Irregular bundles on Hopf surfaces

TL;DR

This paper investigates the structure of hypersurfaces within the moduli spaces of rank 2 vector bundles on Hopf surfaces, focusing on irregular bundles and their geometric properties.

Contribution

It provides a detailed analysis of irregular bundles in the moduli space of rank 2 vector bundles on Hopf surfaces, highlighting their geometric and topological features.

Findings

Identification of hypersurfaces formed by irregular bundles

Characterization of the geometric structure of these hypersurfaces

Insights into the moduli space topology for rank 2 bundles

Abstract

We discuss the hypersurfaces of the moduli spaces of rank $2$ vector bundles on a classical Hopf surface formed by irregular bundles.

Equations

$$w(E)=w_1(E)+w_2(E)+w_3(E)+w_4(E)$$

$${\mathcal{M}}_n=\{E\text{stable bundle on}X,\mathrm{rk}(E)=2,\det (E)\cong {\mathcal{O}}_X,c_2(E)=n\}.$$

$${\mathbb{P}}^1\times T^{\ast }\to {\mathbb{P}}^1\times {\mathbb{P}}^1$$

$$(b,\lambda )\stackrel{\sigma }{\mapsto }(b,\{\lambda ,i(\lambda )\}).$$

$$D\subset {\mathbb{P}}^1\times {\mathbb{P}}^1=\pi (X)\times {\mathrm{Pic}}^0(T)/\pm 1.$$

$${\mathcal{M}}_n(0):=\{E\in {\mathcal{M}}_n,\forall T_{x}E{|}_{T_x}\text{ is semistable}\}.$$

$${\mathcal{M}}_n^{\mathrm{reg}}(0)=\{E\in {\mathcal{M}}_n:\forall T_x,E{|}_{T_x}\ncong L_0\oplus i^{\ast }(L_0)\text{with}{L}_0^{\otimes 2}\cong {\mathcal{O}}_{T_x}\}.$$

$${\mathcal{M}}_n^{\mathrm{irreg}}(0):={\mathcal{M}}_n(0)\setminus {\mathcal{M}}_{\delta ,n}^{\mathrm{reg}}(0).$$

$${\mathcal{M}}_n(0)=G^{-1}(G({\mathcal{M}}_n(0))),{\mathcal{M}}_n^{\mathrm{reg}}(0)=G^{-1}(G({\mathcal{M}}_n^{\mathrm{reg}}(0))),$$

$${\mathcal{M}}_n^{\mathrm{irreg}}(0)=G^{-1}(G({\mathcal{M}}_n^{\mathrm{irreg}}(0))).$$

$${\mathcal{M}}_n(\ge 1):={\mathcal{M}}_n\setminus {\mathcal{M}}_n(0).$$

$$D_i:={\pi }_2^{-1}(a_i).$$

$$Z_U(i):=U\cap D_i,i=1,2,3,4.$$

$$h^0({\mathbb{P}}^1\times {\mathbb{P}}^1,{\mathcal{I}}_{(2p,D_i)}(n,1))=h^0({\mathbb{P}}^1\times {\mathbb{P}}^1,{\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(n,1))-2$$

$$H_i:={\tilde{H}}_i\cap |{\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(n,1){|}_{sm},$$

$$m_1,\dots ,m_{s(i)}\text{or}{m}_1(E),\dots ,m_{s(i)}(E)$$

$$w_i(E):=\sum _{j=1}^{s(i)}(m_j-1),$$

$$w(E):=w_1(E)+w_2(E)+w_3(E)+w_4(E).$$

$$0\to {\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(n-1,1)\to {\mathcal{I}}_Z(n,1)\to {\mathcal{I}}_{Z,D_i}(n,1)\to 0$$

$$0\to {\mathcal{I}}_q(n-1,1)\to {\mathcal{I}}_{Z\cup \{q\}}(n,1)\to {\mathcal{I}}_{Z,D_i}(n,1)\to 0$$

$$0\to {\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(n-1,1)\to {\mathcal{I}}_{Z\cup \{q\}}(n,1)\to {\mathcal{I}}_{Z\cup \{q\},D_i}(n,1)\to 0$$

$$0\to {\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}\to {\mathcal{I}}_Z(n,1)\to {\mathcal{I}}_{Z,D}(n,1)\to 0$$

$$0\to {\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}\to {\mathcal{I}}_W(n,1)\to {\mathcal{I}}_{W,D}(n,1)\to 0$$

$$W:=(2p,D_1)\cup (2q,D_2)\cup (2a,D_3).$$

$$W(p^{\prime },q^{\prime },a^{\prime }):=(2p^{\prime },D_1)\cup (2q^{\prime },D_2)\cup (2a^{\prime },D_3).$$

$$U,D_1\setminus \{p,u\},D_2\setminus \{q,v\},R_1\setminus \{p,v\},R_2\setminus \{q,u\},$$

$${\mathcal{A}}_n:={\mathcal{B}}_n\setminus (H_1\cup H_2\cup H_3\cup H_4)=G({\mathcal{M}}_n^{\mathrm{reg}}(0))$$

Full text

Keywords: Moduli spaces, vector bundles, Hopf surfaces.

1. 1 Statements of results
2. 2 Bundles without jumps
3. 3 Regular v irregular
4. 4 Irregular profiles
5. 5 Ramifications and weights
6. 6 The topology of $\mathcal{M}_{n}^{\operatorname{reg}}(0)$
7. 7 Nontrivial determinants

1. Statements of results

Let $\pi\colon X\to\mathbb{P}^{1}$ be a classical Hopf surface with $T$ as its elliptic fiber. For all $n>0$, the moduli space $\mathcal{M}_{n}$ of $SL(2,\mathbb{C})$ holomorphic bundles with $c_{2}=n$ on $X$ contains an open dense subset $\mathcal{M}_{n}(0)$ formed by bundles without jumps. The goal of this note is to describe some of the geometric features of $\mathcal{M}_{n}(0)$.

A bundle $E\in\mathcal{M}_{n}$ is called irregular when its restriction to any fiber $T$ of $X$ has automorphism group of dimension 4 (Def. 1), this happens whenever the restriction of $E|_{T}=L\oplus L$ with $L^{2}=\mathcal{O}$, that is when $L$ is one of the 4 thetas. We then compare regular and irregular bundles without jumps. We first show:

Theorem** (1).**

Fix $E\in\mathcal{M}_{n}(0)$, $n\geq 2$.

• (a)

If $E\in\mathcal{M}_{n}^{\operatorname{reg}}(0)$, then $C_{E}$ is smooth and connected.

• (b)

If $E\in\mathcal{M}_{n}^{\operatorname{irreg}}(0)$, then $C_{E}$ is singular and irreducible.

Accordingly, we define a concept of weight of a bundle (Def. 1)

[TABLE]

intended as a measure of irregularity, and prove:

Theorem** (2).**

$E\in\mathcal{M}_{n}(0)$* is irregular if and only if $w(E)>0$.*

Having identified $\mathcal{M}_{n}^{\operatorname{irreg}}(0)$ as the set of all bundles $E\in\mathcal{M}_{n}^{sg}$ without jumps and having singular spectral curve $C_{E}$ (Not. 4), we prove:

Theorem** (3).**

Assume $n>1$. For all $E\in\mathcal{M}_{n}^{\operatorname{irreg}}(0)$ we have:

1. (1)

$w(E)\leq 2n-2$. 2. (2)

$E$* has $(n,0,\dots,0)$ has its $i$-th profile if and only if $w_{i}(E)\leq n-1$ for all $i$ and $w_{i}(E)=n-1$ .* 3. (3)

Fix $i\in\{1,2,3,4\}$, a positive integer $\ell$ and a restricted profile $(m_{1},\dots,m_{\ell})$. There exists a non-empty locally closed analytic subset $\Gamma\in\mathcal{M}_{n}^{\operatorname{irreg}}(0)$ such that all $G(E)\in|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|$ have $(m_{1},\dots,m_{\ell})$ as $i$-th profile and this set of graphs $G(E)$ has codimension at most $-\ell+m_{1}+\cdots+m_{\ell}$ in $|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|$.

Theorem** (4).**

Fix an ordering $i_{1},i_{2},i_{3},i_{4}$ of the set $\{1,2,3,4\}$.

• (a)

For all $n\geq 2$ there exists a codimension $2$ locally closed analytic subset $\Gamma\subset\mathcal{M}_{n}(0)$ such that $w_{i_{1}}(E)=w_{i_{2}}(E)=1$ and $w_{i_{3}}(E)=w_{i_{4}}(E)=0$ for all $E\in\Gamma$.

• (b)

For all $n\geq 4$ there exists a codimension $3$ locally closed analytic subset $\Delta\subset\mathcal{M}_{n}(0)$ such that $w_{i_{1}}(E)=w_{i_{2}}(E)=w_{i_{3}}(E)=1$ and $w_{i_{4}}(E)=0$ for all $E\in\Delta$.

Theorem** (5).**

Fix an integer $n\geq 2$. Fix an ordering $i_{1},i_{2},i_{3},i_{4}$ of $\{1,2,3,4\}$. Then there exists an irreducible locally closed analytic subset of dimension $2n+2$ of $\mathcal{M}_{n}(0)$ such that for all $E\in\Gamma$ the restricted $i_{1}$-th and $i_{2}$-th profiles of $E$ are $(n)$, while $w_{i_{3}}(E)=w_{i_{4}}(E)=0$.

We then describe some features of the topology of $\mathcal{M}_{n}^{\operatorname{reg}}(0)$, obtaining:

Theorem** (6).**

Fix an integer $n\geq 2$. We have:

• (i)

$H^{*}(\mathcal{M}_{n}^{\operatorname{reg}}(0),\mathbb{Z})=H^{*}(\mathcal{A}_{n},\mathbb{Z})\otimes H^{*}((S^{1})^{4n-2},\mathbb{Z}),$* and*

• (ii)

$H^{i}(\mathcal{M}_{n}^{\operatorname{reg}}(0),\mathbb{Z})=0$*  for all $i\geq 6n$.*

Theorem** (7).**

Let $Y\subset\mathcal{M}_{n}(0)$ be an irreducible compact complex analytic subspace. Then $G(Y)$ is a single point.

Corollary** (8).**

Let $Y\subset\mathcal{M}_{n}^{\operatorname{reg}}(0)$ be an irreducible compact complex analytic subspace. Then $Y$ is contained in one of the $(2n-1)$-dimensional tori which are the fibers of the graph map.

One might also consider the moduli spaces $\mathcal{M}_{n,\delta}$ of bundles on $X$ having determinant $\delta$ in place of trivial determinant. We conclude the paper by showing (Prop. 2) that $\mathcal{M}_{n,\delta}$ and $\mathcal{M}_{n}$ are biholomorphic, therefore all results proved here have analogous statements for the cases of nontrivial determinant.

Our interest in bundles on Hopf surfaces originated from the survey of E. Witten about instantons on $S^{3}\times S^{1}$ [W].

Remark 1*.*

The real 4-manifold $S^{3}\times S^{1}$ admits the structure of an elliptically fibered complex surface diffeomorphic to a classical Hopf surface, which we denote by $X$ in this work. Using [Bu, Thm. 1], the moduli space of $SU(2)$ instantons on $S^{3}\times S^{1}$ of charge $n$ can be identified with the moduli space of stable $SL(2,\mathbb{C})$ bundles on $X$ with $c_{2}=n$. Here we denote by $\mathcal{M}_{n}$ the moduli space of stable rank 2 vector bundles on a classical Hopf surface $X$ with $c_{2}=n$ and trivial determinant. In the literature about the gauge theory of instantons on $S^{4}$ and on ruled surfaces, strong homological stability of moduli spaces were proven using the description of jumping lines [BHMM, HM] and of local contributions to instanton charges concentrated at exceptional curves [Ga1]. It is now know whether the Atiyah–Jones conjecture [AJ] holds true over $S^{3}\times S^{1}$. For the case of instantons on $S^{3}\times S^{1}$, the corresponding complex geometry presents a completely new phenomenon. Namely, one finds out that the second Chern classes of vector bundles on $X$ can not be calculates via the standard jumping fibers techniques used for rational surfaces. The reason is that for all $n>0$, the moduli space $\mathcal{M}_{n}$ of $SL(2,\mathbb{C})$ holomorphic bundles on $X$ contains a dense open (with analytic complement) subset $\mathcal{M}_{n}(0)$ formed by bundles without any jumps.

Here we concentrate in discussing geometric aspects of the set $\mathcal{M}_{n}(0)$ of bundles without jumps, by dividing it into the subsets of regular and irregular bundles. The second author discusses connectedness of $\mathcal{M}_{n}$ in [Ga2], but we do not use that result here.

2. Bundles without jumps

Let $\pi\colon X\to\mathbb{P}^{1}$ be a classical Hopf surface with $T$ as its elliptic fiber. We have $\operatorname{Pic}^{0}(T)$ (often written as $T^{\ast}$) which is non-canonically isomorphic to $T$. Observe that, since $T$ is a torus, there are $4$ such elements $L\in\mathrm{Pic}^{0}(T)$ satisfying $L^{\otimes 2}\cong\mathcal{O}_{T}$, these are called half-periods. Each of them is informally referred to as a “theta”(as in theta characteristic), and the expression “there are $4$ thetas”  is commonly used. Hence, the 4 thetas of $T$ are the 4 elements $L$ of $\mathrm{Pic}^{0}(T)$ such that $L^{\otimes 2}\cong\mathcal{O}_{T}$. Informally, thinking of the torus as obtained from $\mathbb{C}^{2}/\Lambda$ where $\Lambda$ is the integer lattice, these can be though of as corresponding to the points $(0,0),(\frac{1}{2},0),(0,\frac{1}{2}),(\frac{1}{2},\frac{1}{2})$.

Let $F$ be a rank $2$ vector bundle on $T$ with $\det(F)\cong\mathcal{O}_{T}$. If $F$ is not semistable, then $F\cong R\oplus R^{\vee}$ for some line bundle $R$ on $T$ with $\deg(R)\neq 0$. Now assume that $F$ is semistable and decomposable, i.e. assume $F\cong L\oplus L^{\vee}$ for some degree [math] line bundle on $T$, then there are 2 possibilities:

• •

If $L\neq L^{\ast}$, then $\dim\operatorname{End}(F)=2$.

• •

If $L\cong L^{\ast}$, then $\dim\operatorname{End}(F)=4$.

Notation 1*.*

A bundle $E\in\mathcal{M}_{n}$ is called regular it its restriction to every fiber of $\pi$ has automorphism group of minimal dimension, i.e. $\dim\mbox{Aut}(E|_{T})=2\,\forall T$. Note that, for a fiber $T$, we have $L=L^{\ast}$ if and only if $L^{\oplus 2}=\mathcal{O}_{T}$, i.e. if and only if $L$ is a theta; in which case the bundle is called irregular over $T$. A bundle $E$ over $X$ is called irregular if its restriction to any fiber of $\pi$ is irregular.

Let $\mathcal{M}_{n}$ denote the moduli space of all rank $2$ stable vector bundles on $X$ with trivial determinant and $c_{2}(E)=n$, i.e.

[TABLE]

The variety $\mathcal{M}_{n}$ is smooth non-empty of dimension $4n$ [BH, Prop. 3.4.4], see also [BMo, Prop. 4.2] for the case of nontrivial determinant. Since here we consider only the case of trivial determinant, the spectral involution on $T^{*}$ may be defined as $i(\lambda)=-\lambda$, with corresponding double cover

[TABLE]

[TABLE]

Given a rank 2 bundle $E$ on $X$, let $V$ be the universal Poincaré line bundle on $X\times\mathbb{C}^{*}$ and consider the first derived image $R^{1}\pi^{\prime}_{*}(E\otimes V)$ where $\pi^{\prime}$ the projection $\pi^{\prime}=\pi\times id\colon X\times\mathbb{C}^{*}\to\mathbb{P}^{1}\times\mathbb{C}^{*}$. The sheaf $R^{1}\pi^{\prime}_{*}(E\otimes V)$ is supported on a divisor on $\mathbb{P}^{1}\times\mathbb{C}^{*}$, which descends to a divisor

[TABLE]

When $c_{2}(E)=n$, the divisor $G(E)\mathrel{\mathop{:}}=D$ belongs to the linear system $|\mathcal{O}(n,1)|$ over $\mathbb{P}^{1}\times\mathbb{P}^{1}$ and is called the graph of $E$, see[BH, Prop. 3.2.3].

Let $\mathcal{M}_{n}(0)$ denote the set of all $E\in\mathcal{M}_{n}$ with no jumps, i.e.

[TABLE]

Let $\mathcal{M}_{n}^{\operatorname{reg}}(0)$ denote the subset of all regular bundles in $\mathcal{M}_{n}(0)$, that is, those whose restrictions to fibers are semistable but not sums of half-periods, i.e.

[TABLE]

Remark 2*.*

The image $G(\mathcal{M}_{n}(0))$ of $\mathcal{M}_{n}(0)$ is the Zariski open set of all graphs $D\in|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|$ which are smooth (or, equivalently in this case, irreducible), i.e. the set of all $D\in|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|$ giving a degree $n$ morphism $\mathbb{P}^{1}\to\mathbb{P}^{1}$.

Since $\mathcal{M}_{\delta,1}$ is well-understood, from now on we consider $n>1$.

Assume $n>1$ and set

[TABLE]

Since the notions of irregularity and of having no jumps are completely described in terms of graphs, we have the equalities

[TABLE]

and

[TABLE]

Remark 3*.*

The set $G(\mathcal{M}_{n}\setminus\mathcal{M}_{n}(0))$ is an irreducible quasi-projective variety of dimension $2n$, i.e. of codimension $1$ in $\mathbb{P}^{2n+1}$. It consists of singular graphs.

Remark 4*.*

The set $G(\mathcal{M}_{n}^{\operatorname{irreg}}(0))$ is a quasi-projective variety and each irreducible component of it has dimension $2n$, i.e. it has codimension $1$ in $\mathbb{P}^{2n+1}$. Since $G$ is surjective and since there are $4$ thetas, we get $4$ different “types”of irregular bundles, one for each of the $4$ thetas.

For each $A\in G(\mathcal{M}_{n}^{\operatorname{reg}}(0))$ the fiber $G^{-1}(A)$ is well-understood, it is isomorphic to the Jacobian of the spectral curve of $A$ and in particular it is a smooth and connected projective manifold of dimension $2n-1$.

Recall that for $n>1$ the graph map $G\colon\mathcal{M}_{n}\to\mathbb{P}^{2n+1}$ is surjective, that $\mathbb{U}:=G(\mathcal{M}_{n}^{\operatorname{reg}}(0))$ is a Zariski open subset of $\mathbb{P}^{2n+1}$ and that for each $a\in\mathbb{U}$ the set $G^{-1}(a)$ is isomorphic to an Abelian variety of dimension $2n-1$.

3. Regular v irregular

Let $\mathcal{M}_{n}(0)$ be the set of all $E\in\mathcal{M}_{n}$ with no jumps, then all elements $E\in\mathcal{M}_{n}(0)$ such that $G(E)$ is a smooth element of $|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|$. Set

[TABLE]

Since being singular is a closed analytic condition, $\mathcal{M}_{n}(\geq 1)$ is a closed analytic subset of $\mathcal{M}_{n}$, while $\mathcal{M}_{n}(0)$ is an open subset of $\mathcal{M}_{n}$. It is easy to check that $\mathcal{M}_{n}(\geq 1)$ is the set of all $E\in\mathcal{M}_{n}$ with at least one jump.

Recall from notation 4 that $\mathcal{M}_{n}^{sg}$ is the set of all $E\in\mathcal{M}_{n}(0)$ such that the spectral curve $C_{E}$ is singular, while $\mathcal{M}_{n}^{sm}$ is the set of all $E\in\mathcal{M}_{n}(0)$ such that the spectral curve $C_{E}$ of $E$ is smooth.

Note that $\mathcal{M}_{n}(0)$ is set of all $E\in\mathcal{M}_{n}$ such that $G(E)$ has no vertical component, i.e. it is an irreducible element of $|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|$. We have used the notation $|\mathcal{O}(n,1)|_{sm}$ for the set of such irreducible elements. Observe that in both cases $G(\mathcal{M}_{n}^{\operatorname{reg}}(0))\subset|\mathcal{O}(n,1)|_{sm}$ and $G(\mathcal{M}_{n}^{\operatorname{irreg}}(0))\subset|\mathcal{O}(n,1)|_{sm}$.

For any $E\in\mathcal{M}_{n}(0)$ let $C_{E}$ denote its spectral curve. Let $i$ denote the involution $L\mapsto L^{*}$ on $T^{*}$. We have $T^{*}/i\cong\mathbb{P}^{1}$ and the quotient map $T\to\mathbb{P}^{1}$ is ramified over $4$ points, $a_{1},a_{2},a_{3},a_{4}$, with are the images of of the $4$ thetas of $T^{*}$. We get $4$ element $\mathbb{P}^{1}\times\{a_{i}\}\in|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(0,1)|$, $i=1,2,3,4$.

Observe that $\mathcal{M}_{n}^{\operatorname{reg}}(0)$ is the set of all $E\in\mathcal{M}_{n}(0)$ such that $G(E)$ is transversal to each $\mathbb{P}^{1}\times\{a_{i}\}$, for $i=1,2,3,4$, i.e. if $E\in\mathcal{M}_{n}^{\operatorname{reg}}(0)$ then $G(E)$ intersects $\Sigma=\mathbb{P}^{1}\times\{a_{1},a_{2},a_{3},a_{4}\}$ at points. Observe also that $\mathcal{M}_{n}^{\operatorname{irreg}}(0)=\mathcal{M}_{n}(0)\setminus\mathcal{M}_{n}^{\operatorname{reg}}(0)$, and that all that elements of $\mathcal{M}_{n}^{\operatorname{irreg}}(0)$ have irreducible graphs.

Theorem 1**.**

Fix $E\in\mathcal{M}_{n}(0)$, $n\geq 2$.

• (a)

If $E\in\mathcal{M}_{n}^{\operatorname{reg}}(0)$, then $C_{E}$ is smooth and connected.

• (b)

If $E\in\mathcal{M}_{n}^{\operatorname{irreg}}(0)$, then $C_{E}$ is singular and irreducible.

Proof.

By assumption $G(E)\cong\mathbb{P}^{1}$. By definition $u:C_{E}\to G(E)$ is a double covering ramified exactly at the $4n$ points of $H_{i}\cap G(E)$ with $i=1,2,3,4$. Hence $C_{E}\setminus u^{-1}(G(E))$ is smooth and it is a 2 to 1 unramified covering. This covering comes from a covering $u^{\prime}\colon\mathbb{P}^{1}\times T^{*}\to\mathbb{P}^{1}\times T^{*}/i$ which is ramified exactly over $\Sigma$. We get a universal double covering over all smooth curves. We get that each $C_{E}$ is a certain effective divisor of $\mathbb{P}^{1}\times T^{*}$ with arithmetic genus $2n-1$. Hence $C_{E}$ is smooth if and only if it is over its points which are mapped bijectively onto the points of $G(E)\cap\Sigma$ and this is the case if and only if $E\in\mathcal{M}_{n}^{\operatorname{reg}}(0)$. Since $G(E)\cap\Sigma$ we get that $C_{E}$ is connected. Hence we get (a).

For part (b), assume that $C_{E}$ is not irreducible. Since $G(E)\cong\mathbb{P}^{1}$ and $C_{E}\to G(E)$ is a degree $2$ morphism, $C_{E}$ has $2$ irreducible components, both of them smooth and isomorphic to $\mathbb{P}^{1}$, say $C_{E}=J_{1}\cup J_{2}$. Remember that $C_{E}\subset\mathbb{P}^{1}\times T^{*}$. Hence, $\pi_{2}(J_{i})$ is a single point. Hence $G(E)$, which is the quotient of $C_{E}$ by the involution, is not an element of $|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|$, a contradiction. Since $C_{E}$ is irreducible of arithmetic genus $2n-1$ and the 2-to 1 map $C_{E}\to G(E)$ has at most $4n-1$ ramification points, $C_{E}$ is singular. ∎

Lemma 1**.**

If $G(E)\in|\mathcal{O}(n,1)|_{sm}$, then $E$ is regular.

Proof.

Take $x\in\mathbb{P}^{1}$ such that $E_{|\pi^{-1}(x)}$ is an element of $T^{*}/i$ associated to a theta, i.e. it corresponds to a point $a\in\Sigma\cap G(E)$. We need to prove that $E_{|\pi^{-1}(x)}$ is indecomposable, i.e. that $h^{1}(\pi^{-1}(x),E_{|\pi^{-1}(x)})=1$. The transversality of $G(E)$ and $\Sigma$ implies that $C_{E}$ is smooth at its points, $a^{\prime}$, over $a$ and that the torsion sheaf $R^{1}$ on $\mathbb{P}^{1}\times T$ is locally free of rank $1$ at the point $a^{\prime}$ as an $\mathcal{O}_{C_{E}}$-sheaf [T, Remark 2.8]. ∎

The following lemma is well-known, we just state it for completeness.

Lemma 2**.**

Consider a graph $A$ such that $A=G(E)$ for some $E\in\mathcal{M}_{n}^{\operatorname{reg}}(0)$. Then $G^{-1}(A)$ is isomorphic to the Jacobian $J(C_{E})$ of the spectral curve $C_{E}$ and hence it is connected, smooth, compact and isomorphic to an Abelian variety of dimension $2n-1$.

Proof.

By Lemma 1 $C_{E}$ is a smooth and connected curve of genus $2n-1$. Hence its Jacobian $J(C_{E})$ is connected, smooth, compact and isomorphic to an Abelian variety of dimension $2n-1$. Lemma 1 says that any bundle with $A$ as its graph is regular. In our case the base, $B$, of the Hopf fibration is $\mathbb{P}^{1}$. Since $J(\mathbb{P}^{1})$ is a singleton, it is sufficient to quote [FMW, Th. 5.14]. ∎

4. Irregular profiles

We fix $\delta\in\mathrm{Pic}(X)=\mathrm{Pic}^{0}(X)\cong\mathbb{C}^{\ast}$. Let $I\subset T^{\ast}$ be the set of all thetas with respect to the involution $i_{\delta}$. We we fix an order for the $4$ elements of $I$, we write them as $L_{1},L_{2},L_{3},L_{4}$, and write $a_{1},a_{2},a_{3},a_{4}$ for the $4$ elements of $\mathbb{P}^{1}$ associated to them, see (1). Let $\pi_{2}\colon\mathbb{P}^{1}\times\mathbb{P}^{1}\to\mathbb{P}^{1}$ denote the projection onto the second factor. Set

[TABLE]

We get $4$ elements of $|\mathcal{O}_{\mathbb{P}^{1}}(1,0)|$.

Let $|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|_{sm}$ denote the set of smooth elements in $|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|$. Note that $E\in\mathcal{M}_{n}(0)$, i.e. $E$ has no jumps, if and only if $G(E)\in|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|_{sm}$. For each $U\in|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|_{sm}$ we get $4$ degree $n$ zero-dimensional schemes

[TABLE]

For any $E\in\mathcal{M}_{n}(0)$, taking $G(E)$ in place of $U$, we get $4$ zero-dimensional schemes $Z_{G(E)}(i)$, $i=1,2,3,4$.

Remark 5*.*

Note that $E$ is irregular if and only if at least one of the 4 schemes $Z_{G(E)}(i)$, $i=1,2,3,4$, is not reduced, i.e., it is not formed by $n$ distinct points.

Notation 2*.*

For each $i=1,2,3,4$, denote by $H_{i}$ the set of all divisors $D\in|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|_{sm}$ such that $D$ is tangent to $D_{i}$ (the point of tangency with $D_{i}$ is not fixed), and let $\mathcal{H}_{i}\mathrel{\mathop{:}}=G^{-1}(H_{i})$.

Fix $p\in D_{i}$ and let $(2p,D_{i})$ denote the degree $2$ effective divisor of $D_{i}$ with $p$ as its reduction. Note that $(2p,D_{i})$ is a degree $2$ connected zero-dimensional scheme. Since $n\geq 1$, $\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(1,1)$ is very ample and $\deg(2p,D_{i})=2$, we have

[TABLE]

and a general $D\in|\mathcal{I}_{(2p,D_{i})}(n,1)|$ is smooth. Varying the point $p\in D_{i}$ we get a non-empty hypersurface $\tilde{H}_{i}$ of $|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|$ (see Theorem 3 and its proof for more details, it implies for instance that $\mathcal{H}_{i}\neq\mathcal{H}_{j}$ if $i\neq j$). Observe that

[TABLE]

that is, $H_{i}$ is the set of all smooth elements of $\tilde{H}_{i}$ (these are the same $H_{i}$ from Notation 2). Note that for all $n\geq 1$ the set $H_{i}$ is a Zariski open subset of $\tilde{H}_{i}$.

Observe also that for $n>1$ and for $i=1,2,3,4$, we have that $\mathcal{H}_{i}$, the set of all $E\in\mathcal{M}_{n}^{\operatorname{irreg}}(0)$ such that $Z_{G(E)}(i)$ is not reduced. Fix $D\in|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|$. For each $i\in\{1,2,3,4\}$ the scheme $D\cap D_{i}$ is a zero-dimensional scheme of degree $n$.

Definition 1.

Let $m_{1}\geq\cdots\geq m_{x}>0$ be the degrees of the connected components of $D\cap D_{i}$ (in decreasing order).

• •

We say that $(m_{1},\dots,m_{x})$ is the $i$-th profile of $D$ and of all bundles $E$ with $G(E)=D$.

• •

We say that the integer $\sum(m_{i}-1)=m_{1}+\cdots+m_{x}-x$ is the $i$-th weight of $D$ and of all bundles $E$ with $G(E)=D$.

• •

The length $\ell$ of the profile $D\cap D_{i}$ and of all bundles $E$ with $G(E)=D$ is the maximal integer $y$ such that $m_{y}\geq 2$, with the convention $\ell=0$ if $m_{1}=1$, i.e. if $D$ is transversal to $D_{i}$.

• •

The reduced $i$-th profile of $D$ is the set of integers $m_{1},\dots,m_{\ell}$.

If $\ell>0$, these numbers form a non-decreasing sequence of integers $\geq 2$ with $\sum_{i}m_{i}\leq n$. Note that the $i$-th weight is uniquely determined by the reduced profile of $m_{1},\dots,m_{s}$: it is [math] if $\ell=0$, while it is $-\ell+m_{1}+\cdots+m_{\ell}$ if $\ell>0$. For instance, for $n=2$ the possible profiles are $2$ and $1,1$; while for $n=3$ the profiles are $3$, $2,1$ and $1,1,1$.

Notation 3*.*

For any such $E$ with no jump we denote by

[TABLE]

the multiplicities of the connected components of $Z_{G(E)}(i)$ in non-decreasing order. By definition, the $i$th-weight $w_{i}(E)$ of $E\in\mathcal{M}_{n}(0)$ is the integer

[TABLE]

and the weight $w(E)$ of $E\in\mathcal{M}_{n}(0)$ is the integer

[TABLE]

Theorem 2**.**

$E\in\mathcal{M}_{n}(0)$* is irregular if and only if $w(E)>0$.*

Proof.

This is true because $\mathcal{M}_{n}^{\operatorname{irreg}}(0)=\bigcup_{i=1}^{4}\mathcal{H}_{i}.$ ∎

For simplicity we write $h^{0}(\mathcal{I}_{Z}(x,y))$ instead of $h^{0}(\mathbb{P}^{1}\times\mathbb{P}^{1},\mathcal{I}_{Z}(x,y))$ and the same for higher cohomology groups. We write $\mathcal{M}_{n}$ for the moduli of bundles with trivial determinant and $c_{2}=n$.

Remark 6*.*

We explain here our motivation for the introduction of the profiles of an element of $|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|_{sm}$ and hence of a bundle $E\in\mathcal{M}_{n}(0)$. Since we always assume that $n\geq 2$, the graph map $G$ is surjective [BH, Th. 5.2.2]. Hence the description of a spectral curve $C$ covering a graph $D\in|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|_{sm}$ is an important invariant for each bundle $E$ with $G(E)=D$. Fix $D\in|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|_{sm}$ and set $S_{i}:=D\cap\mathbb{P}^{1}\times\{a_{i}\}$, $i=1,2,3,4$. A spectral curve $C$ of a smooth graph is irreducible (Lemma 3). It is a double covering $u\colon C\to D$ ramified exactly over the points of $S:=S_{1}\cup S_{2}\cup S_{3}\cup S_{4}$. The curve $C$ is contained in the smooth surfaces $\mathbb{P}^{1}\times T^{*}$ and it is smooth outside $S$. Since $p_{a}(C)=2n-1$, the integer $\#S$ is a measure of the singularities of $C$. Since $u\colon C\to D$ is a double covering with $D\cong\mathbb{P}^{1}$, $C$ is a “hyperelliptic-like” curve whose singular points have multiplicity $2$, but the singularity type of $C$ at $\tilde{p}$ depends on the spectral profile. Fix $p\in S$, say $p\in S_{i}$, and let $\tilde{p}$ be the only point of $C$ with $u(\tilde{p})=p$. Call $m_{p}$ the multiplicity of $p$ in $D\cap D\times\{a_{i}\}$. If $m_{p}=1$, then $C$ is smooth at $\tilde{p}$. If $m_{p}=2$, then $C$ has an ordinary double point at $\tilde{p}$.

Lemma 3**.**

Let $C\subset\mathbb{P}^{1}\times T^{*}$ be the spectral curve of a smooth graph $D\in|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|_{sm}$. Then $D$ is irreducible.

Proof.

By the definition of spectral curve there is a morphism $u\colon C\to D$ generically of degree $2$. Since $D\in|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|(0)$, it has no vertical component and hence $C$ has no vertical component, i.e. the restriction of $u$ to any irreducible component is dominant. Assume that $C$ is reducible. Since $\deg(u)=2$, $C$ has exactly $2$ irreducible components, say $C=C_{1}\cup C_{2}$ with $u_{|C_{i}}C_{i}\to D$ a degree $1$ morphism between irreducible curves with a smooth target $D$. Hence $u_{|C_{i}}\colon C_{i}\to D$ is an isomorphism. Thus $C_{i}\cong\mathbb{P}^{1}$. We have $C_{i}\subset\mathbb{P}^{1}\times T^{*}$. Since $C_{i}\cong\mathbb{P}^{1}$ and $T^{*}$ is a smooth elliptic curve, the restriction to $C_{i}$ of the projection $\mathbb{P}^{1}\times T^{*}\to T^{*}$ is constant. Call $o_{i}$ its image. Thus $C_{i}=\mathbb{P}^{1}\times\{o_{i}\}$. Since $u\colon C\to D$ is generically unramified, $o_{1}\neq o_{2}$. Hence $C_{1}$ and $C_{2}$ are connected components, contradicting the fact that $D\cap\mathbb{P}^{1}\times\{o_{1}\}\neq\emptyset$ and hence $u$ has at least one ramification point. ∎

Notation 4*.*

Let $\mathcal{M}_{n}^{sm}$ denote the set of all $E\in\mathcal{M}_{n}(0)$ such that the spectral curve $C_{E}$ of $E$ is smooth, and let $\mathcal{M}_{n}^{sg}$ be the set of all $E\in\mathcal{M}_{n}(0)$ such that $C_{E}$ is singular. With our notation the elements of $\mathcal{M}_{n}^{sg}$ have no jumps.

It was proven in section 3 that $\mathcal{M}_{n}^{sm}=\mathcal{M}_{n}^{\operatorname{reg}}(0)$ and $\mathcal{M}_{n}^{sg}=\mathcal{M}_{n}^{\operatorname{irreg}}(0)$, and we now explore these equalities.

Theorem 3**.**

Assume $n>1$. For all $E\in\mathcal{M}_{n}^{\operatorname{irreg}}(0)$ we have:

1. (1)

$w(E)\leq 2n-2$. 2. (2)

$E$* has $(n,0,\dots,0)$ has its $i$-th profile if and only if $w_{i}(E)\leq n-1$ for all $i$ and $w_{i}(E)=n-1$ .* 3. (3)

Fix $i\in\{1,2,3,4\}$, a positive integer $\ell$ and a restricted profile $(m_{1},\dots,m_{\ell})$. There exists a non-empty locally closed analytic subset $\Gamma\in\mathcal{M}_{n}^{\operatorname{irreg}}(0)$ such that all $G(E)\in|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|$ have $(m_{1},\dots,m_{\ell})$ as $i$-th profile and this set of graphs $G(E)$ has codimension at most $-\ell+m_{1}+\cdots+m_{\ell}$ in $|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|$.

Proof.

Here we use the Zariski topology of the projective space $|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|$, so being general in it means “outside finitely many proper algebraic subset” (their union has lower dimension).

Fix a smooth $D\in|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|$. Hence $D\cong\mathbb{P}^{1}$. The projection onto the second factor $\pi_{2}\colon\mathbb{P}^{1}\times\mathbb{P}^{1}\to\mathbb{P}^{1}$ induces a degree $n$ morphism $f\colon D\to\mathbb{P}^{1}$. Since $D\cong\mathbb{P}^{1}$, the Riemann–Hurwitz formula gives that, counting multiplicities, the ramification divisor $\mathcal{R}\subset D$ is an effective divisor of degree $2n-2$ ([GH, pp. 216–219], [H, Cor. IV.2.4]). Fix $p\in D$. Since $D$ is irreducible and of bidegree $(n,1)$, the scheme $Z_{p}:=D\cap\pi_{2}^{-1}(f(p))$ has degree $n$. Note that $p\in\mathcal{R}_{\operatorname{red}}$ if and if $Z_{p}$ is not formed by $n$ distinct points and the multiplicity of $p$ in $\mathcal{R}$ is how we computed the multiplicity of the $i$-th profile of $E$. Since $w(E)$ only counts the contribution of $4$ of the fibers of $f$, we get (1) and (2).

Now we use that $G$ is surjective [BH, Th. 5.2.2]. We see that to prove item (3) it is sufficient to prove the “corresponding” statement for $|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|_{sm}$. Fix $\ell$ distinct points $p_{1},\dots,p_{\ell}$. Let $Z\subset D_{i}$ be the connected zero-dimensional subscheme of $D_{i}$ with $Z_{\operatorname{red}}=\{p_{1},\dots,p_{s}\}$ and the connected component of $Z$ with $p_{h}$ as its reduction has degree $m_{h}$. Set $m:=m_{1}+\cdots+m_{\ell}$.

(a) In this step we prove that $h^{1}(\mathbb{P}^{1}\times\mathbb{P}^{1},\mathcal{I}_{Z}(n,1))=0$; hence $h^{0}(\mathcal{I}_{Z}(n,1))=h^{0}(\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1))-m$. Since $Z\subset D_{i}$, we have an exact sequence

[TABLE]

The Künneth formula give $h^{1}(\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n-1,1))=0$. Since $\deg(Z)\leq n$, $D_{i}\cong\mathbb{P}^{1}$ and $\deg(\mathcal{O}_{D_{i}}(n,1))=n$, we have $h^{1}(D_{i},\mathcal{I}_{Z,D_{i}}(n,1))=0$.

Take a general $D\in|\mathcal{I}_{Z}(n,1)|$. In this step we prove that $D$ is smooth, transversal to each $D_{j}$, $j\neq i$, and that $(m_{1},\dots,m_{\ell})$ is the restricted $i$-th profile of $D$. Let $\mathcal{B}$ denote the set-theoretic base locus of $|\mathcal{I}_{Z}(n,1)|$. By the theorem of Bertini to prove that $D$ is smooth outside $\{p_{1},\dots,p_{\ell}\}$ it is sufficient to prove that $\{p_{1},\dots,p_{\ell}\}=\mathcal{B}$ ([GH, p. 137], [H, III.10.9], [Jo, 6.3]), i.e. that $h^{0}(\mathcal{I}_{Z\cup\{q\}}(n,1))=h^{0}(\mathcal{I}_{Z}(n,1))-1$ for all $q\in\mathbb{P}^{1}\times\mathbb{P}^{1}\setminus\{p_{1},\dots,p_{\ell}\}$.

(b1) Take $q\in\mathbb{P}^{1}\times\mathbb{P}^{1}\setminus D_{i}$. Since $q\notin D_{i}$ and $Z\subset D_{i}$, we have the exact sequence

[TABLE]

We saw that $h^{1}(D_{i},\mathcal{I}_{Z,D_{i}}(n,1))=0$. Since $n\geq 0$, $\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n-1,1)$ is globally generated and hence $h^{1}(\mathcal{I}_{q}(n-1,1))=0$. Thus, the long cohomology exact sequence of (3) gives $h^{1}(\mathcal{I}_{Z\cup\{q\}}(n,1))=0$, proving that $q\notin\mathcal{B}$.

(b2) Take $q\in D_{i}\setminus\{p_{1},\dots,p_{\ell}$. Since $Z\cup\{q\}\subset D_{i}$, we have an exact sequence

[TABLE]

Since $\deg(Z\cup\{q\})=\deg(Z)+1\leq n+1$, $D_{i}\cong\mathbb{P}^{1}$ and $\deg(\mathcal{O}_{D_{i}}(n,1))=n$, we have $h^{1}(D_{i},\mathcal{I}_{Z\cup\{q\},D_{i}}(n,1))=0$. Hence $q\notin\mathcal{B}$.

(b3) By steps(b1) and(b2), $D$ is smooth outside the finite set $\{p_{1},\dots,p_{\ell}\}$. Since we are using the Zariski topology, any finite intersection of non-empty Zariski open subsets of the projective space $|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|$ is non-empty and hence Zariski dense in $|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|$. Since being smooth is an open condition, it is sufficient to find $A\in|\mathcal{I}_{Z}(n,1)|$ smooth at all points of $\{p_{1},\dots,p_{\ell}\}$. Take a general $B\in|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n-1,1)|$ and set $A:=D_{i}\cup B$. Since $Z\subset D_{i}$, $Z\subset A$. Since $B$ is general and $\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n-1,1)$ is globally generated, $B\cap\{p_{1},\dots,p_{\ell}\}=\emptyset$. Hence $A$ is smooth at all points of $\{p_{1},\dots,p_{\ell}\}$.

(b4) In this step we prove that $D$ is transversal to each $D_{j}$, $j\neq i$. Since we are using the Zariski topology in which finite intersections of non-empty subsets of $|\mathcal{I}_{Z}(n,1)|$ are non-empty, it is sufficient to find $A\in|\mathcal{I}_{Z}(n,1)|$ which is transversal to each $D_{j}$, $j\neq i$. Take as $A$ the union $D_{i}$ and $n$ distinct elements $R_{1},\dots,R_{n}$ of $|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(1,0)|$. We have $D_{i}\cap D_{j}=\emptyset$ and each $R_{h}$ is transversal to all elements of $|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(1,0)|$.

(b5) In this step we prove that $m_{1},\dots,m_{\ell}$ is the restricted profile of $D$. We saw in step(b2) that $\mathcal{B}\cap D_{i}=\{p_{1},\dots,p_{\ell}\}$. Hence $|\mathcal{I}_{Z}(n,1)|$ induces a base point free linear system $\mathcal{A}$ on $D_{i}$. By the theorem of Bertini its general element is formed by distinct points outside $\{p_{1},\dots,p_{\ell}\}$ ([GH, p. 137], [H, III.10.9], [Jo, 6.3]). Hence $D$ is transversal outside $\{p_{1},\dots,p_{\ell}\}$, i.e. the restricted $i$-th profile of $D$ has exactly $\ell$ entries. Since $\{p_{1},\dots,p_{\ell}\}$ is a finite set and we are using the Zariski topology, it is sufficient to prove that for each $h\in\{1,\dots,\ell\}$ a general $D$ has the property that $D\cap D_{i}$ has multiplicity $m_{h}$ at $p_{h}$. Hence it is sufficient to prove that it has multiplicity $\leq m_{h}$ at $p_{h}$, given that by the definition of $Z$ every smooth element $D\in|\mathcal{I}_{Z}(n,1)|$ has intersection with $D_{i}$ of multiplicity at least $m_{h}$ at $p_{h}$. Set $Z^{\prime}=Z\cup Z((m_{h}+1)p)$. Since $Z(m_{h}p_{h})\subseteq Z$, we have $\deg(Z^{\prime})=\deg(Z)+1$. As in step (b2) we see that $h^{1}(\mathcal{I}_{Z^{\prime}}(n,1))=0$. Hence having contact with multiplicity $>m_{h}$ only occurs in codimension one in $|\mathcal{I}_{Z}(n,1)|$. Hence the general $D$ has $m_{1},\dots,m_{\ell}$ as its restricted $i$-th profile.

(c) By step (b) we get that the set of all $D\in|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|_{sm}$ containing $Z$ is a nonempty subset of codimension $m$ and that a general element $D$ of it has the property that $w_{j}(D)=0$ for all $j\neq i$ and $m_{1},\dots,m_{\ell}$ is the restricted $i$-profile of $D$. Note that $2$ elements $D,D^{\prime}\in|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|_{sm}$ such that $D\cap D_{i}\neq D^{\prime}\cap D_{i}$ are distinct. Hence taking the union for all choices of $s$ distinct points of $D_{i}$ we get item (3). ∎

5. Ramifications and weights

Proposition 1**.**

Fix an integer $n\geq 2$. Take a general $D\in|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|$. Then $\pi_{2|D}:D\to\mathbb{P}^{1}$ has $2n-2$ distinct ramification points and their images are $2n-2$ distinct points of $\mathbb{P}^{1}$.

Proof.

Fix $p\in\mathbb{P}^{1}\times\mathbb{P}^{1}$ and let $J$ be the only element of $|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(1,0)|$ containing $p$. Let $Z\subset J$ be the degree $3$ zero-dimensional subscheme of $J$ with degree $3$. Since $n\geq 2$, step(b2) of the proof of Theorem 3 with $J$ instead of $D_{i}$ gives $h^{1}(\mathcal{I}_{Z}(n,1))=0$. Hence the set of all $D\in|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|$ with a non-ordinary ramification point at $p$ has codimension $3$ in $|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|$. Since $\dim\mathbb{P}^{1}\times\mathbb{P}^{1}=2$, we get that a general $D\in|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|$ has only ordinary ramification points, i.e., it has $2n-2$ distinct ramification points. If $2\leq n\leq 3$ the theorem of Bezout shows that no irreducible $D\in|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|$ may have two distinct ramification points contained in the same element of $|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(1,0)|$. Now assume $n\geq 4$. Fix $q\in J$ such that $q\neq p$ and let $A\subset J$ be the degree $4$ zero-dimensional scheme with $\{p,q\}$ as its reduction and both connected component of $A$ of degree $2$. Since $n\geq 3$, the proof of step(b2) of Theorem 4 below gives that $h^{1}(\mathcal{I}_{A}(n,1))=0$, i.e. the set of all $D\in|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|$ with both $p$ and $q$ as some of their ramification points has codimension $4$ in $|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|$. Since $\dim|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(1,0)|=1$ and for each $J\in\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(1,0)|$ the set of all subsets of $J$ with cardinality $2$ has dimension $2$, a general $D\in|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|$ has $2n-2$ distinct ramification points with $2n-2$ distinct images in $\mathbb{P}^{1}$. ∎

Theorem 4**.**

Fix an ordering $i_{1},i_{2},i_{3},i_{4}$ of the set $\{1,2,3,4\}$.

• (a)

For all $n\geq 2$ there exists a codimension $2$ locally closed analytic subset $\Gamma\subset\mathcal{M}_{n}(0)$ such that $w_{i_{1}}(E)=w_{i_{2}}(E)=1$ and $w_{i_{3}}(E)=w_{i_{4}}(E)=0$ for all $E\in\Gamma$.

• (b)

For all $n\geq 4$ there exists a codimension $3$ locally closed analytic subset $\Delta\subset\mathcal{M}_{n}(0)$ such that $w_{i_{1}}(E)=w_{i_{2}}(E)=w_{i_{3}}(E)=1$ and $w_{i_{4}}(E)=0$ for all $E\in\Delta$.

Proof.

To simplify the notation we use $j$ instead of $i_{j}$. The other permutations only need more double or triple indices.

We always work in $|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|_{sm}$. The surjectivity of the graph map $G$ [BH, Th. 5.2.2] gives that it sufficient to find $\Gamma_{1}\subset|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|$, $n\geq 2$, (resp. $\Delta_{1}\subset|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|$, $n\geq 4$) such that $\dim\Gamma_{1}=\dim|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|-2$ (resp. $\dim\Delta_{1}=\dim|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|-3$) such that all $C\in\Gamma_{1}$ (resp. $C\in\Delta_{1}$) have this property. By Lemma 1 there is $A\in|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|_{sm}$ such that $\pi_{2|A}$ has $2n-2$ ramification points with $2n-2$ different images $a_{1},\dots,a_{2n-2}\in\mathbb{P}^{1}$. The group $\mathrm{Aut}(\mathbb{P}^{1})$ acts uniquely 3-transitively on $\mathbb{P}^{1}$, i.e. for any $2$ triples $(b_{1},b_{2},b_{3})$, $(c_{1},c_{2},c_{3})$ of distinct points of $\mathbb{P}^{1}$ there is a unique $\gamma\in\mathrm{Aut}(\mathbb{P}^{1})$ such that $\gamma(b_{j})=c_{j}$ for all $j$.

(a) In this step we prove part (a). We saw that there is $\alpha_{2}\in\mathrm{Aut}(\mathbb{P}^{1})$ such that $\alpha_{2}(a_{1})=\pi_{2}(D_{1})$ and $\alpha_{2}(a_{2})=\pi_{2}(D_{2})$. Let $\alpha$ denote the automorphism of $\mathbb{P}^{1}\times\mathbb{P}^{1}$ which acts as the identity on the first factor and as $\alpha_{2}$ on the second factor. For all $E\in\mathcal{M}_{n}$ such that $G(E)=\alpha(A)$ we have $w_{1}(E)=w_{2}(E)=1$, because all ramification points of $D:=\alpha(A)$ are ordinary, exactly one of them is contained in $D_{1}$ and exactly one of them is contained in $D_{2}$. If $n=2$ we also have $w_{3}(E)=w_{4}(E)=0$, because $w(E)\leq 2$ for all $E\in\mathcal{M}_{2}(0)$ (Theorem 3). However, for $n>2$ to get $w_{3}(E)=w_{4}(E)=0$ we need to use some dimensional count. Let $p$ (resp. $q$) denote the ramification point of $D$ contained in $D_{1}$ (resp. $D_{2}$). Let $(2p,D_{1})$ (resp. $(2q,D_{2})$) denote the degree $2$ zero-dimensional subscheme of $D_{1}$ (resp. $D_{2}$) with $p$ (resp. $q$) as its reduction. Set $Z:=(2p,D_{1})\cup(2q,D_{2})$. Since $p$ and $q$ are ramification points of $D$, $Z$ is a degree $4$ subscheme of $D$. Hence there is an exact sequence of sheaves

[TABLE]

The Künneth formula gives $h^{1}(\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}})=0$. Since $\deg(Z)=4$, $D\cong\mathbb{P}^{1}$, $n\geq 2$ and $\deg(\mathcal{O}_{D}(n,1))=2n$, we have $h^{1}(D,\mathcal{I}_{Z,D}(n,1))=0$, i.e. $\dim|\mathcal{I}_{Z,D}(n,1)|=\dim|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|-4$.

Take $p^{\prime}\in D_{1}$ and $q^{\prime}\in D_{2}$ and set $Z(p^{\prime},q^{\prime}):=(2p^{\prime},D_{1})\cup(2q^{\prime},D_{2})$. By the semicontinuity theorem for cohomology [H, Th. III.13.8] we have $h^{1}(\mathcal{I}_{Z(p^{\prime},q^{\prime})}(n,1))=0$ for all $(p^{\prime},q^{\prime})$ in a Zariski neighborhood of $(p,q)$ in $D_{1}\times D_{2}$. Hence there is an irreducible locally closed and codimension $2$ algebraic subset $\Gamma_{1}$ of $|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|_{sm}$ such that $D\in\Gamma_{1}$ and all $X\in\Gamma_{1}$ have an ordinary ramification point contained in $D_{1}$ and an ordinary ramification point contained in $D_{2}$.

Since $D$ has only ordinary ramification points and $p$ and $q$ are its only ramification points contained in $D_{1}\cup D_{2}$, there is a Zariski open neighborhood $\Gamma_{2}$ of $D$ in $\Gamma_{1}$ such that all $C\in\Gamma_{1}$ have only ordinary ramification points, $D_{1}$ contains only one ramification point of $C$ and $D_{2}$ contains only one ramification point of $C$.

For $j=3,4$ let $\Gamma(j)$ denote the set of all $C\in\Gamma_{2}$ which are transversal to $D_{j}$. Since transversality is an open condition for the Zariski topology, $\Gamma(j)$ is Zariski open in $\Gamma_{2}$. Note that any bundle $E$ with $G(E)\in\Gamma(3)\cap\Gamma(4)$ satisfies $w_{1}(E)=w_{2}(E)=1$ and $w_{3}(E)=w_{4}(E)=0$. Assume for the moment $\Gamma(3)\neq\emptyset$ and $\Gamma(4)\neq\emptyset$. Since $\Gamma_{2}$ is irreducible of dimension $\dim|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|-2$, the same is true for $\Gamma(3)\cap\Gamma(4)$. Hence part (a) holds if $\Gamma(3)\neq\emptyset$ and $\Gamma(4)\neq\emptyset$. Assume for instance $\Gamma(3)=\emptyset$, i.e. assume that all $C\in\Gamma_{2}$ are tangent to $D_{3}$. In particular this holds for $D$. Call $a$ the point of $D_{3}\cap D$ at which $D_{3}$ and $D$ are tangent. Let $(2a,D_{3})$ denote the degree $2$ zero-dimensional subscheme of $D_{3}$ with $a$ as its reduction. Set $W:=Z\cup(2a,D_{3})$. Since $p$, $q$ and $a$ are ramification points of $D$, $W\subset D$. Hence there is an exact sequence of sheaves

[TABLE]

The Künneth formula gives $h^{1}(\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}})=0$. Since $\deg(Z)=4$, $D\cong\mathbb{P}^{1}$, $n\geq 3$ and $\deg(\mathcal{O}_{D}(n,1))=2n$, we have $h^{1}(D,\mathcal{I}_{W,D}(n,1))=0$, i.e. $\dim|\mathcal{I}_{W,D}(n,1)|=\dim|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|-6$. Since $\dim(D_{1}\times D_{2}\times D_{3})=3$, varying the points $(p^{\prime},q^{\prime},a^{\prime})\in D_{1}\times D_{2}\times D_{3}$, we get that $\Gamma_{2}\setminus\Gamma(3)$ has codimension at least $1$ in $\Gamma_{2}$. Hence $\Gamma(3)\neq\emptyset$, contradicting one of our assumptions.

(b) In this step we prove part (b). Let $\gamma_{2}$ be the only element of $\mathrm{Aut}(\mathbb{P}^{1})$ such that $\gamma_{2}(a_{1})=\pi_{2}(D_{1})$, $\gamma_{2}(a_{2})=\pi_{2}(D_{2})$ and $\gamma_{2}(a_{3})=\pi_{2}(D_{3})$. Let $\gamma$ be the automorphism of $\mathbb{P}^{1}\times\mathbb{P}^{1}$ which acts as the identity on the first factor and as $\gamma_{2}$ on the second factor. Set $X:=\gamma(A)$.

Let $p$ (resp. $q$, resp $a$) denote the contact locus of $X$ and $D_{1}$ (resp. $D_{2}$, resp. $D_{3}$). Note that $w_{1}(E)=w_{2}(E)=w_{3}(E)=1$ for all bundles $E$ such that $G(E)=X$. Hence to prove part (b) it is sufficient to count the curves $X^{\prime}$ near $X$ and tangent to $D_{j}$ for $j=1,2,3$ and prove that the general such curve is not tangent to $D_{4}$. Let $p$ (resp. $q$, resp. $a$) be the point of contact of $X$ and $D_{1}$ (resp. $D_{2}$, resp. $D_{3}$). As in step (a) let $(2p,D_{1})$ (resp. $(2q,D_{2})$, resp. $(2a,D_{3})$) denote the degree $2$ connected zero-dimensional subscheme of $D_{1}$ (resp. $D_{2}$, resp. $D_{3}$) with $p$ (resp. $q$, resp. $a$) as its reduction. Set

[TABLE]

Since $X$ ramifies at $p$, $q$ and $a$, $W\subset X$. We get an exact sequence similar to (6) with $X$ instead of $D$. As in step (a) we get $h^{1}(\mathcal{I}_{W}(n,1))=0$, i.e. $\dim|\mathcal{I}_{W}(n,1)|=\dim|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|-6$. For any$(p^{\prime},q^{\prime},a^{\prime})\in D_{1}\times D_{2}\times D_{3}$ and set

[TABLE]

By the semicontinuity theorem for cohomology [H, Th. III.13.8] we have $h^{1}(\mathcal{I}_{W(p^{\prime},q^{\prime},a^{\prime})}(n,1))=0$, i.e. $\dim|\mathcal{I}_{W(p^{\prime},q^{\prime},a^{\prime})}(n,1)|=\dim|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|-6$ for all $(p^{\prime},q^{\prime},a^{\prime})$ in a non-empty open subset of $(p,q,a)$ in $D_{1}\times D_{2}\times D_{3}$. Since $\dim(D_{1}\times D_{2}\times D_{3})=3$, we get a codimension $3$ analytic subset $\Delta^{\prime}$ of $\mathcal{M}_{n}(0)$ such $w_{1}(E)=w_{2}(E)=w_{3}(E)=1$.

To prove step (b) we need to prove that, outside a lower dimensional analytic subset of $\Delta^{\prime}$, the bundles satisfy $w_{4}(E)=0$, i.e. their graph is transversal to $D_{4}$. As in step (a) we have a locally closed irreducible subset $\Delta_{2}$ of $|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|(0)$ with codimension $3$, containing $X$ and formed by smooth curves with ordinary ramifications, $2n-2$ images of the ramification points. We need to prove that a general $X^{\prime}\in\Delta_{2}$ is transversal to $D_{4}$. Assume that this is not true. In particular $X$ is tangent to $D_{4}$. Since the ramification points of $X$ have different, there is a unique $b\in D_{4}\cap X$ at which $X$ and $D_{4}$ are tangent. Let $(2b,D_{4})$ denote the degree $2$ zero-dimensional subscheme of $D$ with $b$ as its reduction. Set $W^{\prime}:=W\cup(2b,D_{4})$. Note that $\deg(W^{\prime})=8$ and $W^{\prime}\subset X$. We get an exact sequence of sheaves similar to (6) with $X$ instead of $D$ and $W^{\prime}$ instead of $X$. Since $n\geq 4$, $X\cong\mathbb{P}^{1}$, $\deg(W^{\prime})=8$ and $\deg(\mathcal{O}_{X}(n,1))=2n$, we have $h^{1}(X,\mathcal{I}_{W^{\prime},X}(n,1))=0$ and hence $h^{0}(\mathcal{I}_{W^{\prime}}(n,1))=h^{0}(\mathcal{I}_{W}(n,1)-2$. Since $\dim D_{4}=1$, repeating the last part of step (a) we get a contradiction. ∎

Example 1*.*

Take $n=2$. Let $E$ be any irregular bundle. By Theorem 3 we have $1\leq w(E)\leq 2$ and $w_{i}(E)\leq 1$ for all $i$. Theorems 3 and 4 show that for all possible quadruples $(x_{1},x_{2},x_{3},x_{4})$ of integers with $0\leq x_{i}\leq 1$ and $1\leq x_{1}+x_{2}+x_{3}+x_{4}\leq 2$ there is $E\in\mathcal{M}_{\delta,2}$ such that $w_{i}(E)=x_{i}$ for all $i$. We also got that the part corresponding to $(x_{1},x_{2},x_{3},x_{4})$ has dimension at least $8-x_{1}-x_{2}-x_{3}-x_{4}$.

Theorem 5**.**

Fix an integer $n\geq 2$. Fix an ordering $i_{1},i_{2},i_{3},i_{4}$ of $\{1,2,3,4\}$. Then there exists an irreducible locally closed analytic subset of dimension $2n+2$ of $\mathcal{M}_{n}(0)$ such that for all $E\in\Gamma$ the restricted $i_{1}$-th and $i_{2}$-th profiles of $E$ are $(n)$, while $w_{i_{3}}(E)=w_{i_{4}}(E)=0$.

Proof.

Since the general case is similar, we only do the case $i_{j}=j$ for $j=1,2,3,4$. Note that if $E\in\mathcal{M}_{\delta,n}(0)$ has restricted profile $(n)$ for $D_{1}$ and $D_{2}$, then $w_{1}(E)=w_{2}(E)=n-1$ and hence $w_{3}(E)=w_{4}(E)=0$ by Theorem 3. Hence we do not have the most difficult part of the proof of Theorems 3 and 4, transversality with respect to some $D_{j}$.

Since the case $n=2$ is covered by Example 1, we assume $n\geq 3$.

Since the graph map is surjective and all its fibers have dimension at least $2n-1$, it is sufficient to find a $3$-dimensional complex analytic family $\Gamma_{1}\subset|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|$ such that each $D\in\Gamma_{1}$ has profile $(n)$ with respect to $D_{1}$ and $D_{2}$.

Fix $p=(p_{1},p_{2})\in D_{1}$ and $q=(q_{1},q_{2})\in D_{2}$ such that $p_{1}\neq q_{1}$. For any integer $x>0$ let $(xp,D_{1})$ (resp. $(xq,D_{2})$) denote the connected and degree $x$ zero-dimensional subcheme of $D_{1}$ (resp. $D_{2}$) with $p$ (resp. $q$) as its reduction. Set $Z:=(np,D_{1})\cup(nq,D_{2})$. We have $\deg(Z)=2n$ and hence $\dim|\mathcal{I}_{Z}(n,1)|\geq\dim|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|-2n=1$. Take a general $D\in|\mathcal{I}_{Z}(n,1)|$. Since the set of all such pairs $(p,q)\in D_{1}\times D_{2}$ has dimension $2$ and smoothness is an open condition for the Zariski topology, to conclude the proof of the theorem it is sufficient to prove that $D$ is smooth.

Fix $a\in D_{1}\setminus\{p\}$ and $b\in D_{2}\setminus\{q\}$. Let $R_{1}$ be the only element of $|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}|$ containing $p$ and $R_{2}$ the only element of $|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}|$ containing $q$. Since $p_{1}\neq q_{1}$, we have $R_{1}\neq R_{2}$, $q\notin R_{1}$ and $p\notin R_{2}$. Set $U:=\mathbb{P}^{1}\times\mathbb{P}^{1}\setminus(D_{1}\cup D_{2}\cup R_{1}\cup R_{2})$. Set $u:=D_{1}\cap R_{2}$ and $v:=D_{2}\cap R_{1}$.

(a) In this step we prove that $h^{0}(\mathcal{I}_{Z}(n,1))=2$, i.e. that $\dim|\mathcal{I}_{Z}(n,1)|=1$. To prove that $h^{0}(\mathcal{I}_{Z}(n,1))=2$ it is sufficient to prove that $h^{0}(\mathcal{I}_{Z\cup\{a,b\}}(n,1))=0$. Assume $h^{0}(\mathcal{I}_{Z\cup\{a,b\}}(n,1))>0$ and take $X\in|\mathcal{I}_{Z\cup\{a,b\}}(n,1)|$.

Since $\{a\}\cup(np,D_{1})$ and $\{b\}\cup(nq,D_{2})$ have degree $n+1$ and since $D_{1}$ and $D_{2}$ are irreducible, the theorem of Bezout gives $D_{1}\cup D_{2}\subseteq X$. Since $h^{0}(\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)(-D_{1}-D_{2}))=h^{0}(\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,-1))=0$, we get a contradiction.

(b) By the theorem of Bertini ([GH, p. 137], [H, III.10.9], [Jo, 6.3]) to prove that a general $D\in|\mathcal{I}_{Z}(n,1)|$ is smooth outside $\{p,q\}$ it is sufficient to prove that $\{p,q\}$ is the set-theoretic base locus $\mathcal{B}$ of $|\mathcal{I}_{Z}(n,1)|$. Recall that $p_{1}\neq q_{1}$ by assumption. Since $D_{1}\cap D_{2}=\emptyset$, we have $p_{2}\neq q_{2}$. Recall that $\mathrm{Aut}(\mathbb{P}^{1})$ is simply $3$-transitive, i.e. for all triples of $(b_{1},b_{2},b_{3})$ and $(c_{1},c_{2},c_{3})$ of distinct points of $\mathbb{P}^{1}$ there is a unique $h\in\mathrm{Aut}(\mathbb{P}^{1})$ such that $h(b_{i})=c_{i}$ for all $i=1,2,3$. Hence the subset $\mathbb{G}$ of all $u\in\mathrm{Aut}(\mathbb{P}^{1})\times\mathrm{Aut}(\mathbb{P}^{1})$ such that $u(p)=p$ and $u(q)=q$, is an algebraic linear group isomorphic to $(\mathbb{C}^{\ast})^{2}$ and whose actions on $\mathbb{P}^{1}\times\mathbb{P}^{1}$ has the following $9$ orbits:

[TABLE]

and the $4$ singletons $u,\,\,v,\,\,p,$ and $q$.

Assume $\mathcal{B}\neq\{p,q\}$, take $z\in\mathcal{B}\setminus\{p,q\}$ and let $U_{z}$ be the orbit of $\mathbb{G}$ containing $z$. Since each $h\in\mathbb{G}$ fixes scheme-theoretically $z$, we have $U_{z}\subset\mathcal{B}$. If $U_{z}=U$, then we get $h^{0}(\mathcal{I}_{Z}(n,1))=0$, a contradiction.

Now assume $U_{z}=u$, i.e. $z=u$. Hence $h^{0}(\mathcal{I}_{Z\cup\{u\}}(n,1))=h^{0}(\mathcal{I}_{Z}(n,1))=2$. Since $\deg(D_{1}\cap(Z\cup\{u\})=n+1$, the theorem of Bezout gives $D_{1}\subset\mathcal{B}$. Since $Z\cup D_{1}=(nq,D_{2})$, $\mathcal{I}_{D_{1}}(n,1)\cong\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,0)$ and $q\notin D_{1}$, we get $h^{0}(\mathcal{I}_{Z}(n,1))=h^{0}(\mathcal{I}_{(nq,D_{2})}(n,0))$. Since the scheme $\pi_{1}((nq,D_{1}))$ is the degree $n$ zero-dimensional subscheme of $\mathbb{P}^{1}$ with $q_{1}$ as its reduction, we have $h^{0}(\mathcal{I}_{(nq,D_{2})}(n,0))=h^{0}(\mathbb{P}^{1},\mathcal{I}_{\pi_{1}((nq,D_{1})}(n,0))=h^{0}(\mathbb{P}^{1},\mathcal{O}_{\mathbb{P}^{1}})=1$. Hence $h^{0}(\mathcal{I}_{Z}(n,1))=1$, a contradiction. Similarly, we exclude the case $z=v$.

Now assume $U_{z}=D_{1}\setminus\{p,u\}$. Since $\mathcal{B}$ is closed in $\mathbb{P}^{1}\times\mathbb{P}^{1}$, $D_{1}\subset\mathcal{B}$. Hence $u\in\mathcal{B}$. We excluded this case. Similarly, we exclude the case $U_{z}=D_{2}\setminus\{q,v\}$.

Now assume $U_{z}=R_{1}\setminus\{p,v\}$. Since $\mathcal{B}$ is closed in $\mathbb{P}^{1}\times\mathbb{P}^{1}$, $R_{1}\subset\mathcal{B}$. Hence $v\in\mathcal{B}$. We excluded this case. Similarly, we exclude the case $U_{z}=R_{2}\setminus\{q,u\}$.

(c) In this step we conclude the proof of the theorem proving that $D$ is smooth at $p$ and at $q$. Since $|\mathcal{I}_{Z}(n,1)|$ is an irreducible variety, $\{p,q\}$ is a finite set and smoothness is an open condition for the Zariski topology, it is sufficient to prove that a general $D^{\prime}\in|\mathcal{I}_{Z}(n,1)|$ is smooth at $p$ and a general $D^{\prime\prime}\in|\mathcal{I}_{Z}(n,1)|$ is smooth at $q$. We only prove that a general $D^{\prime}\in|\mathcal{I}_{Z}(n,1)|$ is smooth at $p$, since the smoothness at $q$ only requires notational modifications. Assume that a general $D^{\prime}\in|\mathcal{I}_{Z}(n,1)|$ is singular at $p$. Let $(2p,R_{1})$ denote the degree $2$ zero-dimensional subscheme of $R_{1}$ with $p$ as its reduction. Set $W\mathrel{\mathop{:}}=Z\cup(2p,R_{1})$. Since a general $D^{\prime}\in|\mathcal{I}_{Z}(n,1)|$ is singular at $p$, $W\subset D^{\prime}$ and hence $h^{0}(\mathcal{I}_{W}(n,1))=h^{0}(\mathcal{I}_{Z}(n,1))=2$. Since $D^{\prime}$ contains $W$, it contains the degree $2$ subscheme $(2p,R_{1})$. Since $\mathcal{O}_{R_{1}}(n,1)$ is the degree $1$ line bundle on $R_{1}$, the base locus $\mathcal{B}$ of $|\mathcal{I}_{Z}(n,1)|$ contains $R_{1}$. We excluded this case in step (b).

∎

6. The topology of $\mathcal{M}_{n}^{\operatorname{reg}}(0)$

Fix an integer $n\geq 2$. Recall that $\dim|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|=2n+1$. Let $\mathcal{C}_{n}\subset|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|$ denote the set of all singular $D\in|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|$. As usual in Complex Analysis and in Algebraic Geometry, $\Delta$ is a hypersurface of $|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|$. It is easy to check that $\mathcal{C}_{n}$ contains a codimension one subset formed by all $A\cup B$ with $A\in|\mathcal{O}_{\mathbb{P}^{1}}(1,0)|$ (which has dimension $1$) and $B$ and element of $|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n-1,1)|$. The set $\mathcal{C}_{n}$ is irreducible [GKZ, Ch. 1] and hence it is given by a unique equation. Therefore, the set $\mathcal{B}_{n}:=|\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(n,1)|\setminus\mathcal{C}_{n}$ is a smooth and connected affine variety of complex dimension $2n+1$.

By [AF, Hl] we obtain that $\mathcal{B}_{n}$ is homotopy equivalent to a finite CW complex of real dimension at most $\dim_{\mathcal{C}}\mathcal{B}_{n}=2n+1$. Let $H_{1}\cup H_{2}\cup H_{3}\cup H_{4}$ be the set of smooth, but irregular graphs, i.e. (since $n\geq 2$) the images of $\mathcal{M}_{n}^{\operatorname{irreg}}(0)$. Hence

[TABLE]

and using [AF, Hl] we get that it is a smooth affine variety with the homotopy type of a finite CW complex of real dimension at most $2n+1$.

The graph map $G\colon\mathcal{M}_{n}^{\operatorname{reg}}(0)\to\mathcal{A}_{n}$ is a smooth submersion whose fibers are compact differential manifolds diffeomorphic to $(S^{1})^{4n-2}$, where $S^{1}$ is the unit circle. It then follows that the cohomology ring $H^{*}(\mathcal{M}_{n}^{\operatorname{reg}},\mathbb{Z})$ is the tensor product of the cohomology ring $H^{*}(\mathcal{A}_{n},\mathbb{Z})$ and the cohomology ring $H^{*}((S^{1})^{4n-2},\mathbb{Z})$. We restate this fact as a theorem.

Theorem 6**.**

Fix an integer $n\geq 2$. We have:

• (i)

$H^{*}(\mathcal{M}_{n}^{\operatorname{reg}}(0),\mathbb{Z})=H^{*}(\mathcal{A}_{n},\mathbb{Z})\otimes H^{*}((S^{1})^{4n-2},\mathbb{Z}),$* and*

• (ii)

$H^{i}(\mathcal{M}_{n}^{\operatorname{reg}}(0),\mathbb{Z})=0$*  for all $i\geq 6n$.*

Proof.

The graph map $G\colon\mathcal{M}_{n}^{\operatorname{reg}}(0)\to\mathcal{A}_{n}$ is a $C^{\infty}$-submersion with compact fibers. The fibers of $G$ are biholomorphic to compact complex $(2n-1)$-tori and hence they are diffeomorphic to $(S^{1})^{4n-2}$. Since $G$ is a $C^{\infty}$-submersion with compact fibers it is locally trivial on the base $\mathcal{A}_{n}$, i.e. for each $p\in\mathcal{A}_{n}$ there is an Euclidean open subset $U$ such that $G^{-1}(U)$ is fiberwise diffeomorphic to $U\times(S^{1})^{4n-2}$ , i.e. it commutes with the projection $U\times(S^{1})^{4n-2}\to U$. Since being a Serre fibration is a local condition on the base and their products are Serre fibrations, $G$ is a Serre fibration. Thus, part (a) follows from a theorem of Leray and Hirsch [Dc, 17.8.1].

Since $\mathcal{A}_{n}(\operatorname{reg})$ is an affine variety, it has the homotopy type of a finite CW-complex of (real!) dimension at most $\dim\mathcal{A}_{n}=2n+1$ [AF, Hl], $H^{i}(\mathcal{A}_{n},\mathbb{Z})=0$ for all $i\geq 2n+2$. Hence part (ii) follows from part (i) (alternatively, one could use the Leray–Serre spectral sequence of $G$ of the cohomology with $\mathbb{Z}$-coefficient [Di, Cor. 2.3.4] or [MC, Thm. I.5.2]. ∎

Obviously Theorem 6 may be extended to other coefficient abelian groups instead of $\mathbb{Z}$, just quoting [Di] or [MC]

Theorem 7**.**

Let $Y\subset\mathcal{M}_{n}(0)$ be an irreducible compact complex analytic subspace. Then $G(Y)$ is a single point.

Proof.

The set $G(Y)$ is an irreducible and compact complex space contained in the affine variety $\mathcal{B}_{n}$ parametrizing all smooth graphs. Thus $G(Y)$ is a point. ∎

Corollary 8**.**

Let $Y\subset\mathcal{M}_{n}^{\operatorname{reg}}(0)$ be an irreducible compact complex analytic subspace. Then $Y$ is contained in one of the $(2n-1)$-dimensional tori which are the fibers of the graph map.

Proof.

By Theorem 7, $Y$ is contained in a fiber of the graph map. Since $G^{-1}(G(y))$ is a compact torus of complex dimension $2n-1$, we get the corollary. ∎

7. Nontrivial determinants

Let $\delta\in\mathbb{C}^{*}$. In this section we denote the moduli space of rank 2 bundles with $c_{2}=n$ and determinant $\delta$ by $\mathcal{M}_{n,\delta}$ and set $\mathcal{M}_{n}\mathrel{\mathop{:}}=\mathcal{M}_{n,\mathcal{O}_{X}}$ for the case of trivial determinant. We have $K_{X}\cong\mathcal{O}_{X}(-2)$.

Remark 7*.*

Let $E$ be a stable (or just a simple) vector bundle or rank $r$ on $X$. Serre-duality gives $h^{2}(End(E))=h^{0}(End(E)\otimes\mathcal{O}_{X}(-2))=0$. Hence, the local deformation space of $E$ is smooth of dimension $h^{1}(X,End(E))-1$. So, $\mathcal{M}_{\delta,E}$ is smooth and equidimensional and (Riemann–Roch) $\dim\mathcal{M}_{n}=4n$.

Proposition 2**.**

For any $\delta\in\mathbb{C}^{*}$, and all $n\geq 1$ the complex spaces $\mathcal{M}_{n,\delta}$ and $\mathcal{M}_{n}$ are biholomorphic.

Proof.

Both complex spaces are moduli spaces. Hence it is sufficient to prove that $\mathcal{M}_{n,\delta}$ is a moduli space for rank $2$ stable vector bundles on $X$ with $c_{2}=n$ and trivial determinant. Fix $c\in\mathbb{C}^{*}$ such that $c^{2}=\delta$. Let $v\colon\mathcal{E}\to X\times S$ be a family of rank 2 stable vector bundles on $X$ with trivial determinant and $c_{2}=n$ parametrized by the complex space $S$. Let $\pi_{1}\colon X\times S\to X$ denote the projection. The family $\mathcal{E}\otimes\pi_{1}^{\ast}(\mathcal{O}_{X}(c))$ is a family of rank $2$ stable vector bundles on $X$ with $c_{2}=n$ and determinant isomorphic to $\delta$. Hence there is a holomorphic map $f_{v}\colon S\to\mathcal{M}_{n,\delta}$, which to each $s$ in $S$ assigns the bundle $E\otimes\pi_{1}^{\ast}(\mathcal{O}_{X}(c))$. By the definition of coarse moduli space the rule $v\mapsto f_{v}$ shows that $\mathcal{M}_{n,\delta}$ is (coarse) moduli space for $\mathcal{M}_{n}$ and hence $\mathcal{M}_{n,\delta}$ and $\mathcal{M}_{n}$ are biholomorphic. ∎

We conclude that the theorems proved here all have analogous statements for the cases of nontrivial determinant.

Acknowledgements

E. Ballico is a member of GNSAGA of INdAM (Italy). E. Gasparim is a senior associate of the Abdus Salam International Centre for Theoretical Physics, Trieste (Italy).