersection of two smooth hypersurfaces of $\mathbb{P}^{3}$ of degrees $5$ and $3$. Thus by [1, Page 139, C-4], $C$ does not have a $g^{1}_{5}$. In particular, $C$ is not hyperelliptic, trigonal or tetragonal (i.e. $C$ does not have a $g^{1}_{2},g^{1}_{3},g^{1}_{4}$ respectively). Also note that $K_{C}=\mathcal{O}_{C}(4)$. Now for $D=\mathcal{O}_{C}(3)$, $K_{C}\otimes\mathcal{O}_{C}(-D)=\mathcal{O}_{C}(1)$, and this gives an embedding of $C\hookrightarrow\mathbb{P}^{3}$. So by [1, Page 221, B-4], $C$ is not a bi-elliptic.

Proposition 4.2**.**

With the notations as above, for $28\leq d\leq 32$, there exists a base point free line bundle of degree $d$ on $C$.

Proof.

Let us denote by $W^{r}_{d}(C)$ the Brill-Noether loci of degree $d$ line bundles $L$ on $C$ with $h^{0}(C,L)\geq r+1$.

Case I: $d=32$.

In this case, $\text{dim}W^{1}_{32}(C)=31$. If $W^{1}_{32}(C)-W^{2}_{32}(C)(\neq\varnothing)$ does not contain any base point free line bundle, then tensoring by the ideal sheaf of the base locus, we obtain a family of base-point free line bundles with exactly two sections of dimension $31$ and is contained in $\bigcup_{e\leq 31}W^{1}_{e}(C)$. Now $\text{dim}W^{0}_{e}(C)=e\leq 31=g$, and $W^{1}_{e}(C)$ is a proper closed subset of $W^{0}_{e}(C)$. So $\text{dim}W^{1}_{e}(C)\leq e-1\leq 30$. Thus $\text{dim}(\bigcup_{e\leq 31}W^{1}_{e}(C))<31$, a contradiction.

Case II: $d=31$.

In this case, $\text{dim}W^{1}_{31}(C)\geq 29$. Applying Martens’ theorem [1, Chapter IV, Theorem 5.1], we see that $\text{dim}(\bigcup_{2\leq e\leq 30}W^{1}_{e}(C))<29$. Note that, for any $L\in\text{Pic}(C)$ with $h^{0}(C,L)=2$, if $B$ is the base locus of $|L|$, then $\text{deg}(B)\leq\text{deg}(L)-2$. Now arguing as above, we get a base point free line bundle on $C$ of degree $31$ with exactly two sections.

Case III: $d=30$.

This follows from the existence of base point free complete $g^{1}_{30}$ on $C$ [9, Theorem 1.4].

Case IV: $d=29$.

We have $\text{dim}W^{1}_{29}(C)\geq 25$. By Mumford’s theorem [1, Chapter IV, Theorem 5.2], we get $\text{dim}(\bigcup_{2\leq e\leq 28}W^{1}_{e}(C))<25$. Now we argue as in Case I to conclude.

Case V: $d=28$.

Here $\text{dim}W^{1}_{28}(C)\geq 23$. Then using Keem’s theorem [1, Page 200], we get $\text{dim}(\bigcup_{e\leq 27}W^{1}_{e}(C))<23$. Arguing as in Case I, we conclude that there exists a base point free line bundle of degree $28$ on $C$ with exactly two global sections. ∎

Now we are ready to prove

Theorem 4.3**.**

With the notations as in the beginning of this section, $B^{1}_{2}-B^{2}_{2}\subset\mathcal{M}(c_{2})$ is non-empty for $28\leq c_{2}\leq 32$.

Proof.

Combining Propositions 4.1 and 4.2, we see that, for $28\leq c_{2}\leq 32$, $B^{1}_{2}\subset\mathcal{M}(c_{2})$ is non-empty. Moreover, for $28\leq d\leq 32$, we can find base point free line bundles $A$ on $C$ with $h^{0}(C,A)=2$ and $h^{0}(C,\mathcal{O}_{C}(C)\otimes A^{*})=0$. Now using (3), we conclude. ∎

Proposition 4.4**.**

With the notations as in the beginning of this section, let $E\in B_{2}^{1}-B_{2}^{2}\subset\mathcal{M}(c_{2}),28\leq c_{2}\leq 32$ be a general element constructed as in Proposition 4.1. Then $E$ is a smooth point in $\mathcal{M}(c_{2})$.

Proof.

Since the morphism $\mathcal{M}_{X,H}(2;3H,c_{2})\xrightarrow{\otimes\mathcal{O}_{X}(-1)}\mathcal{M}_{X,H}(2;H,c_{2}-10)$ is an isomorphism, it is sufficient to prove that $E\otimes\mathcal{O}_{X}(-1)$ is a smooth point of $\mathcal{M}_{X,H}(2;H,c_{2}-10)$. Set $F:=E\otimes\mathcal{O}_{X}(-1)$. If $F$ is not a smooth point of $\mathcal{M}_{X,H}(2;H,c_{2}-10)$, we have $H^{0}(X,AdF\otimes K_{X})\neq 0$. Thus there is a non-zero map $\phi:F\to F\otimes K_{X}$. Two cases can occur:

Case I: $\phi$ drops rank every where.

Case II: $\phi$ does not drop rank every where.

Following [12], we call the Case I as singularity of first kind, and the Case II as singularity of second kind. If Case I occurs, we will have $h^{0}(X,F)\neq 0$ (see [12, Proposition 5.1]), which is impossible. Also, from [13, Lemma 10.1], we see that the dimension of the singular locus of the second kind is at most $8$ 111In [12, Corollary 5.1], the authors have estimated the bound for singularity of second kind as $\leq 13$. But in [13, Lemma 10.1], the authors have improved the above bound and shown that it is $\leq 8$.

But the dimension of the family of globally generated line bundles $A$ on $C$ with $28\leq\text{deg}(A)\leq 32,h^{0}(C,A)=2$ and $h^{0}(C,\mathcal{O}_{C}(C)\otimes A^{*})=0$ is greater than $8$. Hence a general $E$, as constructed in Proposition 4.1, will be a smooth point in $\mathcal{M}_{X,H}(2;3H,c_{2})$. ∎

Let $E\in B_{2}^{1}-B_{2}^{2}$ be a smooth point in $\mathcal{M}(c_{2}),28\leq c_{2}\leq 32,$ as constructed in Proposition 4.1. Existence of such a smooth point is assured by Proposition 4.4. Then $E$ fits into the following exact sequence

[TABLE]

(see equation (3)). Tensoring this exact sequence by $E^{*}\otimes K_{X}$ we get

[TABLE]

Note that $h^{0}(X,E^{*}\otimes K_{X})=h^{2}(X,E)=0$, by our assumption, and since $E$ is a smooth point in the moduli space, $H^{0}(X,\mathcal{E}nd(E)\otimes K_{X})\simeq H^{0}(X,K_{X})$. Thus, by taking cohomology long exact sequence corresponding to the exact sequence (5), we get

[TABLE]

Note that the Petri map $\mu$ in (1) is same as the map $\eta$ above. Now we will show that the map $\eta$ is injective, and this will in turn imply that the bundle $E$ is a smooth point in $B_{2}^{1}$.

Dualizing the exact sequence (4) and restricting to the curve $C$, we obtain the exact sequence (on $C$)

[TABLE]

which induces the following short exact sequence (on $C$)

[TABLE]

Tensoring the sequence (7) by $\mathcal{O}_{C}(C)\otimes A^{*}\otimes K_{X}|_{{}_{C}}$ and taking cohomology long exact sequence we get

[TABLE]

By our assumption on $A$, $H^{0}(C,\mathcal{O}_{C}(C)\otimes A^{*})=0$. Since $\text{degree}(A^{*}\otimes K_{X}|_{{}_{C}})<0$, we have $H^{0}(C,\mathcal{O}_{C}(C)\otimes A^{*\otimes 2}\otimes K_{X}|_{{}_{C}})=0$. Thus from the above exact sequence $H^{0}(C,K_{X}|_{{}_{C}})\simeq H^{0}(C,\mathcal{O}_{C}(C)\otimes A^{*}\otimes(E^{*}\otimes K_{X})|_{{}_{C}}).$ It is easy to see that $H^{0}(X,K_{X})\simeq H^{0}(C,K_{X}|_{{}_{C}})$. Consequently, the map $\eta$ in (6) is injective.

We summarize the above discussion as

Theorem 4.5**.**

$B_{2}^{1}\subset\mathcal{M}(c_{2}),28\leq c_{2}\leq 32,$* contains a smooth point, and hence the irreducible component containing it has the expected dimension.*

5. Non-emptiness of $B_{2}^{0}$

Let $X$ and $\mathcal{M}(c_{2})$ be as in the previous section. Let $\text{Hilb}^{c_{2}}(X)^{\text{lci}}$ denotes the open subscheme of the Hilbert scheme $\text{Hilb}^{c_{2}}(X)$ consisting of length $c_{2}$ subschemes of $X$ which are locally complete intersections. Given any point $Z\in\text{Hilb}^{c_{2}}(X)^{\text{lci}}$, we have an exact sequence

[TABLE]

where $\mathcal{I}_{Z}$ is the ideal sheaf of $Z$ and $E$ is a rank two torsion-free sheaf on $X$. The space of such (isomorphic classes of) extensions is parametrized by $\mathbb{P}\text{Ext}^{1}(\mathcal{I}_{Z}\otimes\mathcal{O}_{X}(3),\mathcal{O}_{X})$. By duality, $\text{Ext}^{1}(\mathcal{I}_{Z}\otimes\mathcal{O}_{X}(3),\mathcal{O}_{X})=H^{1}(X,\mathcal{I}_{Z}\otimes\mathcal{O}_{X}(4))^{*}$. Since $h^{0}(X,\mathcal{O}_{X}(4))=35$, for a general element of length $c_{2}\geq 36$ in $\text{Hilb}^{c_{2}}(X)^{\text{lci}}$, we have $h^{0}(X,\mathcal{I}_{Z}\otimes\mathcal{O}_{X}(4))=0$. Thus a general element in $\text{Hilb}^{c_{2}}(X)^{\text{lci}},c_{2}\geq 36$, satisfies the Cayley-Bacharach property for $\mathcal{O}_{X}(4)$, and consequently, we get that the corresponding extension $E$ is locally free. Also any such vector bundle $E$ is $H$-stable. Thus for $c_{2}\geq 36,B_{2}^{0}\subset\mathcal{M}(c_{2})$ is non-empty.

The following two results give a bound for $\text{dim}B_{2}^{0}$.

Lemma 5.1**.**

([14, Proposition 1.1]222In [14], Proposition $1.1$ was proved with the assumption that $c_{1}(E)=H$, and the bound author got there is $\text{dim}B_{2}^{0}\leq 3c_{2}-11$, for $c_{2}\geq 10$. The same calculation, in this case, gives us the stated bound) With the notations as above, for $c_{2}\geq 36,\text{dim}B_{2}^{0}\leq 3c_{2}-36$.

Lemma 5.2**.**

([12, Corollary 3.1]) With the notations as above, every irreducible component of $B_{2}^{0}$ has dimension $\geq 3c_{2}-h^{0}(X,\mathcal{O}_{X}(3)\otimes K_{X})-1.$

Since $h^{0}(X,\mathcal{O}_{X}(3)\otimes K_{X})=35,$ we see that, for $c_{2}\geq 36$, every irreducible component of $B_{2}^{0}$ has dimension exactly $3c_{2}-36$. On the other hand, the Brill-Noether number $\rho_{2}^{0}=3c_{2}-36$. We summarize the above discussion as

Proposition 5.3**.**

With the notations as above, $B_{2}^{0}\subset\mathcal{M}(c_{2})$ is non-empty for $c_{2}\geq 36$, and every irreducible component of $B_{2}^{0}$ has the expected dimension.

Acknowledgement: We would like to thank Prof. P. Newstead for his valueable comments and pointing out the gap in the earlier version. We also would like to thank D.S. Nagaraj, V. Balaji, P. Sastry for their encouragement and helpful discussion. First named author would like to thank IISER Trivandrum for their hospitality during the stay where this work started.