2. Preliminaries

Let X be a nonsingular irreducible complex projective algebraic surface. This section contains a brief summary about stable torsion-free sheaves on surfaces, and we recall some basic facts on Grasmannians of vector bundles and m-elementary transformations see [Reference Fantechi9, Reference Friedman10, Reference Huybrechts and Lehn14] for more details.

2.1. Grassmannian

We will collect here the principal properties of Grassmannians of vector bundles necessary for our purpose. For a fuller treatment, we refer the reader to [Reference Eisenbud and Harris8, Reference Tyurin25].

Let E be a vector bundle of rank n on X. Let $p_E\,:\, \mathbb{G}(E,m)\rightarrow X$ be the Grassmannian bundle of rank m quotients of E whose fiber at $x\in X$ is the Grassmannian $\mathbb{G}(E_{x},m)$ of m-dimensional quotients of $E_x$ , that is

\begin{equation*}\mathbb{G}(E,m)=\{(x,W)\ |\ x\in X,\ E_x\rightarrow W\rightarrow 0 \}.\end{equation*}

Let

\begin{equation*}0\rightarrow S_E\rightarrow p^{*} E \rightarrow Q_E\rightarrow 0\end{equation*}

be the tautological exact sequence over $\mathbb{G}(E,m)$ where $S_E$ and $Q_E$ denote the universal subbundle of rank $n-m$ and universal quotient of rank m, respectively. The tangent bundle of $\mathbb{G}(E,m)$ is the vector bundle $T\mathbb{G}(E,m)=Hom(S_E,Q_E)$ and hence $T_x\mathbb{G}(E,m)=Hom(S_{E_x},Q_{E_x})$ . Moreover, we have the following exact sequence:

\begin{equation*}0\rightarrow T_{p_E}\rightarrow T\mathbb{G}(E,m)\rightarrow p^{*}_E TX\rightarrow 0\end{equation*}

where $T_{p_E}$ is the relative tangent bundle to the fibers and $T_{p_E}=S^{*}_E\otimes Q_E$ .

2.2. Torsion-Free sheaves

Let H be an ample divisor on X. For a torsion-free sheaf $\mathcal{E}$ on X with Chern classes $c_i \in H^{2i}(X,\mathbb{Z})$ , $i=1,2$ one sets

\begin{equation*} \mu_H(\mathcal{E}) \,:\!=\,\frac{\text{deg}_H(\mathcal{E})}{\text{rk}(\mathcal{E})}, \,\,\,\,\,\, P_m(\mathcal{E})\,:\!=\, \frac{\chi(\mathcal{E} \otimes H^m)}{\text{rk}\, (\mathcal{E})}, \end{equation*}

where $\text{deg}_H(\mathcal{E})$ is the degree of $\mathcal{E}$ defined by $c_1(\mathcal{E}).H$ and $\chi(\mathcal{E} \otimes H^m)$ denotes the Hilbert polynomial defined by $\sum ({-}1)^ih^i(X, \mathcal{E} \otimes H^m)$ .

Definition 2.1. Let H be an ample divisor on X. A torsion-free sheaf $\mathcal{E}$ on X is H-stable (resp. stable) if for all nonzero subsheaf $\mathcal{F}\subset \mathcal{E}$

\begin{eqnarray*} \mu_H(\mathcal{F})\lt \mu_H(\mathcal{E}) \ \ ( resp. \,\, P_m(\mathcal{F}) \lt P_m(\mathcal{E})). \end{eqnarray*}

We want to emphasize that both notions of stability depend on the ample divisor we fix on the underlying surface X and it is easily seen that H-stability implies stability.Footnote 1

Recall that any H-stable (resp. stable) torsion-free sheaf is simple, i.e. if $\mathcal{E}$ is H-stable (resp. stable), then $\dim Hom(\mathcal{E},\mathcal{E})=1$ . We will denote by $M_{X,H}(n;\, c_1,c_2)$ the moduli space of H-stable vector bundles on X of rank n and fixed Chern classes $c_1, c_2$ and by $\mathfrak{M}_{X,H}(n;\,c_1,c_2)$ the moduli space of stable torsion-free sheaves on X. Since locally free is an open property and H-stability implies stability, it follows that $M_{X,H}(n;\, c_1,c_2)$ is an open subset of $\mathfrak{M}_{X,H}(n;\,c_1,c_2)$ . In general an universal family on $X \times M_{X,H}(n;\,c_1,c_2)$ (resp. on $X \times \mathfrak{M}_{X,H}(n;\,c_1,c_2)$ ) does not exist, the existence of such universal family is guaranteed by the following criterion.

Lemma 2.2. [Reference Huybrechts and Lehn14, Corollary 4.6.7] Let X be a nonsingular surface and let H be an ample divisor on X. Let $n,c_1, c_2$ fixed values for the rank and Chern classes. If $gcd(n, c_1.H, \frac{1}{2}c_1.(c_1-K_X)-c_2)=1$ , then there is an universal family on $X \times M_{X,H}(n;\,c_1,c_2)$ (resp. $X \times \mathfrak{M}_{X,H}(n;\,c_1,c_2)$ ).

2.3. m-elementary transformations

Definition 2.3. Let E be a locally free sheaf on X of rank n and Chern classes $c_1, c_2$ and let

(2.1) \begin{equation} 0\rightarrow E^{\prime}\rightarrow E \rightarrow \mathcal{O}^{m}_{x}\rightarrow 0\end{equation}

be an exact sequence of sheaves, where $\mathcal{O}^{m}_{x}=\oplus_{i=1}^{m} \mathcal{O}_{x}$ is the sum of skyscraper sheaf with support on $x\in X.$ The coherent sheaf E ^′ is called the m-elementary transformation of E at $x \in X$ .

Notice that even though E is locally free, its elementary transformation E ^′ is a torsion free sheaf not locally free. Moreover if E is H-stable then E ^′ is also H-stable. However, if E is stable then E ^′ is not necessarily stable (see for instance [Reference Coskun and Huizenga6, Remark 1]).

The m-elementary transformations have been used for several authors to construct many vector bundles on a higher dimensional projective variety and to determine topological and geometric properties of the moduli space of sheaves. For instance, Maruyama did a general study of elementary transformations of sheaves in his master’s and doctoral theses [Reference Maruyama16, Reference Maruyama17]. In [Reference Narasimhan and Ramanan20] Narasimhan and Ramanan used elementary transformations of vector bundles on curves to introduce certain subvarieties in the moduli space of vector bundles which they called Hecke cycles. Brambila-Paz and the first author also used m-elementary transformations to describe a nonsingular open set of the Hilbert scheme of the moduli space of vector bundles on a curve [Reference Brambila-Paz and Mata-Gutiérrez2]. Coskun and Huizenga have used elementary transformations to study priority sheaves since that they are well-behaved under elementary modifications [Reference Coskun and Huizenga3–Reference Coskun and Huizenga5].

We now collect some other basic properties related with m-elementary transformations in the following result.

Proposition 2.4. Let H be an ample divisor on X. Let E be a vector bundle on X of rank n and Chern classes $c_1,c_2$ ,and let E^′ be a m-elementary transformation of E at $x \in X,$ i.e. we have

(2.2) \begin{equation} 0 \rightarrow E^{\prime} \rightarrow E \rightarrow \mathcal{O}_x^m \rightarrow 0.\end{equation}

Then,

1. (i) $rk(E^{\prime})=n$ , $c_1(E^{\prime})=c_1$ , $c_2(E^{\prime})= c_2+m$ and $\chi(E^{\prime})=\chi(E)-m.$

2. (ii) E^′ is a torsion-free sheaf not locally free.

3. (iii) If E is H-stable, then E^′ is H-stable. Hence, E^′ is stable.

Proof.

1. (i) The proof follows directly from the exact sequence and Riemann–Roch Theorem.

2. (ii) Clearly E ^′ is torsion free since E is a vector bundle. Now, suppose that E ^′ is locally free, by [Reference Friedman10, Chapter 4, Lemma 3], it follows that $E =E^{\prime}$ which is impossible because $c_2(E^{\prime}) =c_2+m$ . Therefore E ^′ is a torsion-free sheaf not locally free.

3. (iii) Let F be subsheaf of E ^′ and assume that E is H-stable. It is clear that F is a subsheaf of E and by item (i), it follows that

   \begin{equation*}\mu_H(F) \lt \mu_H(E) = \mu_H(E^{\prime}).\end{equation*}
   Hence E ^′ is H-stable and therefore stable.

Remark 2.5. The class of extensions (2.2) are parameterized by $\mathbb{G}(E_x,m)$ . Furthermore, any $W \in \mathbb{G}(E_x,m)$ defines a surjective linear transformation $\tilde{\alpha}_{W}\,:\,E_{x}\rightarrow W\rightarrow 0$ which determines a surjective morphism of sheaves $\alpha_{W}\,:\,E\rightarrow \mathcal{O}^{m}_{x}$ . If $E^W$ denotes $\text{ker}(\alpha_W)$ then we have the exact sequence:

(2.3) \begin{eqnarray}0\rightarrow E^{W}\rightarrow E \rightarrow \mathcal{O}^{m}_{x}\rightarrow 0.\end{eqnarray}

The following result will be used in the next sections:

Lemma 2.6. Let E be a vector bundle on X and let $\mathcal{O}_{x}$ be the skyscraper sheaf with support on $x\in X$ . Then, for any integer $m\geq 1$ we have

\begin{equation*}{Ext}^{i}\left(\mathcal{O}^{m}_{x},E\right)=0, \ \ \ \ i\neq 2.\end{equation*}

For a deeper discussion of m-elementary transformations, we refer to reader to [Reference Brambila-Paz and Mata-Gutiérrez2, Reference Coskun and Huizenga3].

2.4. Hecke cycles on the moduli space of vector bundles on curves

Let X be a smooth projective curve, and let $x \in X$ be a point. For any vector bundle E on X, the m-elementary transformation

(2.4) \begin{equation} 0\rightarrow E^{\prime}\rightarrow E \rightarrow \mathcal{O}_{x}^m\rightarrow 0 \end{equation}

determines a vector bundle E ^′, where $\text{deg}(E^{\prime})=\text{deg}(E)-m$ and $\text{rk}(E^{\prime})=\text{rk}(E)$ . If E is general in the moduli space $M_X(n,d)$ of stable vector bundles of rank n and degree d, then E ^′ is stable (see [Reference Brambila-Paz and Mata-Gutiérrez2, Proposition 2.4]).

In [Reference Narasimhan and Ramanan20] Narasimhan and Ramanan considered the m-elementary transformations of type

\begin{equation*}0\rightarrow E^{\prime}\rightarrow E \rightarrow \mathcal{O}_{x}\rightarrow 0 \end{equation*}

to prove that, for a general $E\in M_X(n,d)$ (for an explicit description of the general open set in $M_X(n,d)$ see [Reference Narasimhan and Ramanan20, Lemma 5.5]), the pair (E, x) determines a closed embedding

(2.5) \begin{equation} \Phi_{(E,x)}\,:\,\mathbb{P}\left(E^{*}_x\right)\rightarrow M_X(n,d-1).\end{equation}

(see, [Reference Narasimhan and Ramanan20, Lemma 5.8]) and therefore $\mathbb{P}\left(E^{*}_x\right)$ can be considered as a subscheme of the moduli space $M_X(n,d-1)$ . These projective subschemes are called Hecke cycles. Every Hecke cycle determines a point in the Hilbert scheme $\text{Hilb}_{M_X(n,d-1)}$ . Narasimhan and Ramanan proved that there is an open subscheme in $M_X(n,d)$ which is isomorphic to an open subscheme of $\text{Hilb}_{M_X(n,d-1)}$ (see, [Reference Narasimhan and Ramanan20, Theorem 5.13]).

Later, in [Reference Brambila-Paz and Mata-Gutiérrez2] the authors generalize the ideas of Narasimhan and Ramanan and they considered m-elementary transformations, $m\gt1$ in order to prove that, if $E\in M_X(n,d)$ is general (for an explicit description of the general open set in $M_X(n,d)$ see [Reference Brambila-Paz and Mata-Gutiérrez2, Proposition 2.4]), then E ^′ is stable. Moreover, every pair (E, x) determines a closed embedding

(2.6) \begin{equation} \Phi_{(E,x)}\,:\,\mathbb{G}(E_x,m)\rightarrow M_X(n,d-m)\end{equation}

(see [Reference Brambila-Paz and Mata-Gutiérrez2, Proposition 3.1]) and therefore $\mathbb{G}(E_x,m)$ can be considered as a Grassmannian subvariety in the moduli space $M_X(n,d-m)$ which is called m-Hecke cycles. Hence, they concluded that $\text{Hilb}_{M(n,d-m)}$ has an irreducible component $\mathcal{HG}$ of dimension $(n^2-1)(g-1)+1$ where every m-Hecke cycle determines a smooth point (see, [Reference Brambila-Paz and Mata-Gutiérrez2, Theorem 1.1]).

The principal significance of [Reference Narasimhan and Ramanan20, Lemma 5.8] and [Reference Brambila-Paz and Mata-Gutiérrez2, Proposition 3.1] is that the morphisms (2.5) and (2.6) are closed embeddings. It allows determine m-Hecke cycles and geometric and topological properties of the Hilbert scheme $\text{Hilb}_{M_X(n,d-m)}$ .

3. On the moduli space of torsion free sheaves

The aim of this section is to define an embedding from $\mathbb{G}(E_x,m)$ into the moduli space $\mathfrak{M}_{X,H}(n;\,c_1,c_2+m)$ of torsion-free sheaves. Generalizing some techniques of [Reference Brambila-Paz and Mata-Gutiérrez2, Reference Narasimhan and Ramanan20] we establish a closed embedding $\phi_z\,:\, \mathbb{G}(E_x, m) \to \mathfrak{M}_{X,H}(n;\,c_1,c_2+m)$ and an injective algebraic morphism $\Psi\,:\,X \times M_{X,H}(n;\,c_1, c_2) \to \text{Hilb}_{\mathfrak{M}_{X,H}(n;\,c_1,c_2+m)},$ where $z =(x,E) \in X \times M_{X, H}(n;\,c_1,c_2)$ and $Hilb_{\mathfrak{M}_{X,H}(n;\,c_1,c_2+m)}$ denotes the Hilbert scheme of the moduli space $\mathfrak{M}_{X,H}(n;\,c_1,c_2)$ . Moreover, we construct an irreducible variety properly contained in $\mathfrak{M}_{X,H}(n;\,c_1,c_2+m)-M_{X,H}(n;\,c_1,c_2+m)$ .

The following Lemma deals with m-elementary transformations, specifically we compute the dimension of the morphisms of a m-elementary transformation E ^′ of E. The important point to note here is that E is a vector bundle. Here and subsequently, E denotes a vector bundle on X.

Proof. Let $f\,:\,E^{\prime}\rightarrow E$ be a not zero homomorphism. By [Reference Friedman10, Proposition 7, Chapter 4] the morphism f is injective and hence we have the sequence

\begin{equation*}0 \rightarrow E^{\prime} \rightarrow E \rightarrow E/E^{\prime} \rightarrow 0.\end{equation*}

By [Reference Hartshorne12, Proposition 6.4.], we have the following long exact sequence

\begin{equation*}\begin{aligned}0 \rightarrow & \text{Hom} (E/E^{\prime}, E) \rightarrow \text{Hom}(E,E) \rightarrow \text{Hom}(E^{\prime},E) \rightarrow \\ & \text{Ext}^1(E/E^{\prime}, E) \rightarrow \text{Ext}^1(E,E) \rightarrow \text{Ext}^1(E^{\prime},E) \rightarrow \cdots \end{aligned}\end{equation*}

Note that $E/E^{\prime}$ has support in a finite number of points because $c_1(E)=c_1(E^{\prime})$ , hence $\text{Hom}(E/E^{\prime},E)=0.$ On the other hand Lemma 2.6, implies that $\text{Ext}^{1}(E/E^{\prime},E)=0$ . Since E is a H-stable vector bundle, it follows that

\begin{equation*} \dim\, \text{Hom}(E,E) = \dim\, \text{Hom}(E^{\prime},E) = 1\end{equation*}

as we desired.

Set $z\,:\!=\,(x,E)\in X \times M_{X,H}(n;\,c_1,c_2)$ and let m be a fixed natural number with $m\lt n$ . Let $\pi_E\,:\, \mathbb{G}(E,m) \rightarrow X$ be the Grassmannian bundle associated to E and for any $x \in X$ denote by $\mathbb{G}(E_x,m)$ the Grassmannian of m-quotients of $E_x$ . On $\mathbb{G}(E,m)$ , we have the tautological exact sequence

(3.1) \begin{eqnarray}0 \rightarrow S_E \rightarrow \pi_E^*E \rightarrow Q_E \rightarrow 0,\end{eqnarray}

where $S_E$ is the universal subbundle and $Q_E$ is the universal quotient bundle. Note that for any $x \in X$ , if we restrict (3.1) to $\mathbb{G}(E_x,m)$ then we obtain

(3.2) \begin{equation} 0 \rightarrow S_{E_{x}} \rightarrow \mathcal{O}_{\mathbb{G}} \times E_x \rightarrow Q_{E_x} \rightarrow 0.\end{equation}

Let us denote by $\mathbb{G}(z) \,:\!=\, \mathbb{G}(E_x,m)$ . Consider on $X \times \mathbb{G}(z)$ , the surjective morphism $\alpha\,:\, p_1^*E \longrightarrow p_1^*\mathcal{O}_x \otimes p_2^*Q_{E_x}$ associated to the canonical surjective morphism $\alpha_x\,:\, \mathcal{O}_{\mathbb{G}} \times E_{x} \rightarrow Q_{E_x}$ in (3.2) under the isomorphism:

\begin{eqnarray*}H^0\left(X \times \mathbb{G}(z), p_1^{*}E^* \otimes p_1^{*}\mathcal{O}_x \otimes p_{2}^*Q_{E_x}\right)&\cong & H^0\left( \mathbb{G}(z), p_{2_*} (p_1^{*}E^* \otimes p_1^*\mathcal{O}_x ) \otimes Q_{E_x}\right)\\[3pt]&\cong & H^0\left( \mathbb{G}(z), p_{2_*}p_1^*(E_{{x}}^*) \otimes Q_{E_x}\right) \\[3pt]&\cong & H^0\left( \mathbb{G}(z), \left(\mathcal{O}_{\mathbb{G}} \times E_x^*\right) \otimes Q_{E_x}\right) \\[3pt]&\cong & H^0\left( \mathbb{G}(z), \mathcal{H}om\left(\mathcal{O}_{\mathbb{G}} \times E_x, Q_{E_x}\right)\right),\end{eqnarray*}

where the second isomorphism is given by projection formula (see, [Reference Mumford19], p. 76). Here, taking the kernel of the surjective morphism $\alpha\,:\, p_1^*E \longrightarrow p_1^*\mathcal{O}_x \otimes p_2^*Q_{E_x},$ we get the exact sequence

(3.3) \begin{eqnarray}0 \longrightarrow \mathcal{F}_z\longrightarrow p_1^*E \longrightarrow p_1^*\mathcal{O}_x \otimes p_2^*Q_{E_x} \longrightarrow 0\end{eqnarray}

on $X \times \mathbb{G}(z)$ .

Lemma 3.2. Let $z=(x,E)\in X\times M_{X,H}(n;\, c_1,c_2)$ and $W\in\mathbb{G}(z)$ , then

\begin{eqnarray*} \mathcal{T}or^1\left(\mathcal{O}_{\{x\} \times \mathbb{G}}, \mathcal{O}_{X \times \{W\}}\right)=0. \end{eqnarray*}

Proof. Restricting the exact sequence

\begin{eqnarray*}0\to I_{\{x\} \times \mathbb{G}}\to \mathcal{O}_{X \times \mathbb{G}} \to \mathcal{O}_{\{ x\}\times \mathbb{G}}\to 0\end{eqnarray*}

to $X \times \{W\}$ , we get

\begin{eqnarray*} 0 \to \mathcal{T}or^1\left(\mathcal{O}_{\{x\} \times \mathbb{G}}, \mathcal{O}_{X \times \{W\}}\right)\to I_{\{x\} \times \mathbb{G}}\vert_{X\times \{W\}}\to \mathcal{O}_{X} \to \mathcal{O}_{ x}\to 0\end{eqnarray*}

As is well-known $p_1^{*}I_x\cong I_{\{x\}\times \mathbb{G}}$ and $ I_{\{x\} \times \mathbb{G}}\vert_{X\times \{W\}}\cong I_x$ . Then it follows that

\begin{equation*} \mathcal{T}or^1\left(\mathcal{O}_{\{x\} \times \mathbb{G}}, \mathcal{O}_{X \times \{W\}}\right)=0.\end{equation*}

With the above notation and as consequence of Lemma 3.2, we have the following result.

Proof. Let $W\in \mathbb{G}(z)$ . Restricting the exact sequence (3.3) to $X\times \{W\}$ , we get the exact sequence

(3.4) \begin{eqnarray}0 \longrightarrow E^{W} \longrightarrow E \longrightarrow \mathcal{O}_x \otimes W \longrightarrow 0\end{eqnarray}

over X. Hence, $E^{W}$ is a torsion-free sheaf of rank n called the m-elementary transformation of E in x defined by W. Since $c_1( \mathcal{O}_x \otimes W)=0$ and E is H-stable, it follows that $E^W$ is H-stable and therefore stable with $c_1(E^W)=c_1(E)$ (see Proposition 2.4). Moreover, by Whitney sum and $c_2( \mathcal{O}_x \otimes W)=-\dim\, (W)=-m$ we get $c_2(E^{W})=c_2(E)+m$ which completes the proof.

The classification map of $\mathcal{F}_z$ is given by

\begin{equation*}\begin{aligned}\phi_z\,:\, \mathbb{G}(z)&\rightarrow \mathfrak{M}_{X,H}(n;\,c_1,c_2+m)\\[4pt]W&\mapsto E^{W},\end{aligned}\end{equation*}

where $E^W$ was defined in the above Proposition. The following result shows that the morphism $\phi_z$ is a closed embedding. For the proof of the proposition, we follow the techniques and ideas of [Reference Narasimhan and Ramanan20, Lemma 5.10], and [Reference Brambila-Paz and Mata-Gutiérrez2, Proposition 3.1] who proved a similar result for vector bundles on curves.

Proof. We first prove that the morphism $\phi_z$ is injective. Assume that there exist $W_1,W_2\in \mathbb{G}(z) $ such that $\psi\,:\, E^{W_1}\rightarrow E^{W_{2}}$ is an isomorphism, we claim that $W_1=W_2$ . Recall that for any $i=1,2$ , we have the following exact sequence

By Lemma 3.1 we have $\dim\,\text{Hom}(E^{W_1},E)=1$ , it follows that there exist $\lambda \in \mathbb{C}^{*}$ such that $ \lambda f_1=f_2\circ \psi $ . Hence, $\text{Im}\,f_{1,x}=\text{Im}\,f_{2,x}$ which implies $W_{1}=W_{2}$ . Therefore, $\phi_{z}$ is injective.

We now proceed to show the injectivity of the differential map $d\phi_z\,:\, T_W\mathbb{G}(z) \to \mathfrak{M}_{X,H}(n;\,c_1,c_2+m)$ . By [Reference Narasimhan and Ramanan20, Lemma 5.10], its infinitesimal deformation map in $W\in \mathbb{G}(z)$ is, up to the sign, the composition of the natural map $T_W\mathbb{G}(z) \rightarrow \text{Hom} \left(E^{W}, \mathcal{O}_{x}\otimes W\right)$ with the boundary map $\text{Hom} \left(E^{W}, \mathcal{O}_{x}\otimes W\right)\rightarrow \text{Ext}^{1}(X, E^{W},E^{W})$ given by the long exact sequence

\begin{eqnarray*} 0\rightarrow \text{Hom}\left(E^{W},E^{W}\right)\rightarrow \text{Hom}(E^{W},E)\rightarrow \text{Hom} \left(E^{W}, \mathcal{O}_{x}\otimes W\right) \rightarrow \text{Ext}^{1}\left(E^{W},E^{W}\right)\rightarrow\cdots\end{eqnarray*}

obtained from (3.4). Notice that $\text{Hom}\left(E^{W},E^{W}\right)\cong \mathbb{C}$ because $E^{W}$ is an H-stable free torsion sheaf. Moreover, $\text{Hom}(E^{W},E)\cong \mathbb{C}$ by Lemma 3.1. Therefore, the coboundary morphism

\begin{equation*}\delta\,:\, \text{Hom} \left(E^{W}, \mathcal{O}_{x}\otimes W\right)\rightarrow \text{Ext}^{1}\left(E^{W},E^{W}\right)\end{equation*}

is injective.

As in [Reference Brambila-Paz and Mata-Gutiérrez2, Reference Narasimhan and Ramanan20], a consequence of the above result is that we determine a collection of closed subschemes in $\mathfrak{M}_{X,H}(n;\,c_1,c_2+m)$ and a collection of points in its Hilbert scheme (see, [Reference Narasimhan and Ramanan20, Definition 5.12]). From a stable vector bundle E on X, we constructed the family $\mathcal{F}_z$ of stable torsion-free sheaves. Analogously, if we start with a family $\mathcal{E}$ of stable vector bundles on X parameterized by T, then we can construct a family of of stable torsion-free sheaves $\mathcal{F}$ . In the next paragraphs, we describe the construction when $\mathcal{E}$ is the universal family of stable vector bundles parameterized by $M_{X,H}(n;\,c_1, c_2)$ .

Let H be an ample divisor on X. As is well-known if $\text{gcd}\left(n, c_1.H, \frac{1}{2}c_1.(c_1-K_X)-c_2\right)=1,$ then there exists a universal family $\mathcal{U}$ of vector bundles parameterized by $M_{X,H}(n;\,c_1,c_2)$ (see Lemma 2.2). Under this conditions, we will determine a family $\mathcal{F}$ of stable torsion-free sheaves parameterized by $\mathbb{G}(\mathcal{U},m)$ which extends to $\mathcal{F}_z$ (see Proposition 3.3).

Let $\mathcal{U}$ be the universal family of vector bundles parameterized by $M_{X,H}(n;\,c_1,c_2)$ , hence $p\,:\,\mathcal{U}\rightarrow X\times M_{X,H}(n;\,c_1, c_2)$ is a vector bundle. We denote by $\pi_{\mathcal{U}}\,:\,\mathbb{G}(\mathcal{U},m) \rightarrow X\times M_{X,H}(n;\, c_1, c_2) $ the Grassmannian bundle of quotients associated to $\mathcal{U}$ . An element of $\mathbb{G}(\mathcal{U}, m)$ is a pair ((x, E),W), where $(x,E) \in X \times M_{X,H}(n;\,c_1,c_2)$ and $W\in \mathbb{G}(E_x,m)$ . The tautological exact sequence over $\mathbb{G}(\mathcal{U}, m)$ is

(3.5) \begin{eqnarray}0 \rightarrow S_\mathcal{U} \rightarrow \pi_{\mathcal{U}}^*\,\mathcal{U}\stackrel{\alpha} \rightarrow Q_{\mathcal{U}} \rightarrow 0,\end{eqnarray}

where $Q_{\mathcal{U}}$ denotes the universal quotient bundle of rank m over $\mathbb{G}(\mathcal{U},m)$ . We now consider the graph of the following composition

$\Gamma\,:\!=\,\Gamma_{p_{1}\circ \pi_{U}}$ as a subvariety of $X\times \mathbb{G}(\mathcal{U},m)$ . Then we have the following result.

Proof. Taking $\beta\,:\!=\,p_{\mathbb{G}(\mathcal{U})}|_\Gamma$ as the restriction of the projection, we have the following commutative diagram

where $i\,:\,\Gamma \to X\times \mathbb{G}(\mathcal{U})$ is the inclusion map, hence $I_{X \times {g}}\vert_{\Gamma}= i^{*}p_{\mathbb{G}(\mathcal{U})}^{*}(I_{g})=\beta^{*}(I_g).$

From the exact sequence

\begin{eqnarray*}0 \rightarrow I_g \rightarrow \mathcal{O}_{\mathbb{G}(\mathcal{U})} \rightarrow \mathcal{O}_{g}\rightarrow 0,\end{eqnarray*}

we get

\begin{eqnarray*}0 \rightarrow \beta^{*}(I_g) \rightarrow \beta^{*}(\mathcal{O}_{\mathbb{G}(\mathcal{U})}) \rightarrow \beta^{*}(\mathcal{O}_{g})\rightarrow 0,\end{eqnarray*}

Therefore, $\mathcal{T}or^1(I_{X\times \{g\}}, \mathcal{O}_{\Gamma})=0$ and this prove (a).

Now, to prove (b) consider the surjective map $\alpha\,:\, \pi_\mathcal{U}^{*}\,\mathcal{U}\rightarrow Q_{\mathcal{U}}$ given in (3.5) and notice that $\beta^{*}\alpha\,:\,\beta^{*}\pi_\mathcal{U}^{*}\,\mathcal{U}\rightarrow \beta^{*}Q_{\mathcal{U}}$ is also surjective. Since $\beta^{*} \pi_U^{*}(\mathcal{U})\cong (id \times p_{2} \circ \pi_{\mathcal{U}})^{*}\,(\mathcal{U})\vert_{\Gamma}$ and $\beta^{*}Q_{\mathcal{U}}\cong p_{\mathbb{G}((\mathcal{U})}^{*}(Q_{\mathcal{U}})\vert_{\Gamma}$ , we get a surjective morphism

(3.7) \begin{equation} (id \times p_{2} \circ \pi_{\mathcal{U}})^{*}\,(\mathcal{U})\vert_{\Gamma} \rightarrow \mathcal{O}_{\Gamma}\otimes p_{\mathbb{G}(\mathcal{U})}^{*}Q_{\mathcal{U}}.\end{equation}

Hence, from the exact sequence

\begin{eqnarray*}0\to (id \times p_{2} \circ \pi_{\mathcal{U}})^{*}\,\mathcal{U}\ \otimes I_{\Gamma} \to (id \times p_{2} \circ \pi_{\mathcal{U}})^{*}\,\mathcal{U}\ \to (id \times p_{2} \circ \pi_{\mathcal{U}})^{*}\,\mathcal{U}\ \vert_{\Gamma}\to 0\end{eqnarray*}

and the morphism (3.7) we get the surjective map $(id \times p_{2} \circ \pi_{\mathcal{U}})^{*}\,\mathcal{U}\rightarrow \mathcal{O}_{\Gamma}\otimes p_{\mathbb{G}(\mathcal{U})}^{*}Q_{\mathcal{U}}$ which completes the proof.

According to the above Lemma, let us denote by $\mathcal{F}$ the kernel of the surjective morphism (3.6). Hence, we get the exact sequence

(3.8) \begin{eqnarray}0\rightarrow \mathcal{F}\rightarrow (id \times p_{2} \circ \pi_{\mathcal{U}})^{*}\,\mathcal{U}\rightarrow \mathcal{O}_{\Gamma}\otimes p_{\mathbb{G}(\mathcal{U})}^*Q_{\mathcal{U}} \rightarrow 0.\end{eqnarray}

Note that $(id \times p_{2} \circ \pi_{\mathcal{U}})^{*}\,(\mathcal{U}) \vert_{X\times ((x,E),W)}=E$ and $\mathcal{O}_{\Gamma}\otimes p_{\mathbb{G}(\mathcal{U})}^*Q_{\mathcal{U}}\vert_{X \times ((x,E),W)}=\mathcal{O}_x \otimes W$ . Since $ p_{\mathbb{G}(\mathcal{U})}^*Q_{\mathcal{U}}$ is a vector bundle and $\mathcal{T}or^1(I_{X\times \{g\}}, \mathcal{O}_{\Gamma})=0,$ it follows that $\mathcal{T}or^1(I_{X\times \{g\}}, \mathcal{O}_{\Gamma}\otimes p_{\mathbb{G}(\mathcal{U})}^*Q_{\mathcal{U}} )=p_{\mathbb{G}(\mathcal{U})}^*Q_{\mathcal{U}}\otimes \mathcal{T}or^1(I_{X\times \{g\}}, \mathcal{O}_{\Gamma})=0.$ Therefore, restricting the exact sequence (3.8) to $X\times \{((x,E),W)\}$ , we get the exact sequence

\begin{eqnarray*}0 \longrightarrow E^{W} \longrightarrow E \longrightarrow \mathcal{O}_x \otimes W \longrightarrow 0\end{eqnarray*}

over X. Moreover, if we restrict (3.8) to $X\times \mathbb{G}(z)$ , we obtain (3.3).

Hence by similar arguments to Proposition 3.3, we have that $\mathcal{F}$ is a family of stable torsion-free sheaves of rank n of type $(c_1, c_2+m)$ which determines a morphism

\begin{eqnarray*} \Phi\,:\,\mathbb{G}(\mathcal{U},m)&\rightarrow& \mathfrak{M}_{X,H}(n;\,c_1,c_2+m)\\ ((x,E),W) &\mapsto & E^W.\end{eqnarray*}

Note that $\text{Im}\,\Phi$ lies in $\mathfrak{M}_{X,H}(n;\,c_1,c_2+m)- M_{X,H}(n;\,c_1,c_2+m)$ . In the following theorem, we compute the dimension of $\text{Im}\,\Phi$ .

Theorem 3.6. Let m, n natural integers with $1\leq m \lt n $ . Then $\mathfrak{M}_{X,H}(n;\,c_1,c_2+m)-M_{X,H}(n;\,c_1,c_2+m)$ contains an irreducible projective variety Y of dimension $3+\dim\, M_{X,H}(n;\,c_1,c_2)$ such that the general element $F \in Y$ fits into exact sequence

\begin{eqnarray*}0 \to F \to E \to \mathcal{O}_{X,x}\otimes W \to 0,\end{eqnarray*}

where $E\in M_{X,H}(n;\,c_1,c_2)$ , $W\in\mathbb{G}(E_x, m)$ and $x\in X$ . In particular, if $n=2 $ then $\Phi$ is injective and Y is a divisor.

Proof. We will prove that image of $\Phi$ is an irreducible variety of dimension $3 + \dim\, M_{X,H}(n;\,c_1,c_2)$ . For this, it will thus be sufficient to compute the dimension of the fibers of $\Phi$ . Let $F \in \text{Im} \,\Phi$ , then there exists $((x,E),W)\in \mathbb{G}(\mathcal{U},m)$ such that F fits into the following exact sequence

(3.10) \begin{equation}0 \to F \to E \to \mathcal{O}_{X,x}\otimes W \to 0,\end{equation}

where E is a vector bundle and $W\in \mathbb{G}(E_x,m)$ . We claim $\dim\, \text{Ext}^1(\mathcal{O}_{X,x}\otimes W,F)=m^2$ .

From the exact sequence (3.10), we get the long exact sequence

\begin{equation*}\begin{aligned}0 & \to \text{Hom}(\mathcal{O}_{X,x},F) \to\text{Hom}(\mathcal{O}_{X,x},E)\to \text{Hom}(\mathcal{O}_{X,x},\mathcal{O}_{X,x}\otimes W) \to \\[4pt] & \text{Ext}^1(\mathcal{O}_{X,x},F) \to \text{Ext}^1(\mathcal{O}_{X,x},E) \to \text{Ext}^1(\mathcal{O}_{X,x},\mathcal{O}_{X,x}\otimes W) \to \ldots\end{aligned}\end{equation*}

Since $\text{Hom}(\mathcal{O}_{X,x},E)=0$ and by Lemma 2.6 $\text{Ext}^1(\mathcal{O}_{X,x},E)=0$ , it follows that

\begin{equation*}\dim\, \text{Ext}^1(\mathcal{O}_{X,x},F) = \dim\,\text{Hom}(\mathcal{O}_{X,x},\mathcal{O}_{X,x}\otimes W)=m.\end{equation*}

Thus, $\dim\,\text{Ext}^1(\mathcal{O}_{X,x}\otimes W,F)=m^2$ .

We now proceed to compute the dimension of $\text{Im} \, \Phi$ . Let $p_i$ be denote the canonical projection of $X\times \mathbb{G}(E_x, m)$ for $i=1,2$ and consider the sheaf $\mathcal{H}om(p_{1}^*\mathcal{O}_{x}\otimes p_2^{*}\mathcal{Q}_{E_{x}},p_{1}^*F )$ . Taking higher direct image, we obtain on $\mathbb{G}(E_x, m)$ the sheaf:

\begin{eqnarray*}\Lambda \,:\!=\, R^1_{p_{{2}_*}}\mathcal{H}om\left(p_{1}^*\mathcal{O}_{x}\otimes p_2^{*}\mathcal{Q}_{E_{x}},p_{1}^*F\right).\end{eqnarray*}

This $\Lambda$ is locally free over $\mathbb{G}(E_x, m)$ because

\begin{eqnarray*}H^0(\mathcal{H}om(\mathcal{O}_{X,x} \otimes W,F)) \cong\text{Hom}(\mathcal{O}_{X,x} \otimes W,F)=0,\end{eqnarray*}

for any $W\in \mathbb{G}(E_x, m)$ . Hence, the fiber of $\Lambda$ at $W \in \mathbb{G}(E_x, m)$ is $\text{Ext}^1(\mathcal{O}_{X,x} \otimes W,F)$ .

Let $\pi \,:\,\mathbb{P}\Lambda \to \mathbb{G}(E_x, m)$ denote the projectivization of the sheaf $\Lambda$ . By [Reference Gottsche11, Lemma 3.2] there exists an exact sequence:

(3.11) \begin{equation} 0 \to (id\times\pi)^*p_{1}^*F \otimes \mathcal{O}_{X \times \mathbb{P}\Lambda}(1) \to\mathcal{E} \to(id\times\pi)^*(p_{1}^*\mathcal{O}_{X,x} \otimes p_2^{*}\mathcal{Q}_{E_{x}}) \to 0\end{equation}

on $X \times \mathbb{P}\Lambda$ such that, for each $p\in \mathbb{P}\Lambda$ , its restriction to $X \times \{p\}$ is the extension

\begin{eqnarray*}0 \longrightarrow F \longrightarrow \mathcal{E}_{|_p} \longrightarrow\mathcal{O}_{X,x}\otimes W \longrightarrow 0\end{eqnarray*}

where $\mathcal{E}_{|_p}\,:\!=\,\mathcal{E}_{|_{X\times\{p\}}}$ .

The set

\begin{equation*}U\,:\!=\, \{p \in \mathbb{P}\Lambda ~|~\mathcal{E}_{|_p} \text{ is locally free and stable} \} \end{equation*}

is irreducible open set of dimension $ m(n-m)+m^2-1=mn-1.$ Therefore, the dimension of the fiber of $\Phi$ is $mn-1-m^2=m(n-m)-1$ and then we have

\begin{equation*}\begin{aligned} \dim \, \text{Im}\, \Phi &= m(n-m)+2+ \dim\, M_{X,H}(n;\, c_1,c_2)-m(n-m)+1 \\ &= 3+\dim\, M_{X,H}(n;\,c_1,c_2). \end{aligned}\end{equation*}

Note that for rank two case, the morphism $\phi$ is injective because the dimension of $\mathbb{P}\text{Ext}^1(\mathcal{O}_{X,x}\otimes W, F)=0$ and $\mathbb{P}\text{Ext}^1(\mathcal{O}_{X,x}\otimes W, F)$ is irreducible.

By functorial construction, we also have the following algebraic morphism

\begin{eqnarray*} \Psi\,:\, X \times M_{X,H}(n;\,c_1, c_2)&\rightarrow &\text{Hilb}_{\ \mathfrak{M}_{X,H}(n;\,c_1,c_2+m)}\\z=(x,E) &\mapsto &\mathbb{G}(z) \end{eqnarray*}

with $\mathbb{G}(z)\,:\!=\, \phi_z(\mathbb{G}(E_x,m))$ . This construction is essentially the same as the one carried out in [Reference Brambila-Paz and Mata-Gutiérrez2, Reference Narasimhan and Ramanan20].

The injectivity of the function $\Psi\,:\,X \times M_{X,H}(n;\,c_1, c_2)\rightarrow \text{Hilb}_{\ \mathfrak{M}_{X,H}(n;\,c_1,c_2+m)}$ is established in the next proposition. The proof proceeds as [Reference Brambila-Paz and Mata-Gutiérrez2, Proposition 3.2] and we use the following two lemmas.

Lemma 3.7. Let X be an irreducible variety and let

\begin{equation*}0 \to F \to E \to G \to 0\end{equation*}

be an exact sequence of sheaves over X. If E and G are locally free sheaves, then F is locally free.

Proof. Let H be a sheaf on X. We claim that for any locally free sheaf E on X $\mathcal{E}xt^{i}(E,H)=0$ . By [Reference Hartshorne12, Proposition 6.8], we have

\begin{equation*}\mathcal{E}xt^{i}(E,H)_x \cong \text{Ext}^i(E_x, H_x)\end{equation*}

which is zero for any $x\in X$ because [Reference Friedman10, Theorem 17]. Consider the exact sequence

(3.12) \begin{equation} 0 \to F \to E \to G \to 0\end{equation}

where E and G are locally free sheaves. Applying the functor $\mathcal{H}om({-},H)$ to the exact sequence (3.12), we get

\begin{equation*}\begin{aligned} 0 \to & \mathcal{H}om(G,H) \to \mathcal{H}om(E,H) \to \mathcal{H}om(F,H) \to \\[4pt] & \mathcal{E}xt^1(G,H) \to \mathcal{E}xt^1(E,H) \to \mathcal{E}xt^1(F,H) \to \mathcal{E}xt^2(G,H) \to \cdots\end{aligned}\end{equation*}

Note that $\mathcal{E}xt^i(G,H) = \mathcal{E}xt^i(E,H) = 0$ for $i\gt 0$ . Therefore, $\mathcal{E}xt^1(F,H) =0$ from which we conclude that F is locally free as we desired.

Proof. Assume that for $i=1,2$ , there exist $z_i=(x_i,E_i)\in X\times M_{X,H}(n;\,c_1,c_2)$ such that $\mathbb{G}(z_1)= \mathbb{G}(z_2)$ , we want to prove that $E_1\cong E_2$ and $x_1=x_2$ . We recall that for any $z_i=(x_i,E_i)$ there exists a family $\mathcal{F}_{z_i}$ of stable torsion-free sheaves parameterized by $\mathbb{G}(z_i),$ and $\mathcal{F}_{z_i}$ fits into the following exact sequence

(3.13) \begin{eqnarray}0 \longrightarrow \mathcal{F}_{z_i}\longrightarrow p_1^*E_i \longrightarrow p_1^*\mathcal{O}_{x_i} \otimes p_2^*Q_{E_{x_i}} \longrightarrow 0\end{eqnarray}

of sheaves over $X\times \mathbb{G}(z_i),$ where $p_j$ denotes the j-projection over $X\times \mathbb{G}(z_i)$ . From universal properties of moduli space $\mathfrak{M}_{X,H}(n;\,c_1,c_2+m)$ , there exists an isomorphism $\beta\,:\,\mathbb{G}(z_1)\rightarrow \mathbb{G}({z_2})$ that induces the following commutative diagrams

and

i.e. $\phi_{z_1}=\phi_{z_2}\circ\beta$ and $p_1 = p^{\prime}_{1} \circ \, (id_{X} \times \beta)$ . By the universal property of $\mathfrak{M}_{X,H}(n;\,c_1,c_2+m)$ , we have

\begin{equation*}\mathcal{F}_{z_1}\cong (id_{X}\times\beta)^*\mathcal{F}_{z_2}\otimes p_2^*(L)\end{equation*}

for some line bundle L on $\mathbb{G}({z_1)}$ . The following properties are satisfied:

1. (1) L is trivial.

2. (2) $ R^1 {p_1}_*\left(\mathcal{F}_{z_1}\right)= R^1 {p^{\prime}_1}_*\left(\mathcal{F}_{z_2}\right)= 0$ .

3. (3) ${p_1}_{*}\mathcal{F}_{z_1}= {p^{\prime}_1}_{*} \mathcal{F}_{z_2}$ .

First we proved that $\mathcal{F}_{z_i}|_{\{y\}\times \mathbb{G}(z_i)}\cong E_y\otimes \mathcal{O}_{\mathbb{G}(z_i)}$ is trivial for any $y\neq x_i$ . Restricting the exact sequence (3.13), we obtain

\begin{eqnarray*} 0 \rightarrow \mathcal{T}or^1\left(\mathcal{O}_{\mathbb{G}},p_1^*\mathcal{O}_{x_i} \otimes p_2^*Q_{E_{x_i}}\right) \rightarrow \mathcal{F}_{z_i}|_{y\times \mathbb{G}(z_i)}\rightarrow p_1^{*}(E_i)|_{y\times \mathbb{G}(z_i)}\rightarrow 0.\end{eqnarray*}

Note that $p_1^{*}(E_i)|_{y\times \mathbb{G}(z_i)}\cong E_y \otimes \mathcal{O}_{\mathbb{G}(z_i)}$ and $\mathcal{F}_{z_i}|_{y\times \mathbb{G}(z_i)}$ are vector bundle of the same rank, then by Lemma 3.7 we have $\mathcal{T}or^1\left(\mathcal{O}_{\mathbb{G}},p_1^*\mathcal{O}_{x_i} \otimes p_2^*Q_{E_{x_i}}\right)=0$ and $\mathcal{F}_{z_i}|_{y\times \mathbb{G}(z_i)}\cong E_y \otimes \mathcal{O}_{\mathbb{G}(z_i)}$ . On the other hand

\begin{eqnarray*}\left(id_X \times \beta\right)^{*}\left(\mathcal{F}_{z_2}\right)|_{y\times \mathbb{G}(z_1)}=\beta^{*}\left(\mathcal{F}_{z_2}|_{y\times \mathbb{G}(z_2)}\right)=\beta^{*}\left(E_y\otimes \mathcal{O}_{G(z_2)}\right)=E_y\otimes \mathcal{O}_{G(z_1)}.\end{eqnarray*}

Therefore,

\begin{eqnarray*}E_y\otimes \mathcal{O}_{G(z_1)}=\mathcal{F}_{z_1}|_{y \times \mathbb{G}(z_1)}\cong \left((id_{X}\times\beta)^*\mathcal{F}_{z_2}\otimes p_2^*(L)\right)|_{y\times \mathbb{G}(z_1)}= E_y\otimes \mathcal{O}_{G(z_1)}\otimes L.\end{eqnarray*}

Thus, L is trivial [Reference Newstead22, p. 12] and this prove (1). Moreover

\begin{eqnarray*} \mathcal{F}_{z_1}|_{x_1 \times \mathbb{G}(z_1)}\cong \left(\left(id_{X}\times\beta\right)^*\mathcal{F}_{z_2}\right)|_{x_1\times \mathbb{G}(z_1)}= \beta^{*}\left(E_{x_1}\otimes \mathcal{O}_{G(z_2)}\right)=E_{x_1}\otimes \mathcal{O}_{\mathbb{G}(z_1)}. \end{eqnarray*}

And for any $y \in X$ we have

\begin{eqnarray*}R^1 {p_1}_{*}\left(\mathcal{F}_{z_1}\right)_y= H^1\left(\mathcal{F}_{z_1}|_{y\times \mathbb{G}(z_1)}\right)=H^1\left(E_y \otimes \mathcal{O}_{\mathbb{G}(z_1)}\right)=0.\end{eqnarray*}

Similarly, we can prove that $\mathcal{F}_{z_2}|_{x_2 \times \mathbb{G}(z_2)}\cong E_{x_2}\otimes \mathcal{O}_{\mathbb{G}(z_2)}$ and

$R^1 {p_1}^{\prime}_*\left(\mathcal{F}_{z_2}\right)=0$ and this prove (2). Since $p_1=p_1^{\prime}\circ (id \times \beta)$ and $\left(id_X \times \beta\right)$ is an isomorphism, we get

\begin{eqnarray*}p_{1*}\left(\mathcal{F}_{z_1}\right)=p_{1*}(id \times \beta)^{*}\left(\mathcal{F}_{z_2}\right)&=&\left.(p^{\prime}_1 \circ (id \times \beta))_{*} (id\times \beta)^{*} \mathcal{F}_{z_2}\right)\\&=& \left. p^{\prime}_{1*}((id \times \beta)_{*}(id \times \beta)^{*}\left(\mathcal{F}_{z_2}\right)\right)\\&=& p^{\prime}_{1_*}\left(\mathcal{F}_{z_2}\right),\end{eqnarray*}

and this proves (3). We now proceed to show that $E_1\cong E_2$ and $x_1=x_2$ . The proof will be divided into three steps:

Step 1: We will show that $E_1\otimes I_{x_1}\cong E_2\otimes I_{x_2}$ .

Taking the direct image of (3.13) by $p_1$ we obtain the following exact sequence:

\begin{eqnarray*}0\rightarrow {p_1}_*\left(\mathcal{F}_{z_1}\right)\rightarrow {p_1}_*\left(p^*_1 E_1\right)\rightarrow {p_1}_*\left(p_1^*\mathcal{O}_{x_1}\otimes p_2^*Q_{E_{1,x_1}}\right)\rightarrow 0\end{eqnarray*}

because $ R^1 {p_1}_*\left(\mathcal{F}_{z_1}\right)=0$ . And we can complete the diagram

Since ${p_1}_*p_1^*(E_1)\cong E_1$ and ${p_1}_*\left(p_1^*\mathcal{O}_{x_1}\otimes p_2^*Q_{E_{1,x_1}}\right)\cong E_{1}\otimes \mathcal{O}_{x_1}$ by projection formula, it follows that ${p_1}_*\mathcal{F}_{z_1}\cong E_1\otimes I_{x_1}$ . We can now proceed analogously to obtain ${p^{\prime}_1}_{*}\mathcal{F}_{z_2}\cong E_2\otimes I_{x_2}$ . Therefore,

\begin{eqnarray*} E_1\otimes I_{x_1}\cong {p_1}_*\mathcal{F}_{z_1} \cong {p^{\prime}_1}_{*}\mathcal{F}_{z_2}\cong E_2\otimes I_{x_2}. \end{eqnarray*}

Step 2: We will show that $E_1 \cong E_2$ ;

Note that the general curve on X does not goes through the points $x_1$ and $x_2$ , hence $E_1|_C\cong (E_1\otimes I_{x_1})|_{C}\cong (E_2\otimes I_{x_1})|_{C}\cong E_2|_C$ for the general curve $C\in |aH|$ . From Lemma 3.8, we conclude that $E_1\cong E_2$ which is the desired conclusion.

Step 3: We show will that $x_1=x_2$ ;

Notice that by step 1 there exists an isomorphism $\lambda\,:\,E_1\otimes I_{x_1} \to E_2\otimes I_{x_2}$ . On the other hand, step 2 provided us an isomorphism $\phi\,:\,E_1\rightarrow E_2$ . Considering the exact sequence

for $i=1,2$ . Moreover $\phi\circ f_{1},$ $f_{2}\circ \lambda\in \text{Hom}(E_1\otimes I_{x_1},E_2)$ , and hence by Lemma 3.1, $\phi\circ f_{1}=t(f_{2}\circ \lambda)$ for some $t\in \mathbb{C}^{*}$ . Without loss of generality, we suppose that $t=1$ therefore we have the following commutative diagram

where $\alpha$ is an isomorphism of skyscraper sheaves supported at $x_1$ and $x_2$ , respectively. Hence $x_1=x_2$ . Therefore, $\Psi$ is injective which establishes the proposition.

We can now state our main result. The theorem computes a bound of the dimension of an irreducible subvariety of the Hilbert scheme $\text{Hilb}_{\mathfrak{M}_{X,H}(n;\,c_1,c_2+m)}$ .

Proof. The proof follows from Proposition 3.9.

4. Application to the moduli space of sheaves on the projective plane

Let us denote by $\mathfrak{M}_{\mathbb{P}^2}(2;\,c_1,c_2)$ the moduli space of rank 2 stable sheaves on the projective plane $\mathbb{P}^2$ with respect to the ample line bundle $\mathcal{O}_{\mathbb{P}^2}(1)$ . By Proposition 3.4, the image $\phi_{z}(\mathbb{P}(z))$ defines a cycle in the Hilbert scheme of $\mathfrak{M}_{\mathbb{P}^2}(2;\,c_1,c_2)$

In this section, we will describe the component of the Hilbert scheme which contains the cycles $\phi_z(\mathbb{P}(E_x))$ . Our computations use some results and techniques of [Reference Hirshowitz and Hulek13, Reference Stromme24].

The following theorem was proved in [Reference Stromme24]

Following the construction given in Section 3, if $z=(x,E) \in \mathbb{P}^2 \times M_{\mathbb{P}^2}(2;\,c_1,c_2-1)$ then, Proposition 3.3, we have a family $\mathcal{F}_z$ of H-stable torsion-free sheaves rank two on $\mathbb{P}^2$ parameterized by $\mathbb{P}(E_x)$ or $\mathbb{P}(z)$ for short. Such family fits in the following exact sequence

(4.1) \begin{eqnarray} 0 \longrightarrow \mathcal{F}_z\longrightarrow p_1^*E \longrightarrow p_1^*\mathcal{O}_x \otimes p_2^*Q_{E_x} \longrightarrow 0,\end{eqnarray}

defined on $\mathbb{P}^2\times \mathbb{P}(z).$ The classification map of $\mathcal{F}_z$ is the morphism

(4.2) \begin{equation} \phi_z\,:\, \mathbb{P}(z) \to \mathfrak{M}_{\mathbb{P}^2}(2;\,c_1,c_2)\end{equation}

defined as $\phi_z(W)=E^{W}$ .

We now use the exact sequence (4.1) and the morphism (4.2) to determine the irreducible component of the Hilbert scheme $\text{Hilb}_{\mathfrak{M}_{\mathbb{P}^2}(2;\,c_1,c_2)}$ of the moduli space $\mathfrak{M}_{\mathbb{P}^2}(2;\,c_1,c_2)$ , $c_1=0$ or $-1$ which contains the cycles $\phi_z(\mathbb{P}(z))$ . This component is denoted by $\mathcal{H}\mathcal{G}$ .

For the proof of the theorem, we first establish the result for two particular cases: $c_1=-1$ and $c_1=0$ .

Proof.

1. (1) Let $z=(x,E) \in \mathbb{P}^2 \times M_{\mathbb{P}^2}(2;\,c_1,r)$ , $c_1=-1$ and $r \geq 1.$ Consider the family $\mathcal{F}_z$ of stable sheaves of rank two given by the exact sequence (4.1). Then, $\mathcal{F}_{z_t} \,:\!=\,(\mathcal{F}_{z})\vert_{\mathbb{P}^2 \times \{t\}}$ is stable for any $t\in \mathbb{P}(z)$ and by Proposition 2.4 its Chern classes are $c_1(\mathcal{F}_{z_t})=-1$ and $c_2\,:\!=\, c_2(\mathcal{F}_{z_t})=r+1 \geq 2$ . Therefore, we have the morphism

   \begin{equation*}\phi_z\,:\, \mathbb{P}(E_x) \longrightarrow \mathfrak{M}_{\mathbb{P}^2}(2;\,c_1,c_2), \,\,\,\, t \mapsto \mathcal{F}_z \vert_t\end{equation*}

   and set $\tau = p_1^*(\mathcal{O}_{\mathbb{P}^2}(1))$ .

   Now we will compute $\phi_z^* \epsilon$ and $\phi_z^* \delta$ .

   Let $l \geq 0 $ , from the exact sequence (4.1) we have

   \begin{equation*}\begin{aligned}0 \to & p_{2_*}\mathcal{F}({-}l\tau) \to p_{2_*}p_1^*E({-}l\tau) \to p_{2_*}p_1^*\mathcal{O}_x({-}l\tau) \otimes p_2^*Q_{E_x} \to \\ & R^1p_{2_*}\mathcal{F}({-}l\tau) \to R^1p_{2_*}p_1^*E({-}l\tau) \to R^1p_{2_*}\left(p_1^*\mathcal{O}_x({-}l\tau) \otimes p_2^*Q_{E_x}\right) \to 0.\end{aligned}\end{equation*}

   Using the projection formula, we get

   \begin{eqnarray*}R^ip_{2_*}p_1^*E({-}l\tau) = \mathcal{O}_{\mathbb{P}(E_x)}\otimes H^i(\mathbb{P}^2,E({-}l)).\end{eqnarray*}

   Since $E({-}l)$ is a stable vector bundle on $\mathbb{P}^2$ with $c_1\leq 0,$ it follows that $p_{2_*}p_1^*E({-}l\tau) =0$ and $R^ip_{2_*}p_1^*E({-}l\tau)$ is a trivial bundle. Moreover, by similar arguments we have

   \begin{eqnarray*}R^ip_{2_*}\left(p_1^*\mathcal{O}_x({-}l\tau) \otimes p_2^*Q_{E_x}\right) \cong Q_{E_x} \otimes p_{2_*}p_1^*\mathcal{O}_x({-}l\tau) \cong Q_{E_x}\otimes H^i\left(\mathbb{P}^2,\mathcal{O}_{\mathbb{P}^2}({-}l)_x\right).\end{eqnarray*}

   Hence $R^1p_{2_*}p_1^*\mathcal{O}_x({-}l\tau) \otimes p_2^*Q_{E_x}=0$ and $p_{2_*}p_1^*\mathcal{O}_x({-}l\tau) \otimes p_2^*Q_{E_x}=Q_{E_x}$ . Therefore, we have the exact sequence

   \begin{eqnarray*}0 \to Q_{E_x} \to R^1p_{2_*}\mathcal{F}({-}l\tau) \to R^1p_{2_*}p_1^*E({-}l\tau) \to 0 \end{eqnarray*}
   where we conclude that $ c_1(R^1p_{2_*}\mathcal{F}({-}l\tau)) = 1$ for any $l\geq 0$ .

   According to [Reference Hirshowitz and Hulek13, Lemmas 3.3 and 3.4], it follows that

   \begin{equation*} \phi_z^*(\epsilon) = c_1\left(R^1p_{2_*}\mathcal{F}\right)- c_1(R^1p_{2_*}\mathcal{F}({-}2\tau)=0 \end{equation*}
   and
   \begin{equation*} \phi_z^*(\delta) = (r+1)c_1\left(R^1p_{2_*}\mathcal{F}\right)-rc_1\left(R^1p_{2_*}\mathcal{F}({-}\tau)\right)= 1. \end{equation*}

   Hence, we conclude that

   \begin{equation*}P(m) = \chi(\mathbb{P}(z), \phi^{*}_z(a\epsilon+b\delta) ) = \chi(\mathbb{P}(z), \mathcal{O}_{\mathbb{P}(E_x)}(mb))\end{equation*}

   as we desired.

2. (2) For the case, $c_1=0$ and $c_2 \geq 3$ odd. Consider $z=(x,E) \in \mathbb{P}^2 \times M_{\mathbb{P}^2}(2;\,c_1,r)$ , $c_1= 0$ and $r \geq 2$ even. From the exact sequence (4.1), we get $\mathcal{F}_{z_t} \,:\!=\,\mathcal{F}_{z_{\vert_{\mathbb{P}^2 \times \{t\}}}}$ is stable for all $t \in \mathbb{P}(E_x)$ and $c_1(\mathcal{F}_{z_t})=0$ , $c_2\,:\!=\, c_2(\mathcal{F}_{z_t})=r+1 \geq 3$ odd. By [Reference Hirshowitz and Hulek13, Lemmas 3.3 and 3.4] we have that

   \begin{equation*}\phi_z^*(\varphi) = c_1\left(R^1p_{2_*}\mathcal{F}({-}\tau)\right)- c_1(R^1p_{2_*}\mathcal{F}({-}2\tau))=0,\end{equation*}

   and

   \begin{equation*} \phi_z^*(\psi) = \frac{1}{2}r\left((r+1)c_1\left(R^1p_{2_*}\mathcal{F}\right)-(r-1)c_1\left(R^1p_{2_*}\mathcal{F}({-}\tau)\right)\right)=c_2-1.\end{equation*}

   which implies

   \begin{equation*}P(m) = \chi(\mathbb{P}(z), \phi^{*}_z(a\varphi+b\psi) ) = \chi(\mathbb{P}(z), \mathcal{O}_{\mathbb{P}(E_x)}(m(c_2-1)b)))\end{equation*}

   and the proof is complete.