Irregular Bundles on Hopf Surfaces

Abstract

We discuss the complex-analytic subsets of the moduli spaces of rank 2 vector bundles on a classical Hopf surface formed by irregular bundles. We stratify the set of irregular bundles by weight (and irregular profiles). We provide the topological result (vanishing of higher cohomology groups) on the part of the moduli spaces parameterizing regular bundles.

1. Motivation

The profound and productive relationship between physics and mathematics has unfolded for centuries, with each discipline pushing the other toward new heights of understanding. While the development of physics often awaits progress regarding a required geometric background, at times the reverse occurs, with new geometries being created based on existing physics concepts. As a formidable example, we mention the celebrated article “An application of gauge theory to four dimensions” [1] by S. Donaldson, who used physics concepts, namely instantons in Yang–Mills theory, to produce new invariants for smooth four-manifolds. His results revolutionized the field of differential topology by proving that the smooth structure of a four-manifold is a finer invariant than its topological structure. More specifically, cohomology classes of instanton moduli spaces emerged as essential tools for calculating the coefficients of the so-called Donaldson polynomials, which enabled the classification of differentiable structures of four-manifolds, with this work giving rise to an entire new school of thought in mathematics that uses instanton moduli spaces as tools for the classification of geometric objects. Donaldson’s work profited from the translation of the concept of instantons on a real four-manifold into the complex algebraic geometry of holomorphic bundles over a complex surface, known as the Kobayashi–Hitchin correspondence.

Yang–Mills theory forms an essential part of modern physics as it provides the underlying framework for understanding strong and weak nuclear forces, which are two of the four fundamental forces of nature. Nevertheless, despite its prominent role in physics, there still remain many basic open questions in Yang–Mills theory. Indeed, E. Witten, arguably the most famous physicist alive, in his recent survey [2], presented various open questions about moduli of instantons over $S^3\times S^1$. Analogously to the work of Donaldson, here too the gauge-theoretical questions about instantons over the real four-manifold $S^3\times S^1$ can be translated into the complex geometry of holomorphic bundles over a complex surface. However, in this case, the complex structure on $S^3\times S^1$ gives rise to the classical Hopf surface, which is a complex-analytic manifold that does not admit an algebraic structure. Our interest in bundles on Hopf surfaces originated from the questions about instantons on $S^3\times S^1$ posed by Witten. We will see over the course of this work that moduli of holomorphic bundles on Hopf surfaces (non-algebraic and non-Käler) lead us to study various new phenomena that were not present in the algebraic case.

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 [3], Th. 1, the moduli space of $SU\left(2\right)$ 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 was proven using the description of jumping lines [4,5] and of local contributions to instanton charges concentrated at exceptional curves [6]. It is now known whether the Atiyah–Jones conjecture [7] 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 cannot be calculated 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 ${\mathcal{M}}_n\left(0\right)$ formed by bundles without any jumps.

We consider rank 2 stable vector bundles on the classical Hopf surface X. See [8], §18, for a description of these connected compact complex manifolds of complex dimension 2. They are not Kähler (and in particular they are not algebraic). Not being projective results in major problems for the study of stable vector bundles on X. Even their usual definition for projective varieties ([9]) makes no sense because X has no ample line bundle. It was P. Gaudoschon who found a way: it is sufficient to fix a metric on X with certain properties, a so-called Gaudoschon metric ([9], §5.3. Let Y be a smooth compact complex surface with a fixed Gauduschon metric. This metric provides the notions of stable and semistable vector bundles. For any fixed line bundle L, any rank $r\ge 2$, and any integer $c_2$ (the second Chern number), there is a complex space $M(Y,L,c_2,r)$ parameterizing all stable rank r vector bundles on Y with rank r, determinant L, and second Chern number $c_2$ ([9], Theorem 5.34). The set ${\mathcal{M}}_n$ of Remark 1 is exactly $M(X;{\mathcal{O}}_X,n,2)$. Stable vector bundles are simple ([9], Corollary 5.31), but the converse often fails. Simple vector bundles have their own moduli space ([9], §5.2), but the moduli space of simple bundles is often not Hausdorff; only each of its points has a Hausdorff open neighborhood ([9], Theorem 5.15).

In Section 2, we state our main results. There is a trichotomy for elements of ${\mathcal{M}}_n$: regular, irregular, or with a jump. We outline important new results on the set of regular bundles and on the set of irregular bundles. Bundles with jumps are obtained from regular or irregular bundles with lower Chern numbers. Hence, inductively, our results provide some information on all stable bundles. Before this paper, it was known that ${\mathcal{M}}_n$ is non-empty, smooth, and of dimension n ([10], Prop. 3.4.3). The complex manifold ${\mathcal{M}}_1$ is described in [10], Theorem 5.4.1. Moreover, it was known that regular bundles exist for all $n\ge 1$ and that they form an open subset of ${\mathcal{M}}_n$. The paper [11] contains the same study for other compact non-algebraic complex surfaces and generalizes [10], Theorem 5.4.1, extending rank 2 stable vector bundles with second Chern class 1 to other two-dimensional elliptic fibrations. The geometry of these compact complex surfaces is described in [8], pp. 200–219.

In Section 3, we provide the notation and preliminary results needed for our proofs.

In Section 4, we characterize the irregular bundles among the bundles with no jumps (Theorems 9 and 10).

In Section 5, we provide a measure (the total weight) of the irregularity of a bundle with no jump and a refining of this weight as a sum of local contributions (the profile of the bundle). We present three existence theorems for irregular bundles with a prescribed profile (Theorems 11–13).

In Section 7, we prove a topological result about the set of regular bundles (Theorem 14) and describe all the compact complex-analytic subsets of the set or regular bundles (Theorem 15).

In Section 8, we prove that similar results are true for rank 2 bundles with nontrivial determinants (Theorem 16).

The last section outlines our conclusions.

As in [10,11,12], our results are for rank 2 vector bundles. At the moment, the higher-rank case seems to be out of reach, except perhaps for a description of regular bundles using [13]. Even for regular bundles, any topological result similar to Theorem 14 would be outstanding.

We thank the referees for their useful suggestions regarding the organization of the paper.

2. Statements of Results

Let $\pi :X\to {\mathbb{P}}^1$ 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 ${\mathcal{M}}_n\left(0\right)$ formed by bundles without jumps. The goal of this note is to describe some of the geometric features of ${\mathcal{M}}_n\left(0\right)$.

A bundle $E\in {\mathcal{M}}_n$ is called irregular when there is a fiber T of X such that the restriction of E to T has automorphism group of dimension 4 (Definition 1); this happens whenever ${E|}_T=L\oplus L$ with $L^2=\mathcal{O}$; that is, when L is one of the 4 thetas. We recall the graph maps $G:{\mathcal{M}}_n\to \left|{\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(n,1)\right|\cong {\mathbb{P}}^{2n+1}$ and their surjectivity for $n\ge 2$ [10]. We then compare regular and irregular bundles without jumps. We first show

Theorem 1.

Fix $E\in {\mathcal{M}}_n\left(0\right)$, $n\ge 2$.

(a) 

If $E\in {\mathcal{M}}_n^{reg}\left(0\right)$, then $C_E$ is smooth and connected.

(b) 

If $E\in {\mathcal{M}}_n^{irreg}\left(0\right)$, then $C_E$ is singular and irreducible.

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

$$w\left(E\right)=w_1\left(E\right)+w_2\left(E\right)+w_3\left(E\right)+w_4\left(E\right)$$

intended as a measure of irregularity and prove

Theorem 2.

$E\in {\mathcal{M}}_n\left(0\right)$ is irregular if and only if $w\left(E\right)>0$.

Having identified ${\mathcal{M}}_n^{irreg}\left(0\right)$ as the set of all bundles $E\in {\mathcal{M}}_n^{sg}$ without jumps and having singular spectral curve $C_E$ (Notation 4), we prove

Theorem 3.

Assume $n>1$. For all $E\in {\mathcal{M}}_n^{irreg}\left(0\right)$, we have

1. 

$w\left(E\right)\le 2n-2$.

2. 

E has $(n,0,\dots ,0)$ as its i-th profile if and only if $w_i\left(E\right)\le n-1$ for all i and $w_i\left(E\right)=n-1$.

3. 

Fix $i\in \{1,2,3,4\}$, a positive integer ℓ and a restricted profile $(m_1,\dots ,m_{\ell })$. There exists a non-empty locally closed analytic subset $\Gamma \in {\mathcal{M}}_n^{irreg}\left(0\right)$ such that all $G\left(E\right)\in |{\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(n,1)|$ have $(m_1,\dots ,m_{\ell })$ as i-th profile, and this set of graphs $G\left(E\right)$ has codimension at most $-\ell +m_1+\dots +m_{\ell }$ 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 $n\ge 2$, there exists a locally closed analytic subset $\Gamma \subset {\mathcal{M}}_n\left(0\right)$ such that $w_{i_1}\left(E\right)=w_{i_2}\left(E\right)=1$ and $w_{i_3}\left(E\right)=w_{i_4}\left(E\right)=0$ for all $E\in \Gamma$, and $G(\Gamma )$ has codimension 2 in $|{\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(n,1)|$.

(b) 

For all $n\ge 4$, there exists a locally closed analytic subset $\Delta \subset {\mathcal{M}}_n\left(0\right)$ such that $w_{i_1}\left(E\right)=w_{i_2}\left(E\right)=w_{i_3}\left(E\right)=1$ and $w_{i_4}\left(E\right)=0$ for all $E\in \Delta$, and $G(\Delta )$ has codimension 3 in $|{\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(n,1)|$.

Theorem 5.

Fix an integer $n\ge 2$. 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 Γ of ${\mathcal{M}}_n\left(0\right)$ such that $dimG(\Gamma )=3$, and for all $E\in \Gamma$ the restricted $i_1$-th and $i_2$-th profiles of E are $\left(n\right)$, while $w_{i_3}\left(E\right)=w_{i_4}\left(E\right)=0$.

We then describe some features of the topology of ${\mathcal{M}}_n^{reg}\left(0\right)$, obtaining

Theorem 6.

Fix an integer $n\ge 2$. We have

(i) 

$H^{\ast }({\mathcal{M}}_n^{reg}\left(0\right),\mathbb{Z})=H^{\ast }({\mathcal{A}}_n,\mathbb{Z})\otimes H^{\ast }({\left(S^1\right)}^{4n-2},\mathbb{Z}).$

(ii) 

$H^i({\mathcal{M}}_n^{reg}\left(0\right),\mathbb{Z})=0$ for all $i\ge 6n$.

Theorem 7.

Let $Y\subset {\mathcal{M}}_n\left(0\right)$ be an irreducible compact complex-analytic subspace. Then, $G\left(Y\right)$ is a single point.

As a consequence, we obtain

Corollary 1.

Let $Y\subset {\mathcal{M}}_n^{reg}\left(0\right)$ 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

Theorem 8.

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

Therefore, all results proved here have analogous statements for the cases of nontrivial determinant.

Here, we concentrate on discussing geometric aspects of the set ${\mathcal{M}}_n\left(0\right)$ 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 [14], but we do not use that result here.

3. Bundles Without Jumps

Let $\pi :X\to {\mathbb{P}}^1$ be a classical Hopf surface with T as its elliptic fiber. We have ${\mathrm{Pic}}^0\left(T\right)$ (often written as $T^{\ast }$), which is non-canonically isomorphic to T. Observe that, since T is a torus, there are 4 such elements $L\in {\mathrm{Pic}}^0\left(T\right)$ 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 ${\mathrm{Pic}}^0\left(T\right)$ such that $L^{\otimes 2}\cong {\mathcal{O}}_T$. Informally, thinking of the torus as obtained from ${\mathbb{C}}^2/\mathsf{\Lambda }$, where $\mathsf{\Lambda }$ is the integer lattice, these can be thought of as corresponding to the points $(0,0),({\displaystyle \frac{1}{2}},0),(0,{\displaystyle \frac{1}{2}}),({\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}})$.

Let F be a rank 2 vector bundle on T with $det\left(F\right)\cong {\mathcal{O}}_T$. The set $\mathrm{End}\left(F\right)$ of all endomorphisms of F is a finite-dimensional complex vector space ([9,10,11,12,13]), and a Zariski open subset of it is formed by the set $\mathrm{Aut}\left(F\right)$ of its automorphisms. Hence, $dim\mathrm{Aut}\left(F\right)$ is a linear complex algebraic group of dimension $dim\mathrm{End}\left(F\right)$. We use Atiyah’s classification of holomorphic vector bundles on the elliptic curve T ([10,15], §3.2, [11]). If F has rank 2 and degree 0, then it is neither stable nor simple, but it may be properly semistable and even polystable. If F is not semistable, then $F\cong R\oplus R^{\ast }$ for some line bundle R on T with $\mathrm{deg}\left(R\right)\ne 0$. Now assume that F is semistable and decomposable; i.e., assume $F\cong L\oplus L^{\ast }$ for some degree 0 line bundle on T, and then there are 2 possibilities:

• If $L\ne L^{\ast }$, then $dim\mathrm{End}\left(F\right)=2$.
• If $L\cong L^{\ast }$, then $dim\mathrm{End}\left(F\right)=4$.

Notation 1.

A bundle E over X is called regular if its restriction to every fiber of π has automorphism group of minimal dimension; i.e., $dimAut\left(E{|}_T\right)=2\forall T$. Observe 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 π is irregular.

Let ${\mathcal{M}}_n$ denote the moduli space of all rank 2 stable vector bundles on X with trivial determinant and $c_2\left(E\right)=n$; i.e.,

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

The variety ${\mathcal{M}}_n$ is smooth non-empty of dimension $4n$ [10], Prop. 3.4.4; see also [12], Prop. 4.2, for the case of nontrivial determinant.

Since here we consider only the case of trivial determinant, the spectral involution on $T^{\ast }$ may be defined as $i\left(\lambda \right)=-\lambda$, with corresponding double cover

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

(1)

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

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

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

When $c_2\left(E\right)=n$, the divisor $G\left(E\right)=D$ belongs to the linear system $\left|\mathcal{O}\right(n,1\left)\right|$ over ${\mathbb{P}}^1\times {\mathbb{P}}^1$ and is called the graph of E; see [10], Prop. 3.2.3.

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

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

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

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

Remark 2.

The image $G\left({\mathcal{M}}_n\left(0\right)\right)$ of ${\mathcal{M}}_n\left(0\right)$ 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 $D\in |{\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(n,1)|$ provides 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

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

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

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

and

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

Remark 3.

The set $G({\mathcal{M}}_n\setminus {\mathcal{M}}_n\left(0\right))$ 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 $G\left({\mathcal{M}}_n^{irreg}\left(0\right)\right)$ 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 obtain 4 different “types”of irregular bundles, one for each of the 4 thetas.

For each $A\in G\left({\mathcal{M}}_n^{reg}\left(0\right)\right)$, the fiber $G^{-1}\left(A\right)$ 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 $G:{\mathcal{M}}_n\to {\mathbb{P}}^{2n+1}$ is surjective, $\mathbb{U}:=G\left({\mathcal{M}}_n^{reg}\left(0\right)\right)$ is a Zariski open subset of ${\mathbb{P}}^{2n+1}$, and that, for each $a\in \mathbb{U}$, the set $G^{-1}\left(a\right)$ is isomorphic to an Abelian variety of dimension $2n-1$.

4. Regular V Irregular

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

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

Since being singular is a closed analytic condition, ${\mathcal{M}}_n(\ge 1)$ is a closed analytic subset of ${\mathcal{M}}_n$, while ${\mathcal{M}}_n\left(0\right)$ is an open subset of ${\mathcal{M}}_n$. It is easy to check that ${\mathcal{M}}_n(\ge 1)$ 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 $E\in {\mathcal{M}}_n\left(0\right)$ such that the spectral curve $C_E$ is singular, while ${\mathcal{M}}_n^{sm}$ is the set of all $E\in {\mathcal{M}}_n\left(0\right)$ such that the spectral curve $C_E$ of E is smooth.

Note that ${\mathcal{M}}_n\left(0\right)$ is set of all $E\in {\mathcal{M}}_n$ such that $G\left(E\right)$ 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 ${\left|\mathcal{O}(n,1)\right|}_{sm}$ for the set of such irreducible elements. Observe that, in both cases, $G\left({\mathcal{M}}_n^{reg}\left(0\right)\right)\subset {\left|\mathcal{O}(n,1)\right|}_{sm}$ and $G\left({\mathcal{M}}_n^{irreg}\left(0\right)\right)\subset {\left|\mathcal{O}(n,1)\right|}_{sm}$.

For any $E\in {\mathcal{M}}_n\left(0\right)$, let $C_E$ denote its spectral curve. Let i denote the involution $L\mapsto L^{\ast }$ on $T^{\ast }$. We have $T^{\ast }/i\cong {\mathbb{P}}^1$, and the quotient map $T\to {\mathbb{P}}^1$ is ramified over 4 points, $a_1,a_2,a_3,a_4$, which are the images of the 4 thetas of $T^{\ast }$. We get 4 element ${\mathbb{P}}^1\times \left\{a_i\right\}\in \left|{\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(0,1)\right|$, $i=1,2,3,4$.

Observe that ${\mathcal{M}}_n^{reg}\left(0\right)$ is the set of all $E\in {\mathcal{M}}_n\left(0\right)$ such that $G\left(E\right)$ is transversal to each ${\mathbb{P}}^1\times \left\{a_i\right\}$ for $i=1,2,3,4$; i.e., if $E\in {\mathcal{M}}_n^{reg}\left(0\right)$, then $G\left(E\right)$ intersects $\mathsf{\Sigma }={\mathbb{P}}^1\times \{a_1,a_2,a_3,a_4\}$ at points. Observe also that ${\mathcal{M}}_n^{irreg}\left(0\right)={\mathcal{M}}_n\left(0\right)\setminus {\mathcal{M}}_n^{reg}\left(0\right)$, and that all the elements of ${\mathcal{M}}_n^{irreg}\left(0\right)$ have irreducible graphs.

Theorem 9.

Fix $E\in {\mathcal{M}}_n\left(0\right)$, $n\ge 2$.

(a) 

If $E\in {\mathcal{M}}_n^{reg}\left(0\right)$, then $C_E$ is smooth and connected.

(b) 

If $E\in {\mathcal{M}}_n^{irreg}\left(0\right)$, then $C_E$ is singular and irreducible.

Proof. 

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

For part (b), assume that $C_E$ is not irreducible. Since $G\left(E\right)\cong {\mathbb{P}}^1$ and $C_E\to G\left(E\right)$ 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 $C_E\subset {\mathbb{P}}^1\times T^{\ast }$. Hence, ${\pi }_2\left(J_i\right)$ is a single point. Hence, $G\left(E\right)$, 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 $C_E\to G\left(E\right)$ has at most $4n-1$ ramification points, $C_E$ is singular. □

Lemma 1.

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

Proof. 

Take $x\in {\mathbb{P}}^1$ such that $E_{|{\pi }^{-1}\left(x\right)}$ is an element of $T^{\ast }/i$ associated with a theta; i.e., it corresponds to a point $a\in \mathsf{\Sigma }\cap G\left(E\right)$. We need to prove that $E_{|{\pi }^{-1}\left(x\right)}$ is indecomposable; i.e., that $h^1({\pi }^{-1}\left(x\right),E_{|{\pi }^{-1}\left(x\right)})=1$. The transversality of $G\left(E\right)$ and $\mathsf{\Sigma }$ 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 [11], Remark 2.8. □

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

Lemma 2.

Consider a graph A such that $A=G\left(E\right)$ for some $E\in {\mathcal{M}}_n^{reg}\left(0\right)$. Then, $G^{-1}\left(A\right)$ is isomorphic to the Jacobian $J\left(C_E\right)$ 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 9, $C_E$ is a smooth and connected curve of genus $2n-1$. Hence, its Jacobian $J\left(C_E\right)$ 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 $J\left({\mathbb{P}}^1\right)$ is a singleton, it is sufficient to quote [13], Theorem 5.14. □

5. Irregular Profiles

We fix $\delta \in \mathrm{Pic}\left(X\right)={\mathrm{Pic}}^0\left(X\right)\cong {\mathbb{C}}^{\ast }$. Let $I\subset T^{\ast }$ be the set of all thetas with respect to the involution $i_{\delta }$. 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 with them; see (1). Let ${\pi }_2:{\mathbb{P}}^1\times {\mathbb{P}}^1\to {\mathbb{P}}^1$ denote the projection onto the second factor. Set

$$D_i={\pi }_2^{-1}\left(a_i\right)={\mathbb{P}}^1\times \left\{a_i\right\}.$$

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 $E\in {\mathcal{M}}_n\left(0\right)$; i.e., E has no jumps if and only if $G\left(E\right)\in |{\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}{(n,1)|}_{sm}$. For each $U\in |{\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}{(n,1)|}_{sm}$, we get 4 degree n zero-dimensional schemes

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

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

Remark 5.

Note that E is irregular if and only if at least one of the 4 schemes $Z_{G\left(E\right)}\left(i\right)$, $i=1,2,3,4$, 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 ${\mathcal{H}}_i=G^{-1}\left(H_i\right)$.

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 $n\ge 1$, ${\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(1,1)$ is very ample and $\mathrm{deg}(2p,D_i)=2$; we have

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

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 11 and its proof for more details; it implies for instance that ${\mathcal{H}}_i\ne {\mathcal{H}}_j$ if $i\ne j$). Observe that

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

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 $n\ge 1$, the set $H_i$ is a Zariski open subset of ${\tilde{H}}_i$.

Observe also that, for $n>1$ and $i=1,2,3,4$, we have that ${\mathcal{H}}_i$; the set of all $E\in {\mathcal{M}}_n^{irreg}\left(0\right)$ such that $Z_{G\left(E\right)}\left(i\right)$ 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 $m_1\ge \dots \ge m_x>0$ 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 $G\left(E\right)=D$.
• We say that the integer $\sum (m_i-1)=m_1+\dots +m_x-x$ is the i-th weight of D and of all bundles E with $G\left(E\right)=D$.
• The length ℓ of the profile $D\cap D_i$ and of all bundles E with $G\left(E\right)=D$ is the maximal integer y such that $m_y\ge 2$, 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 $\ge 2$ with ${\sum }_i{m}_i\le n$. Note that the i-th weight is uniquely determined by the reduced profile of $m_1,\dots ,m_s$: it is 0 if $\ell =0$, while it is $-\ell +m_1+\dots +m_{\ell }$ 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 bundle E with no jump, we denote by

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

the multiplicities of the connected components of $Z_{G\left(E\right)}\left(i\right)$ in non-decreasing order. By definition, the ith-weight $w_i\left(E\right)$ of $E\in {\mathcal{M}}_n\left(0\right)$ is the integer

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

and the weight $w\left(E\right)$ of $E\in {\mathcal{M}}_n\left(0\right)$ is the integer

$$w\left(E\right)=w_1\left(E\right)+w_2\left(E\right)+w_3\left(E\right)+w_4\left(E\right).$$

Theorem 10.

$E\in {\mathcal{M}}_n\left(0\right)$ is irregular if and only if $w\left(E\right)>0$.

Proof. 

This is true because ${\mathcal{M}}_n^{irreg}\left(0\right)={\bigcup }_{i=1}^4{\mathcal{H}}_i.$ □

For simplicity, we write $h^0\left({\mathcal{I}}_Z(x,y)\right)$ 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 $E\in {\mathcal{M}}_n\left(0\right)$. Since we always assume that $n\ge 2$, the graph map G is surjective ([10] Thm. 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 $G\left(E\right)=D$. Fix $D\in |{\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}{(n,1)|}_{sm}$ and set $S_i:=D\cap {\mathbb{P}}^1\times \left\{a_i\right\}$, $i=1,2,3,4$. A spectral curve C of a smooth graph is irreducible (Lemma 3). It is a double covering $u:C\to D$ 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 ${\mathbb{P}}^1\times T^{\ast }$ and is smooth outside S. Since $p_a\left(C\right)=2n-1$, the integer $\#S$ is a measure of the singularities of C. Since $u:C\to D$ 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 $u\left(\tilde{p}\right)=p$. Call $m_p$ the multiplicity of p in $D\cap D\times \left\{a_i\right\}$. 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 $C\subset {\mathbb{P}}^1\times T^{\ast }$ 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 $u:C\to D$ generically of degree 2. Since $D\in |{\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(n,1)|\left(0\right)$, 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 $\mathrm{deg}\left(u\right)=2$, 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, $u_{|C_i}:C_i\to D$ is an isomorphism. Thus, $C_i\cong {\mathbb{P}}^1$. We have $C_i\subset {\mathbb{P}}^1\times T^{\ast }$. Since $C_i\cong {\mathbb{P}}^1$ and $T^{\ast }$ is a smooth elliptic curve, the restriction to $C_i$ of the projection ${\mathbb{P}}^1\times T^{\ast }\to T^{\ast }$ is constant. Call $o_i$ its image. Thus, $C_i={\mathbb{P}}^1\times \left\{o_i\right\}$. Since $u:C\to D$ is generically unramified, $o_1\ne o_2$. Hence, $C_1$ and $C_2$ are connected components, contradicting the fact that $D\cap {\mathbb{P}}^1\times \left\{o_1\right\}\ne \varnothing$, and hence u has at least one ramification point. □

Notation 4.

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

We now explore the equalities ${\mathcal{M}}_n^{sm}={\mathcal{M}}_n^{reg}\left(0\right)$ and ${\mathcal{M}}_n^{sg}={\mathcal{M}}_n^{irreg}\left(0\right)$ proved in Section 4.

Theorem 11.

Assume $n>1$. For all $E\in {\mathcal{M}}_n^{irreg}\left(0\right)$, we have

1. 

$w\left(E\right)\le 2n-2$.

2. 

E$(n,0,\dots ,0)$ has its i-th profile if and only if $w_i\left(E\right)\le n-1$ for all i and $w_i\left(E\right)=n-1$.

3. 

Fix $i\in \{1,2,3,4\}$, a positive integer ℓ, and a restricted profile $(m_1,\dots ,m_{\ell })$. There exists a non-empty locally closed analytic subset $\Gamma \in {\mathcal{M}}_n^{irreg}\left(0\right)$ such that all $G\left(E\right)\in |{\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(n,1)|$ have $(m_1,\dots ,m_{\ell })$ as i-th profile, and this set of graphs $G\left(E\right)$ has codimension at most $-\ell +m_1+\dots +m_{\ell }$ 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 ${\pi }_2:{\mathbb{P}}^1\times {\mathbb{P}}^1\to {\mathbb{P}}^1$ induces a degree n morphism $f:D\to {\mathbb{P}}^1$. Since $D\cong {\mathbb{P}}^1$, the Riemann–Hurwitz formula provides that, counting multiplicities, the ramification divisor $\mathcal{R}\subset D$ is an effective divisor of degree $2n-2$ ([16], pp. 216–219, [17], Cor. IV.2.4). Fix $p\in D$. Since D is irreducible and of bidegree $(n,1)$, the scheme $Z_p:=D\cap {\pi }_2^{-1}\left(f\left(p\right)\right)$ has degree n. Note that $p\in {\mathcal{R}}_{red}$ if and only 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 $w\left(E\right)$ only counts the contribution of 4 of the fibers of f, we obtain (1) and (2).

Now we hold that G is surjective [10], Theorem 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 ℓ distinct points $p_1,\dots ,p_{\ell }$. Let $Z\subset D_i$ be the connected zero-dimensional subscheme of $D_i$ with $Z_{red}=\{p_1,\dots ,p_s\}$, and the connected component of Z with $p_h$ as its reduction has degree $m_h$. Set $m:=m_1+\dots +m_{\ell }$.

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

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

(2)

The Künneth formula indicates that $h^1\left({\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(n-1,1)\right)=0$. Since $\mathrm{deg}\left(Z\right)\le n$, $D_i\cong {\mathbb{P}}^1$ and $\mathrm{deg}\left({\mathcal{O}}_{D_i}(n,1)\right)=n$, 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 $D_j$, $j\ne i$, 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 $\{p_1,\dots ,p_{\ell }\}=\mathcal{B}$ ([16], p. 137, [17], III.10.9, [18], 6.3); i.e., that $h^0\left({\mathcal{I}}_{Z\cup \left\{q\right\}}(n,1)\right)=h^0\left({\mathcal{I}}_Z(n,1)\right)-1$ 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

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

(3)

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

(b2) Take $q\in D_i\setminus \{p_1,\dots ,p_{\ell }$. Since $Z\cup \left\{q\right\}\subset D_i$, we have an exact sequence

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

(4)

Since $\mathrm{deg}(Z\cup \{q\left\}\right)=\mathrm{deg}\left(Z\right)+1\le n+1$, $D_i\cong {\mathbb{P}}^1$ and $\mathrm{deg}\left({\mathcal{O}}_{D_i}(n,1)\right)=n$, we have $h^1(D_i,{\mathcal{I}}_{Z\cup \left\{q\right\},D_i}(n,1))=0$. 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, $B\cap \{p_1,\dots ,p_{\ell }\}=\varnothing$. 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 $D_j$, $j\ne i$. 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 $D_j$, $j\ne i$. 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 $D_i\cap D_j=\varnothing$, 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 $\{p_1,\dots ,p_{\ell }\}$ ([16], p. 137, [17], III.10.9, [18], 6.3). Hence, D is transversal outside $\{p_1,\dots ,p_{\ell }\}$; i.e., the restricted i-th profile of D has exactly ℓ 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 $\le m_h$ 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 $Z^{\prime }=Z\cup Z\left((m_h+1)p\right)$. Since $Z\left(m_h{p}_h\right)\subseteq Z$, we have $\mathrm{deg}\left(Z^{\prime }\right)=\mathrm{deg}\left(Z\right)+1$. As in step (b2), we see that $h^1\left({\mathcal{I}}_{Z^{\prime }}(n,1)\right)=0$. 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 obtain that the set of all $D\in |{\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}{(n,1)|}_{sm}$ containing Z is a non-empty subset of codimension m, that a general element D of it has the property that $w_j\left(D\right)=0$ for all $j\ne i$, and $m_1,\dots ,m_{\ell }$ is the restricted i-profile of D. Note that 2 elements $D,D^{\prime }\in {\left|{\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(n,1)\right|}_{sm}$ such that $D\cap D_i\ne D^{\prime }\cap D_i$ are distinct. Hence, taking the union for all choices of s distinct points of $D_i$, we obtain item (3). □

6. Ramifications and Weights

Proposition 1.

Fix an integer $n\ge 2$. 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 $n\ge 2$, step (b2) of the proof of Theorem 11 with J instead of $D_i$ yields $h^1\left({\mathcal{I}}_Z(n,1)\right)=0$. 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 obtain 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 $2\le n\le 3$, 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 $n\ge 4$. Fix $q\in J$ such that $q\ne p$, 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 $n\ge 3$, the proof of step (b2) of Theorem 12 below provides $h^1\left({\mathcal{I}}_A(n,1)\right)=0$; 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 12.

Fix an ordering $i_1,i_2,i_3,i_4$ of the set $\{1,2,3,4\}$.

(a) 

For all $n\ge 2$, there exists a locally closed analytic subset $\Gamma \subset {\mathcal{M}}_n\left(0\right)$ such that $w_{i_1}\left(E\right)=w_{i_2}\left(E\right)=1$ and $w_{i_3}\left(E\right)=w_{i_4}\left(E\right)=0$ for all $E\in \Gamma$, and $G(\Gamma )$ has codimension 2 in $|{\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(n,1)|$.

(b) 

For all $n\ge 4$, there exists a locally closed analytic subset $\Delta \subset {\mathcal{M}}_n\left(0\right)$ such that $w_{i_1}\left(E\right)=w_{i_2}\left(E\right)=w_{i_3}\left(E\right)=1$ and $w_{i_4}\left(E\right)=0$ for all $E\in \Delta$, and $G(\Delta )$ has codimension 3 in $|{\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(n,1)|$.

Proof of Theorem 12.

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 [10], Theorem 5.2.2, holds that it is sufficient to find ${\Gamma }_1\subset \left|{\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(n,1)\right|$, $n\ge 2$, (resp. ${\Delta }_1\subset \left|{\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(n,1)\right|$, $n\ge 4$) such that $dim{\Gamma }_1=dim\left|{\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(n,1)\right|-2$ (resp. $dim{\Delta }_1=dim\left|{\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(n,1)\right|-3$) 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 $\mathrm{Aut}\left({\mathbb{P}}^1\right)$ 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 $\gamma \in \mathrm{Aut}\left({\mathbb{P}}^1\right)$ such that $\gamma \left(b_j\right)=c_j$ for all j.

(a) In this step, we prove part (a). We saw that there is ${\alpha }_2\in \mathrm{Aut}\left({\mathbb{P}}^1\right)$ such that ${\alpha }_2\left(a_1\right)={\pi }_2\left(D_1\right)$ and ${\alpha }_2\left(a_2\right)={\pi }_2\left(D_2\right)$. 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 $G\left(E\right)=\alpha \left(A\right)$, we have $w_1\left(E\right)=w_2\left(E\right)=1$ because all ramification points of $D:=\alpha \left(A\right)$ 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 $w_3\left(E\right)=w_4\left(E\right)=0$ because $w\left(E\right)\le 2$ for all $E\in {\mathcal{M}}_2\left(0\right)$ (Theorem 11). However, for $n>2$ to result in $w_3\left(E\right)=w_4\left(E\right)=0$, 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

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

(5)

The Künneth formula indicates that $h^1\left({\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}\right)=0$. Since $\mathrm{deg}\left(Z\right)=4$, $D\cong {\mathbb{P}}^1$, $n\ge 2$, and $\mathrm{deg}\left({\mathcal{O}}_D(n,1)\right)=2n$, 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 [17], Theorem III.13.8, we have $h^1\left({\mathcal{I}}_{Z(p^{\prime },q^{\prime })}(n,1)\right)=0$ 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 $\Gamma \left(j\right)$ 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, $\Gamma \left(j\right)$ is Zariski open in ${\Gamma }_2$. Note that any bundle E with $G\left(E\right)\in \Gamma \left(3\right)\cap \Gamma \left(4\right)$ satisfies $w_1\left(E\right)=w_2\left(E\right)=1$ and $w_3\left(E\right)=w_4\left(E\right)=0$. Assume for the moment $\Gamma \left(3\right)\ne \varnothing$ and $\Gamma \left(4\right)\ne \varnothing$. 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 $\Gamma \left(3\right)\cap \Gamma \left(4\right)$. Hence, part (a) holds if $\Gamma \left(3\right)\ne \varnothing$ and $\Gamma \left(4\right)\ne \varnothing$. Assume for instance $\Gamma \left(3\right)=\varnothing$; 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 $W:=Z\cup (2a,D_3)$. p, q, and a are ramification points of D, $W\subset D$. Hence, there is an exact sequence of sheaves

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

(6)

The Künneth formula indicates that $h^1\left({\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}\right)=0$. Since $\mathrm{deg}\left(Z\right)=4$, $D\cong {\mathbb{P}}^1$, $n\ge 3$, and $\mathrm{deg}\left({\mathcal{O}}_D(n,1)\right)=2n$, 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 obtain that ${\Gamma }_2\setminus \Gamma \left(3\right)$ has codimension at least 1 in ${\Gamma }_2$. Hence, $\Gamma \left(3\right)\ne \varnothing$, contradicting one of our assumptions.

(b) In this step, we prove part (b). Let ${\gamma }_2$ be the only element of $\mathrm{Aut}\left({\mathbb{P}}^1\right)$ such that ${\gamma }_2\left(a_1\right)={\pi }_2\left(D_1\right)$, ${\gamma }_2\left(a_2\right)={\pi }_2\left(D_2\right)$, and ${\gamma }_2\left(a_3\right)={\pi }_2\left(D_3\right)$. 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 $X:=\gamma \left(A\right)$.

Let p (resp. q, resp a) denote the contact locus of X and $D_1$ (resp. $D_2$, resp. $D_3$). Note that $w_1\left(E\right)=w_2\left(E\right)=w_3\left(E\right)=1$ for all bundles E such that $G\left(E\right)=X$. 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

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

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 $h^1\left({\mathcal{I}}_W(n,1)\right)=0$; i.e., $dim|{\mathcal{I}}_W(n,1)|=dim|{\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(n,1)|-6$. For any $(p^{\prime },q^{\prime },a^{\prime })\in D_1\times D_2\times D_3$, set

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

By the semicontinuity theorem for cohomology ([17], Thm. III.13.8), we have $h^1\left({\mathcal{I}}_{W(p^{\prime },q^{\prime },a^{\prime })}(n,1)\right)=0$; 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 ${\mathcal{M}}_n\left(0\right)$ such that $w_1\left(E\right)=w_2\left(E\right)=w_3\left(E\right)=1$.

To prove step (b), we need to prove that, outside a lower-dimensional analytic subset of ${\Delta }^{\prime }$, the bundles satisfy $w_4\left(E\right)=0$; i.e., their graph is transversal to $D_4$. As in step (a), we have a locally closed irreducible subset ${\Delta }_2$ of $|{\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(n,1)|\left(0\right)$ 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 are 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 $\mathrm{deg}\left(W^{\prime }\right)=8$ and $W^{\prime }\subset X$. We obtain an exact sequence of sheaves similar to (6) with X instead of D and $W^{\prime }$ instead of X. Since $n\ge 4$, $X\cong {\mathbb{P}}^1$, $\mathrm{deg}\left(W^{\prime }\right)=8$, and $\mathrm{deg}\left({\mathcal{O}}_X(n,1)\right)=2n$, we have $h^1(X,{\mathcal{I}}_{W^{\prime },X}(n,1))=0$ and hence $h^0\left({\mathcal{I}}_{W^{\prime }}(n,1)\right)=h^0({\mathcal{I}}_W(n,1)-2$. Since $dim{D}_4=1$, repeating the last part of step (a), we obtain a contradiction. □

Example 1.

Take $n=2$. Let E be any irregular bundle. By Theorem 11, we have $1\le w\left(E\right)\le 2$ and $w_i\left(E\right)\le 1$ for all i. Theorems 11 and 12 show that, for all possible quadruples $(x_1,x_2,x_3,x_4)$ of integers with $0\le x_i\le 1$ and $1\le x_1+x_2+x_3+x_4\le 2$, there is $E\in {\mathcal{M}}_{\delta ,2}$ such that $w_i\left(E\right)=x_i$ for all i. We also found 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 13.

Fix an integer $n\ge 2$. 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 Γ of ${\mathcal{M}}_n\left(0\right)$ such that $dimG(\Gamma )=3$, and for all $E\in \Gamma$ the restricted $i_1$-th and $i_2$-th profiles of E are $\left(n\right)$, while $w_{i_3}\left(E\right)=w_{i_4}\left(E\right)=0$.

Proof. 

Since the general case is similar, we only complete the case $i_j=j$ for $j=1,2,3,4$. Note that, if $E\in {\mathcal{M}}_{\delta ,n}\left(0\right)$ has restricted profile $\left(n\right)$ for $D_1$ and $D_2$, then $w_1\left(E\right)=w_2\left(E\right)=n-1$, and hence $w_3\left(E\right)=w_4\left(E\right)=0$ by Theorem 11. Hence, we do not have the most difficult part of the proof of Theorems 11 and 12, transversality with respect to some $D_j$.

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

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 ${\Gamma }_1\subset \left|{\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(n,1)\right|$ such that each $D\in {\Gamma }_1$ has profile $\left(n\right)$ 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 $p_1\ne q_1$. For any integer $x>0$, let $(xp,D_1)$ (resp. $(xq,D_2)$) denote the connected and degree x zero-dimensional subscheme of $D_1$ (resp. $D_2$) with p (resp. q) as its reduction. Set $Z:=(np,D_1)\cup (nq,D_2)$. We have $\mathrm{deg}\left(Z\right)=2n$, and hence $dim|{\mathcal{I}}_Z(n,1)|\ge dim|{\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(n,1)|-2n=1$. 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 $a\in D_1\setminus \left\{p\right\}$ and $b\in D_2\setminus \left\{q\right\}$. 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 $p_1\ne q_1$, we have $R_1\ne R_2$, $q\notin R_1$, 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 $h^0\left({\mathcal{I}}_Z(n,1)\right)=2$; i.e., that $dim|{\mathcal{I}}_Z(n,1)|=1$. To prove that $h^0\left({\mathcal{I}}_Z(n,1)\right)=2$, it is sufficient to prove that $h^0\left({\mathcal{I}}_{Z\cup \{a,b\}}(n,1)\right)=0$. Assume $h^0\left({\mathcal{I}}_{Z\cup \{a,b\}}(n,1)\right)>0$ and take $X\in |{\mathcal{I}}_{Z\cup \{a,b\}}(n,1)|$.

Since $\left\{a\right\}\cup (np,D_1)$ and $\left\{b\right\}\cup (nq,D_2)$ have degree $n+1$ and since $D_1$ and $D_2$ are irreducible, the theorem of Bezout yields $D_1\cup D_2\subseteq X$. Since $h^0\left({\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(n,1)(-D_1-D_2)\right)=h^0\left({\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(n,-1)\right)=0$, we get a contradiction.

(b) By the theorem of Bertini ([16], p. 137, [17], III.10.9, [18], 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 $p_1\ne q_1$ by assumption. Since $D_1\cap D_2=\varnothing$, we have $p_2\ne q_2$. Recall that $\mathrm{Aut}\left({\mathbb{P}}^1\right)$ 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 $h\in \mathrm{Aut}\left({\mathbb{P}}^1\right)$ such that $h\left(b_i\right)=c_i$ for all $i=1,2,3$. Hence, the subset $\mathbb{G}$ of all $u\in \mathrm{Aut}\left({\mathbb{P}}^1\right)\times \mathrm{Aut}\left({\mathbb{P}}^1\right)$ such that $u\left(p\right)=p$ and $u\left(q\right)=q$ is an algebraic linear group isomorphic to ${\left({\mathbb{C}}^{\ast }\right)}^2$ whose actions on ${\mathbb{P}}^1\times {\mathbb{P}}^1$ have the following 9 orbits:

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

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

Assume $\mathcal{B}\ne \{p,q\}$, 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 $h^0\left({\mathcal{I}}_Z(n,1)\right)=0$, a contradiction.

Now assume $U_z=u$; i.e., $z=u$. Hence, $h^0\left({\mathcal{I}}_{Z\cup \left\{u\right\}}(n,1)\right)=h^0\left({\mathcal{I}}_Z(n,1)\right)=2$. Since $\mathrm{deg}(D_1\cap (Z\cup \left\{u\right\})=n+1$, the theorem of Bezout yields $D_1\subset \mathcal{B}$. 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 $h^0\left({\mathcal{I}}_Z(n,1)\right)=h^0\left({\mathcal{I}}_{(nq,D_2)}(n,0)\right)$. Since the scheme ${\pi }_1\left((nq,D_1)\right)$ is the degree n zero-dimensional subscheme of ${\mathbb{P}}^1$ with $q_1$ as its reduction, we have $h^0\left({\mathcal{I}}_{(nq,D_2)}(n,0)\right)=h^0({\mathbb{P}}^1,{\mathcal{I}}_{{\pi }_1((nq,D_1)}(n,0))=h^0({\mathbb{P}}^1,{\mathcal{O}}_{{\mathbb{P}}^1})=1$. Hence, $h^0\left({\mathcal{I}}_Z(n,1)\right)=1$, 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 $D^{\prime }\in \left|{\mathcal{I}}_Z(n,1)\right|$ is smooth at p and a general $D^{\prime \prime }\in \left|{\mathcal{I}}_Z(n,1)\right|$ is smooth at q. We only prove that a general $D^{\prime }\in \left|{\mathcal{I}}_Z(n,1)\right|$ is smooth at p since the smoothness at q only requires notational modifications. Assume that a general $D^{\prime }\in \left|{\mathcal{I}}_Z(n,1)\right|$ is singular at p. Let $(2p,R_1)$ denote the degree 2 zero-dimensional subscheme of $R_1$ with p as its reduction. Set $W=Z\cup (2p,R_1)$. Since a general $D^{\prime }\in \left|{\mathcal{I}}_Z(n,1)\right|$ is singular at p, $W\subset D^{\prime }$, and hence $h^0\left({\mathcal{I}}_W(n,1)\right)=h^0\left({\mathcal{I}}_Z(n,1)\right)=2$. 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). □

7. The Topology of ${\mathcal{M}}_n^{reg}\left(0\right)$

Fix an integer $n\ge 2$. Recall that $dim|{\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(n,1)|=2n+1$. Let ${\mathcal{C}}_n\subset \left|{\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(n,1)\right|$ 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 an element of $|{\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(n-1,1)|$. The set ${\mathcal{C}}_n$ is irreducible [19], Ch. 1, and hence it is given by a unique equation. Therefore, the set ${\mathcal{B}}_n:=\left|{\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(n,1)\right|\setminus {\mathcal{C}}_n$ is a smooth and connected affine variety of complex dimension $2n+1$.

By [20,21], we obtain that ${\mathcal{B}}_n$ is homotopy equivalent to a finite CW complex of real dimension at most $2n+1$. Let $H_1\cup H_2\cup H_3\cup H_4$ be the set of smooth but irregular graphs, i.e., (since $n\ge 2$) the images of ${\mathcal{M}}_n^{irreg}\left(0\right)$. Hence,

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

and using [20,21], we find 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 $G:{\mathcal{M}}_n^{reg}\left(0\right)\to {\mathcal{A}}_n$ is a smooth submersion whose fibers are compact differential manifolds diffeomorphic to ${\left(S^1\right)}^{4n-2}$, where $S^1$ is the unit circle. It then follows that the cohomology ring $H^{\ast }({\mathcal{M}}_n^{reg},\mathbb{Z})$ is the tensor product of the cohomology ring $H^{\ast }({\mathcal{A}}_n,\mathbb{Z})$ and the cohomology ring $H^{\ast }({\left(S^1\right)}^{4n-2},\mathbb{Z})$. We restate this fact as a theorem.

Theorem 14.

Fix an integer $n\ge 2$. We have

(i) 

$H^{\ast }({\mathcal{M}}_n^{reg}\left(0\right),\mathbb{Z})=H^{\ast }({\mathcal{A}}_n,\mathbb{Z})\otimes H^{\ast }({\left(S^1\right)}^{4n-2},\mathbb{Z}).$

(ii) 

$H^i({\mathcal{M}}_n^{reg}\left(0\right),\mathbb{Z})=0$ for all $i\ge 6n$.

Proof. 

The graph map $G:{\mathcal{M}}_n^{reg}\left(0\right)\to {\mathcal{A}}_n$ 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 ${\left(S^1\right)}^{4n-2}$. 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 a Euclidean open subset U such that $G^{-1}\left(U\right)$ is fiberwise diffeomorphic to $U\times {\left(S^1\right)}^{4n-2}$; i.e., it commutes with the projection $U\times {\left(S^1\right)}^{4n-2}\to U$. 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 [22], 17.8.1).

Since ${\mathcal{A}}_n\left(reg\right)$ is an affine variety, it has the homotopy type of a finite CW-complex of (real!) dimension at most $dim{\mathcal{A}}_n=2n+1$ [20,21], $H^i({\mathcal{A}}_n,\mathbb{Z})=0$ for all $i\ge 2n+2$. 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 [23], Cor. 2.3.4, or [24], Th. I.5.2). □

Obviously, Theorem 14 may be extended to other coefficient Abelian groups instead of $\mathbb{Z}$, just quoting [23,24]

Theorem 15.

Let $Y\subset {\mathcal{M}}_n\left(0\right)$ be an irreducible compact complex-analytic subspace. Then, $G\left(Y\right)$ is a single point.

Proof. 

The set $G\left(Y\right)$ is an irreducible and compact complex space contained in the affine variety ${\mathcal{B}}_n$, parameterizing all smooth graphs. Thus, $G\left(Y\right)$ is a point. □

Corollary 2.

Let $Y\subset {\mathcal{M}}_n^{reg}\left(0\right)$ 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 15, Y is contained in a fiber of the graph map. Since $G^{-1}\left(G\left(y\right)\right)$ is a compact torus of complex dimension $2n-1$, we obtain the corollary. □

8. Nontrivial Determinants

Let $\delta \in {\mathbb{C}}^{\ast }$. 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 ${\mathcal{M}}_n={\mathcal{M}}_{n,{\mathcal{O}}_X}$ 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 yields $h^2\left(End\left(E\right)\right)=h^0(End\left(E\right)\otimes {\mathcal{O}}_X(-2))=0$. Hence, the local deformation space of E is smooth of dimension $h^1(X,End\left(E\right))-1$. So, ${\mathcal{M}}_{\delta ,E}$ is smooth and equidimensional, and (Riemann–Roch) $dim{\mathcal{M}}_n=4n$.

Theorem 16.

For any $\delta \in {\mathbb{C}}^{\ast }$ and all $n\ge 1$, 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 $c\in {\mathbb{C}}^{\ast }$ such that $c^2=\delta$. Let $v:\mathcal{E}\to X\times S$ 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 ${\pi }_1:X\times S\to X$ denote the projection. The family $\mathcal{E}\otimes {\pi }_1^{\ast }\left({\mathcal{O}}_X\left(c\right)\right)$ 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 $f_v:S\to {\mathcal{M}}_{n,\delta }$, which to each s in S assigns the bundle $E\otimes {\pi }_1^{\ast }\left({\mathcal{O}}_X\left(c\right)\right)$. 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. □

Therefore, we conclude that the theorems proved here all have analogous statements for the cases of nontrivial determinants.

9. Conclusions

The group $T^{\ast }={Pic}^0\left(T\right)$ of degree-zero line bundles on the torus T contains four special elements, namely those line bundles satisfying $L=L^{\ast }$; hence, $L^2={\mathcal{O}}_T$. Since the Hopf surface X is fibered by tori, these four thetas play a significant role in the geometry of the moduli space of rank 2 bundles on X. Each vector bundle E on X has a corresponding spectral curve in ${\mathbb{P}}^1\times T^{\ast }$, which descends to a graph in ${\mathbb{P}}^1\times {\mathbb{P}}^1$. Using such graphs, we introduced the notions of irregular profile and weight for each vector bundle E on X. Having nonzero weight $w\left(E\right)=w_1\left(E\right)+w_2\left(E\right)+w_3\left(E\right)+w_4\left(E\right)>0$ occurs precisely when the bundle E is irregular.

The goal of this work was to describe the nontrivial features of the subset ${\mathcal{M}}_n\left(0\right)\subset {\mathcal{M}}_n$ formed by vector bundles on X with $c_2=n$ that have no jumps. We proved the existence of a hypersurface of ${\mathcal{M}}_n\left(0\right)$ formed by irregular bundles, cut out by the condition $w>0$, and having at least four irreducible components (one for each $w_i\ne 0$). Similarly, imposing 2 (resp. 3) nonzero $w_i$s, we obtained subsets of graphs of codimension 2 (resp. 3).

We then considered the codimension 1 set of graphs ${\mathcal{C}}_n\subset \left|{\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(n,1)\right|$ formed by all $A\cup B$ with $A\in |{\mathcal{O}}_{{\mathbb{P}}^1}(1,0)|$ and B as an element of $|{\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(n-1,1)|$. Since ${\mathcal{C}}_n$ is irreducible, it is provided by a unique equation. Thus, the set ${\mathcal{B}}_n:=\left|{\mathcal{O}}_{{\mathbb{P}}^1\times {\mathbb{P}}^1}(n,1)\right|\setminus {\mathcal{C}}_n$ is a smooth and connected affine variety of complex dimension $2n+1$; consequently, ${\mathcal{B}}_n$ is homotopy equivalent to a finite CW-complex of real dimension at most $2n+1$. Using this observation, we computed the cohomology ring of ${\mathcal{M}}_n^{reg}\left(0\right)$, in particular obtaining that any irreducible compact complex-analytic subspace of ${\mathcal{M}}_n\left(0\right)$ consists of a single point.

We think that the tools and ideas we used and improved may be employed for other elliptic fibrations. Among the possible targets for non-Kälher surfaces, the easiest should be primary Kodaira surfaces. Among the Käler surfaces, the most important ones for physicists are the K3 surfaces because they are Calabi–Yau. We hope that our tools provide the existence and descriptions of irregular stable bundles.