Vector bundles and finite covers

Abstract

Motivated by the problem of finding algebraic constructions of finite coverings in commutative algebra, the Steinitz realization problem in number theory and the study of Hurwitz spaces in algebraic geometry, we investigate the vector bundles underlying the structure sheaf of a finite flat branched covering. We prove that, up to a twist, every vector bundle on a smooth projective curve arises from the direct image of the structure sheaf of a smooth, connected branched cover.

1 Introduction

Associated to a finite flat morphism $\phi {\colon } X \to Y$ is the vector bundle $\phi _* {\mathcal O}_X$ on Y. This is the bundle whose fibre over $y \in Y$ is the vector space of functions on $\phi ^{-1}(y)$ . In this paper, we address the following basic question: which vector bundles on a given Y arise in this way? We are particularly interested in cases where X and Y are smooth projective varieties.

Our main result is that, up to a twist, every vector bundle on a smooth projective curve Y arises from a branched cover $X \to Y$ , with smooth projective X. Let d be a positive integer, and let k be an algebraically closed field with $\operatorname {char} k = 0$ or $\operatorname {char} k> d$ .

Theorem 1.1 (Main)

Let Y be a smooth projective curve over k, and let E be a vector bundle of rank $(d-1)$ on Y. There exists an integer n (depending on E) such that for any line bundle L on Y of degree at least n, there exists a smooth curve X and a finite map $\phi {\colon } X \to Y$ of degree d such that $\phi _* {\mathcal O}_X$ is isomorphic to ${\mathcal O}_Y \oplus E^{\vee } \otimes L^{\vee }$ .

The reason for the ${\mathcal O}_Y$ summand is as follows. Pull-back of functions gives a map ${\mathcal O}_Y \to \phi _* {\mathcal O}_X$ , which admits a splitting by $1/d$ times the trace map. Therefore, every bundle of the form $\phi _* {\mathcal O}_X$ contains ${\mathcal O}_Y$ as a direct summand. The dual of the remaining direct summand is called the Tschirnhausen bundle and is denoted by $E = E_\phi $ (the dual is taken as a convention.) Theorem 1.1 says that on a smooth projective curve, a sufficiently positive twist of every vector bundle is Tschirnhausen.

The reason for needing the twist is a bit more subtle and arises from some geometric restrictions on Tschirnhausen bundles. For $Y = \mathbf {P}^n$ and a smooth X, the Tschirnhausen bundle E is ample by a result of Lazarsfeld [27]. For more general Y and smooth X, it enjoys several positivity properties, as shown in [32, 33]. The precise necessary and sufficient conditions for being Tschirnhausen (without the twist) are unknown and seem to be delicate even when $Y = \mathbf { P}^1$ .

The attempt at extending Theorem 1.1 to higher-dimensional varieties Y presents interesting new challenges. We discuss them through some examples in Section 4. As it stands, the analogue of Theorem 1.1 for higher-dimensional varieties Y is false. We end the paper by posing modifications for which we are unable to find counterexamples.

1.1 Motivation and related work

The question of understanding the vector bundles associated to finite covers arises in many different contexts. We explain three main motivations below.

1.1.1 The realization problem for finite covers

Given a space Y and a positive integer d, a basic question in algebraic geometry is to find algebraic constructions of all possible degree d branched coverings of Y. The prototypical example occurs when $d = 2$ . A double cover $X \to Y$ is given as $X = \operatorname {Spec}({\mathcal O}_Y \oplus L^\vee )$ , where L is a line bundle on Y, and the algebra structure on ${\mathcal O}_Y \oplus L^\vee $ is specified by a map $L^{\otimes -2} \to {\mathcal O}_Y$ of ${\mathcal O}_Y$ -modules. In other words, the data of a double cover consists of a line bundle L and a section of $L^{\otimes 2}$ . In general, a degree d cover $X \to Y$ is given as $X = \operatorname {Spec} ({\mathcal O}_Y \oplus E^\vee )$ , where E is a vector bundle on Y of rank $(d-1)$ . The specification of the algebra structure, however, is much less obvious. For higher d, it is far from clear that simple linear algebraic data determines an algebra structure. In fact, given an E, it is not clear whether there exists a (regular/normal/Cohen-Macaulay) ${\mathcal O}_Y$ -algebra structure on ${\mathcal O}_Y \oplus E^\vee $ : that is, whether E can be realized as the Tschirnhausen bundle of a cover $\phi {\colon } X \to Y$ for some (regular/normal/Cohen-Macaulay) X. We call this the realization problem for Tschirnhausen bundles.

For $d = 3, 4$ and $5$ , theorems of Miranda, Casnati and Ekedahl provide a linear algebraic description of degree d coverings of Y in terms of vector bundles on Y [28, 12]. These descriptions give a direct method for attacking the realization problem for d up to $5$ . For $d \geq 6$ , however, no such description is known, and finding one is a difficult open problem. Theorem 1.1 solves the realization problem for all d up to twisting by a line bundle, circumventing the lack of effective structure theorems.

The realization problem has attracted the attention of several mathematicians, even in the simplest non-trivial case, namely where $Y = \mathbf {P}^1$ [30, 14, 35, 4]. Historically, this problem for $Y = \mathbf {P}^1$ is known as the problem of classifying scrollar invariants. Recall that every vector bundle on $\mathbf {P}^1$ splits as a direct sum of line bundles. Suppose $\phi {\colon } X \to Y = \mathbf {P}^1$ is a branched cover with X smooth and connected. Writing $E_\phi = {\mathcal O}(a_{1}) \oplus \dots \oplus {\mathcal O}(a_{d-1})$ , the scrollar invariants of $\phi $ are the integers $a_1, \dots , a_{d-1}$ . For $d = 2$ , any positive integer $a_1$ is realized as a scrollar invariant of a smooth double cover. For $d = 3$ , a pair of positive integers $(a_1, a_2)$ with $a_1 \leq a_2$ is realized as scrollar invariants of smooth triple coverings if and only if $a_2\leq 2a_1$ [28, § 9]. Although it may be possible to use the structure theorems to settle the cases of $d = 4$ and $5$ , such direct attacks are infeasible for $d \geq 6$ . Nevertheless, the picture emerging from the collective work of several authors [14, 30], and visible in the $d = 3$ case, indicates that if the $a_i$ are too far apart, then they cannot be scrollar invariants.

Theorem 1.1 specialized to $Y = \mathbf {P}^1$ says that the picture is the cleanest possible if we allow twisting by a line bundle.

Corollary 1.2. Let $a_1, \dots , a_{d-1}$ be integers. For every sufficiently large c, the integers $a_1 + c, \dots , a_{d-1}+c$ can be realized as scrollar invariants of $\phi {\colon } X \to \mathbf { P}^1$ , where X is a smooth projective curve.

Before our work, the work of Ballico [4] came closest to a characterization of scrollar invariants up to a shift. He showed that one can arbitrarily specify the smallest $d/2$ of the $(d-1)$ scrollar invariants. Corollary 1.2 answers the question completely: one can in fact arbitrarily specify all of them.

1.1.2 Arithmetic analogues

The realization problem of Tschirnhausen bundles is a well-studied and difficult open problem in number theory. When $\phi {\colon } \operatorname {Spec} {\mathcal O}_L \to \operatorname {Spec} {\mathcal O}_K$ is the map corresponding to the extension of rings of integers of number fields $L/K$ , the isomorphism class of $E_\phi $ is encoded by its Steinitz class, which is the ideal class $\det E \in \operatorname {Cl} (K)$ . Indeed, the structure theorem of projective modules over a Dedekind domain [37] says that every projective module E of rank $(d-1)$ is isomorphic to ${\mathcal O}_K^{d-2} \oplus \det E$ as an ${\mathcal O}_K$ -module. A long-standing unsolved problem in number theory is to prove that, for each fixed degree $d \geq 2$ , every element of the class group is realized as the Steinitz class of some degree d extension of K. The first cases ( $d \leq 5$ ) of this problem follow from the work of Bhargava, Shankar and Wang [9, Theorem 4]. In general, the realization problem for Steinitz classes is open, with progress under various conditions on the Galois group; see [10] and the references therein.

Theorem 1.1 completely answers the complex function field analogue of the realization problem for Steinitz classes.

Corollary 1.3. Suppose Y is a smooth affine curve, and $I \in \operatorname {Pic}(Y)$ . Then I is realized as the Steinitz class of a degree d covering $\phi {\colon } X \to Y$ , with X smooth and connected. That is, there exists $\phi {\colon } X \to Y$ , with X smooth and connected such that

$$\begin{align*}E_\phi \cong {\mathcal O}_Y^{d-2} \oplus I.\end{align*}$$

Proof. Extend E to a vector bundle $E'$ on the smooth projective compactification $Y'$ of Y. Apply Theorem 1.1 to $E'$ , twisting by a sufficiently positive line bundle L on $Y'$ whose divisor class is supported on the complement $Y' \setminus Y$ . We obtain a smooth curve $X'$ and a map $\phi {\colon } X' \to Y'$ whose Tschirnhausen bundle is $E' \otimes L$ ; letting $X = \phi ^{-1}(Y)$ , we obtain the corollary.

We note that the affine covers in the above corollary have full $(S_d)$ monodromy groups, as can easily be deduced from the method of proof of Theorem 1.1.

The analogy between the arithmetic and the geometric realization problems discussed above for affine curves extends further to projective curves, provided we interpret the projective closure of an arithmetic curve like $\operatorname {Spec} {\mathcal O}_K$ in the sense of Arakelov geometry [39]. For simplicity, take $K = \mathbf {Q}$ and $Y = \mathbf {P}^1$ . A vector bundle on a ‘projective closure’ of $\operatorname {Spec} \mathbf {Z}$ in the Arakelov sense is a free $\mathbf {Z}$ -module E with a Hermitian form on its complex fibre $E \otimes \mathbf {C}$ . Let $L/\mathbf {Q}$ be an extension of degree d. The Tschirnhausen bundle $E_\phi $ of $\phi {\colon } \operatorname {Spec} {\mathcal O}_L \to \operatorname {Spec} {\mathcal O}_K$ is naturally an Arakelov bundle, where the Hermitian form is induced by the trace. Thus, the realization problem has a natural interpretation in the Arakelov sense. An Arakelov bundle over $\operatorname {Spec} \mathbf {Z}$ of rank r is just a lattice of rank r, and the set of such lattices (up to isomorphism and scaling) forms an orbifold (a double quotient space), denoted by $\mathcal S_r$ . A theorem of Bhargava and Harron says that for $d \leq 5$ , the (Arakelov) Tschirnhausen bundles are equidistributed in $\mathcal S_{d-1}$ [8, Theorem 1]. Again, one crucial ingredient in their proof is provided by the structure theorems for finite covers. We may view Corollary 1.2 as a (complex) function field analogue, but for all d.

1.1.3 Geometry of Hurwitz spaces

Another source of motivation for Theorem 1.1 concerns the geometry of moduli spaces of coverings, known as Hurwitz spaces. For simplicity, take $k = \mathbf {C}$ , and let Y be a smooth projective curve over k. Denote by $H_{d,g}(Y)$ the coarse moduli space that parameterises primitive covers $\phi {\colon } X \to Y$ , where $\phi $ is a map of degree d and X is a smooth curve of genus g (the cover $\phi $ is primitive if $\phi _* {\colon } \pi _1(X) \to \pi _1(Y)$ is surjective). The space $H_{d,g}(Y)$ is an irreducible algebraic variety [18, Theorem 9.2].

The association $\phi \leadsto E_\phi $ gives rise to interesting cycles on $H_{d,g}(Y)$ , called the Maroni loci. For a vector bundle E on Y, define the Maroni locus $\operatorname {Mar}(E) \subset H_{d,g}(Y)$ as the locally closed subset that parameterises covers with Tschirnhausen bundle isomorphic to E. This notion generalizes the classical Maroni loci for $Y = \mathbf {P}^1$ , which play a key role in describing the cones of various cycles classes on $H_{d,g}(Y)$ in [15] and [31]. It would be interesting to know if the cycle of $\overline {\operatorname {Mar}(E)}$ has similar distinguishing properties, such as rigidity or extremality, more generally than for $Y = \mathbf {P}^1$ . A first step toward this study is to determine when these cycles are non-empty and of the expected dimension. The method of proof of the main theorem yields the following.

Theorem 1.4. Let Y be a smooth projective curve over $\mathbf {C}$ of genus $g_Y$ , and let E be a vector bundle on Y of rank $(d-1)$ . There exists an n (depending on E) such that for every line bundle L on Y of degree at least n, the Maroni locus $\operatorname {Mar}(E \otimes L) \subset H_{d,g}(Y)$ contains an irreducible component having the expected codimension $h^{1}(\operatorname {End} E)$ . Here, g is related to E and L by the formula

$$\begin{align*}\deg E + (d-1) \deg L = g - 1 - d(g_Y-1).\end{align*}$$

Theorem 1.4 is Theorem 3.13 in the main text. Going further, it would be valuable to know whether all the components of $\operatorname {Mar}(E \otimes L)$ are of the expected dimension or, even better, if $\operatorname {Mar}(E \otimes L)$ is irreducible. The results of [16, § 2] imply irreducibility for $Y = \mathbf {P}^1$ and some vector bundles E. But the question remains open in general.

More broadly, the association $\phi \leadsto E_\phi $ allows us to relate $H_{d,g}(Y)$ to the moduli space of vector bundles on Y. Denote by $M_{r,k}(Y)$ the moduli space of semi-stable vector bundles of rank r and degree k on Y. It is well-known that $M_{r,k}(Y)$ is an irreducible algebraic variety [38]. Note that the Tschirnhausen bundle of a degree d and genus g cover of Y has rank $d-1$ and degree $e = g-1-d(g_Y-1)$ . One would like to say that $\phi \leadsto E_\phi $ yields a rational map

$$\begin{align*}H_{d,g}(Y) \dashrightarrow M_{d-1,e},\end{align*}$$

but to say so, we must know the basic fact that a general element $\phi {\colon } X \to Y$ of $H_{d,g}(Y)$ gives a semi-stable vector bundle $E_\phi $ . We obtain this as a consequence of our methods.

Theorem 1.5. Let Y be a smooth projective curve over $\mathbf {C}$ of genus $g_Y \geq 2$ . Set $e = g-1-d(g_Y-1)$ . If g is sufficiently large (depending on Y and d), then the Tschirnhausen bundle of a general degree d and genus g branched cover of Y is stable. Moreover, the rational map $H_{d,g}(Y) \dashrightarrow M_{d-1,e}(Y)$ defined by $\phi \mapsto E_\phi $ is dominant.

The same statement holds for $g_{Y} = 1$ , with ‘stable’ replaced with ‘regular poly-stable’.

Theorem 1.5 is Theorem 3.11 in the main text.

The low-degree cases ( $d \leq 5$ ) of Theorem 1.5 were proved by Kanev [24, 23, 25] using the structure theorems. The crucial new ingredient in our approach is the use of deformation theory to circumvent such direct attacks. The validity of Theorem 1.5 for low g is an interesting open problem. It would be nice to know whether $\phi \mapsto E_\phi $ is dominant as soon as we have $\dim H_{d,g}(Y) \geq \dim M_{d-1,e}(Y)$ .

We also draw the reader’s attention to results, similar in spirit to Theorem 1.5, proved by Beauville, Narasimhan and Ramanan [5]. Motivated by the study of the Hitchin fibration, they study not the pushforward of ${\mathcal O}_X$ itself but the pushforwards of general line bundles on X.

1.2 Strategy of proof

The proof of Theorem 1.1 proceeds by degeneration. To help the reader, we first outline our approach to a weaker version of Theorem 1.1. In the weaker version, we consider not the vector bundle E itself, but its projectivization $\mathbf {P} E$ , which we call the Tschirnhausen scroll. A branched cover with Gorenstein fibres $\phi {\colon } X \to Y$ with Tschirnhausen bundle E factors through a relative canonical embedding $\iota {\colon } X \hookrightarrow \mathbf {P} E$ by the main theorem in [11].

Theorem 1.6. Let E be any vector bundle on a smooth projective curve Y. Then the scroll $\mathbf {P} E$ is the Tschirnhausen scroll of a finite cover $\phi {\colon } X \to Y$ with X smooth.

The following steps outline a proof of Theorem 1.6 that parallels the proof of the stronger Theorem 1.1. We omit the details, since they are subsumed by the results in the paper.

1. 1. First consider the case

   $$\begin{align*}E = L_1 \oplus \dots \oplus L_{d-1},\end{align*}$$
   where the $L_i$ are line bundles on Y whose degrees satisfy
   $$\begin{align*}\deg L_i \ll \deg L_{i+1}.\end{align*}$$
   For such E, we construct a nodal cover $\psi {\colon } X \to Y$ such that $\mathbf {P} E_{\psi } = \mathbf {P} E$ . For example, we may take X to be a nodal union of d copies of Y, each mapping isomorphically to Y under $\psi $ , where the ith copy meets the $(i+1)$ th copy along nodes lying in the linear series $|L_i|$ .

2. 2. Consider $X \subset \mathbf {P} E$ , where X is the nodal curve constructed above. We now attempt to find a smoothing of X in $\mathbf {P} E$ . However, the normal bundle $N_{X/ \mathbf {P} E}$ may be quite negative. Fixing this negativity is the most crucial step.

   To overcome the negativity, we draw motivation from Mori’s idea of making curves flexible by attaching free rational tails. If we view a cover $X \to Y$ as a map from Y to the classifying stack $BS_d$ as done in [1], then attaching rational tails can be interpreted as attaching general rational normal curves to X in the fibres of $\mathbf {P} E \to Y$ . Of course, the classifying stack $BS_d$ is not a projective variety, so the above only serves as an inspiration.

3. 3. Given a general point $y \in Y$ , the d points $\psi ^{-1}(y) \subset \mathbf {P} E_{y} \simeq \mathbf {P}^{d-2}$ are in linear general position, and therefore they lie on many smooth rational normal curves $R_{y} \subset \mathbf {P} E_{y}$ . Choose a large subset $S \subset Y$ , and attach general rational normal curves $R_y$ for each $y \in S$ to X, obtaining a new nodal curve $Z \subset \mathbf {P} E$ .

4. 4. The key technical step is showing that the new normal bundle $N_{Z/\mathbf {P} E}$ is sufficiently positive. Using this positivity, we get that Z is the flat limit of a family of smooth, relative canonically embedded curves $X_t \subset \mathbf {P} E$ . The generic cover $\phi {\colon } X_t \to Y$ in this family satisfies $E_\phi \cong L_1 \oplus \dots \oplus L_{d-1}$ .

5. 5. We tackle the case of an arbitrary bundle E as follows:

   1. (a) We note that every vector bundle E degenerates isotrivially to a bundle of the form $E_0 = L_1 \oplus \dots \oplus L_{d-1}$ treated in the previous steps.

   2. (b) We take a cover $X_0 \to Y$ with Tschirnhausen bundle $E_0$ constructed above. Using the abundant positivity of $N_{X_0/\mathbf {P} E_0}$ , we show that $X_0 \subset \mathbf {P} E_0$ deforms to $X \subset \mathbf {P} E$ . The cover $\phi {\colon } X \to Y$ satisfies $\mathbf {P} E_\phi \cong \mathbf {P} E$ .

We need to refine the strategy above to handle the vector bundle E itself, not just its projectivization. Therefore, we work with the canonical affine embedding of X in the total space of E. The proof of Theorem 1.1 involves carrying out the steps outlined above for the embedding $X \subset E$ relative to the divisor of hyperplanes at infinity in a projective completion of E.

1.3 Conventions

We work over an algebraically closed field k. ‘Scheme’ means a scheme of finite type over k unless stated otherwise. Likewise, ‘point’ means a k-point, unless stated otherwise. The projectivization $\mathbf {P} V$ of a vector bundle V refers to the space of one-dimensional quotients of V. We identify vector bundles with their sheaves of sections. An injection is understood as an injection of sheaves. Given a coherent sheaf F on X and a closed immersion $j {\colon } Z \to X$ , we use $F|_Z$ to denote the pullback $j^*F = j^{-1}(F \otimes _{\mathcal O_X} \mathcal O_Z)$ . In particular, if Z is a point $p \in X$ , then $F|_p$ is the fibre of F at p. As a convention, we use $=$ to denote canonical isomorphisms and $\cong $ to denote non-canonical ones.

2 Vector bundles and their inflations and degenerations

This section contains standard results about vector bundles, their degenerations, and finite covers. The only new ingredient is a careful but straightforward study of elementary transformations (‘inflations’) that decrease higher cohomology (Proposition 2.5). Throughout, Y is a smooth, projective, connected curve over k, an algebraically closed field of arbitrary characteristic.

2.1 Inflations

Let E be a vector bundle on Y. A degree n inflation of E is a vector bundle $E^+$ along with an injective map of sheaves $E \to E^+$ whose cokernel is finite of length n. If the cokernel is supported on a subscheme $S \subset Y$ , then we say that $E \to E^+$ is an inflation of E at S.

Remark 2.1. Let $E \to E^+$ be a degree 1 inflation. In standard parlance, E and $E^+$ are said to be related by an elementary transformation. We use ‘inflation’ only to emphasize the asymmetry in the relationship.

Fix a point $y \in Y$ . Consider an inflation $E \to E^+$ whose cokernel is supported (scheme-theoretically) at y. Then we have an exact sequence

(2.1) $$ \begin{align} 0 \to E \to E^+ \to B \to 0, \end{align} $$

where the cokernel is annihilated by the maximal ideal $m_y \subset {\mathcal O}_Y$ . By applying $\operatorname {Hom}_Y(-,{\mathcal O}_Y)$ to equation (2.1), we get the sequence

$$\begin{align*}0 \to {E^+}^\vee \to E^\vee \to A \to 0,\end{align*}$$

where $A = \operatorname {\mathscr {E}xt}^1_{{\mathcal O}_Y}(B, {\mathcal O}_Y)$ . Note that A is also annihilated by the maximal ideal $m_y$ , and hence we may view it as a k-vector space. We thus get a surjection of k-vector spaces

(2.2) $$ \begin{align} E^\vee|_y \to A. \end{align} $$

We call equation (2.2) the defining quotient of the inflation $E \to E^+$ . Indeed, given equation (2.2), we can reconstruct ${E^+}^\vee \to E^\vee $ as the kernel of the induced map $E^\vee \to A$ and $E \to E^+$ by taking the dual.

Let us describe the defining quotient explicitly in terms of the original map $i {\colon } E \to E^+$ . By applying $B \otimes _k -$ to the exact sequence

$$\begin{align*}0 \to {\mathcal O}_Y(-y) \to {\mathcal O}_Y \to k(y) \to 0\end{align*}$$

we get

$$\begin{align*}0 \to B \otimes {\mathcal O}_Y(-y) \to B \otimes {\mathcal O}_Y \to B \to 0.\end{align*}$$

Now, by applying $\operatorname {Hom}_Y(-,{\mathcal O}_Y)$ , we get

$$\begin{align*}0 \to B^\vee \otimes_k {\mathcal O}_Y \to B^\vee \otimes_k {\mathcal O}_Y(y) \to \operatorname{\mathscr{E}xt}^1_{{\mathcal O}_Y}(B, {\mathcal O}_Y) \to 0, \end{align*}$$

and hence an isomorphism

(2.3) $$ \begin{align} A &= \operatorname{\mathscr{E}xt}^1_{{\mathcal O}_Y}(B, {\mathcal O}_Y)\nonumber\\ &= B^\vee \otimes_k {\mathcal O}_Y(y)|_y \nonumber\\ &= B^\vee \otimes_k N_{y/Y}. \end{align} $$

With this identification of A, the defining quotient in equation (2.2) is a map

$$\begin{align*}q {\colon} E^\vee|_y \to B^\vee \otimes_k N_{y/Y}.\end{align*}$$

For $g \in E^\vee |_y$ , let us describe $q(g) \in B^\vee \otimes _k N_{y/Y}$ as a k-linear map

$$\begin{align*}q(g) {\colon} B \otimes_k m_y/m_y^2 \to k. \end{align*}$$

Let $b \otimes t \in B \otimes m_y/m_y^2$ be given. Let $e^+$ be a section of $E^+$ defined in a neighbourhood of y that maps to b under the projection $E^+ \to B$ . Let $\widetilde t$ be a section of  $m_y$ defined in a neighbourhood of y that maps to $t \in m_y/m_y^2$ . Then the section $\widetilde t \cdot e^+$ of $E^+$ maps to $0$ in B and hence is equal to the image of a section e of E under the map $i {\colon } E \to E^+$ ; that is, $\widetilde t \cdot e^+ = i(e)$ . Then the functional $q(g)$ is defined by

(2.4) $$ \begin{align} q(g) {\colon} b \otimes t \mapsto \langle g ,e|_p \rangle, \end{align} $$

where $\langle -,- \rangle $ is the natural pairing between $E^\vee |_y$ and $E|_y$ . Although the lifts $e^+$ and $\widetilde t$ are not unique, it is easy to check that the right-hand side of equation (2.4) does not depend on the choice of lifts.

2.2 Inflations and higher cohomology

We now study how inflations increase positivity.

Proposition 2.2. Let $E \to E^+$ be an inflation. Then $h^1(Y, E^+) \leq h^1(Y, E)$ . In particular, if $H^1(Y, E) = 0$ , then $H^1(Y, E^+) = 0$ . If, furthermore, E is globally generated, then so is $E^+$ .

Proof. For the statement about $h^1$ , apply the long exact sequence on cohomology to

$$\begin{align*}0 \to E \to E^+ \to E^+/E \to 0,\end{align*}$$

and use that $E^+/E$ has zero-dimensional support. For global generation, consider the sequence

(2.5) $$ \begin{align} 0 \to E(-y) \to E \to E|_y \to 0 \end{align} $$

and the associated long exact sequence on cohomology. It follows that if E is globally generated and $H^1(Y, E) = 0$ , then $H^1(Y, E(-y)) = 0$ for every $y \in Y$ . But then we also have $H^1(Y, E^+) = 0$ and $H^1(Y, E^+(-y)) = 0$ for every $y \in Y$ . From the sequence in equation (2.5) for $E^+$ , we conclude that $E^+$ is also globally generated.

Let $V \subset E^\vee \otimes \Omega _Y|_y$ be the image of the evaluation map

$$\begin{align*}H^0\left(E^\vee \otimes \Omega_Y\right) \to E^\vee \otimes \Omega_Y|_y.\end{align*}$$

Let $q {\colon } E^\vee |_y \to k^n$ be a surjection, and let $E \to E^+_q$ be the inflation with q as its defining quotient.

Proposition 2.3. With the notation above, let $q_V {\colon } V \to k^n \otimes \Omega _Y|_y$ be the restriction of $q \otimes \operatorname {id}$ to $V \subset E^\vee \otimes \Omega _Y|_y$ . Then we have

$$ \begin{align*} h^0(Y, E^+_q) &= h^0(Y, E) + n - \operatorname{rk} q_V \text{ and }\\ h^1(Y, E^+_q) &= h^1(Y, E) - \operatorname{rk} q_V. \end{align*} $$

Proof. We have the exact sequence $0 \to {E^+_q}^\vee \to E^\vee \xrightarrow {q} k^n \to 0$ , where the cokernel is supported at y. Tensoring by $\Omega _Y$ , taking the long exact sequence in cohomology and using Serre duality yields the proposition.

Proposition 2.4. Suppose E is such that $h^1(Y, E) \neq 0$ . Let $E \to E^+$ be a degree 1 inflation at a general point of Y with a general defining quotient. Then

$$\begin{align*}h^1(Y, E^+) = h^1(Y, E) - 1 \text{ and } h^0(Y, E^+) = h^0(Y, E).\end{align*}$$

Proof. If $h^1(E) = h^0(E^\vee \otimes \Omega _Y) \neq 0$ , the space $V \subset E^\vee \otimes \Omega _Y|_y$ defined above is non-zero if $y \in Y$ is general. Then, for a general choice of $q {\colon } E^\vee _q \to k$ , we have $\operatorname {rk} q_V = 1$ . The statement now follows from Proposition 2.3.

The following sharpens the meaning of ‘general’ in Proposition 2.4.

Proposition 2.5. Let $y \in Y$ be such that the image V of the evaluation map

$$\begin{align*}H^0(E^\vee \otimes \Omega_Y) \to E^\vee\otimes \Omega_Y|_y\end{align*}$$

is non-zero. Suppose we have a set Q of surjections $E^\vee _y \to k$ such that the linear span of Q is the entire projective space $\mathbf {P} E^\vee |_y$ . Then for some $q \in Q$ , we have

$$\begin{align*}h^1(Y, E^+_q) \leq h^1(Y, E) - 1\end{align*}$$

Proof. Since $V \neq 0$ and Q spans $\mathbf {P} E^\vee |_y$ , we must have $q_V \neq 0$ for some $q \in Q$ . Then the statement follows from Proposition 2.3.

Corollary 2.6. Let $n \geq h^1(Y, E)$ be a non-negative integer. Choose n general points $y_1, \dots , y_n \in Y$ and general surjections $q_i {\colon } E^\vee |_{y_i} \to k$ . Let $E \to E^+$ be the inflation at $y_1,\dots , y_n$ whose defining quotients are $q_1, \dots , q_n$ . Then

1. 1. $H^1(Y, E^+) = 0$ ;

2. 2. furthermore, if $n \geq 2h^1(Y, E(-y))$ for all $y \in Y$ , then $E^+$ is also globally generated.

Proof. For the first part, apply Proposition 2.4 repeatedly and Proposition 2.2. For the second part, take an arbitrary $y \in Y$ . Let $E'$ be obtained from E by $n/2$ general inflations, as in the first part. Then $H^1(Y, E'(-y)) = 0$ . By upper semi-continuity, there are only finitely many $z \in Y$ for which $H^1(Y, E'(-z)) \neq 0$ . Let $E^+$ be obtained from $E'$ by $n/2$ more general inflations. Then we have $H^1(Y, E^+(-z)) = 0$ for all $z \in Y$ , which implies that $E^+$ is globally generated.

Proposition 2.2 and Corollary 2.6 together imply the following.

Corollary 2.7. Let E be a vector bundle on Y, and let n be large enough. Suppose $E'$ is a coherent sheaf of the same generic rank as E, and $E'$ contains an inflation of E at n general points with n general defining quotients. Then $H^1(Y, E') = 0$ and $E'$ is globally generated.

Remark 2.8. The generality requirement on the quotients $q_i$ in Corollary 2.6 and Corollary 2.7 is only in the sense of not satisfying any linear equations. That is, it is satisfied as long as the $q_i$ are chosen from a set that linearly spans the space of quotients (see Proposition 2.5).

2.3 Nodal curves and inflations of the normal bundle

A common setting for inflations in the paper is the following. Let P be a smooth variety. Let X and R be smooth curves in P that intersect at a point p so that their union Z has a node at p. We analyse the relationship of the normal bundle $N_{Z/P}$ with $N_{X/P}$ and $N_{R/P}$ , following a similar analysis in [20].

We first recall a natural map

(2.6) $$ \begin{align} N_{Z/P}|_p \to N_{p/X} \otimes N_{p/R}. \end{align} $$

To define the map in equation (2.6), consider the multiplication map

$$\begin{align*}I_{R/P} \otimes_{{\mathcal O}_P} I_{X/P} \to I_{Z/P}.\end{align*}$$

The restriction maps ${\mathcal O}_P \to {\mathcal O}_X$ and ${\mathcal O}_P \to {\mathcal O}_R$ yield surjections $I_{R/P} \to I_{p/X}$ and $I_{X/P} \to I_{p/R}$ . The multiplication map above induces a map $a {\colon } I_{p/X} \otimes _{{\mathcal O}_P} I_{p/R} \to I_{Z/P}|_p$ fitting in the diagram

(2.7)

To describe a explicitly, suppose we are given $f \in I_{p/X}$ and $g \in I_{p/R}$ . Then we have

(2.8) $$ \begin{align} a {\colon} f \otimes g \mapsto \widetilde f \cdot \widetilde g, \end{align} $$

where $\widetilde f \in I_{R/P}$ is a lift of f and $\widetilde g \in I_{X/P}$ is a lift of g. The product $\widetilde f \cdot \widetilde g \in I_{Z/P}$ depends on the chosen lifts, but it is easy to check that its image in $I_{Z/P}|_p$ depends only on f and g. The source and target of the map a are supported at p and hence can be treated as $k = {\mathcal O}_P/m_p$ vector spaces. The map in equation (2.6) is the k-linear dual of the map a.

Together with the natural map $T_P|_Z \to N_{Z/P}$ , the map in equation (2.6) yields a right exact sequence

(2.9) $$ \begin{align} T_P|_Z \to N_{Z/P} \to N_{p/X} \otimes N_{p/R} \to 0. \end{align} $$

The sequence in equation (2.9) is identical to the sequence considered before Proposition 1.1 in [20]. In [20], the cokernel of $T_P|_Z \to N_{Z/P}$ is identified with the sheaf $T^1_Z$ , which is indeed isomorphic to $N_{p/X} \otimes N_{p/R}$ [3, Chapter XI, equation (3.8)]. Observe that the kernel of the map $T_P|_p \to N_{Z/P}|_p$ is the two-dimensional space spanned by the subspaces $T_X|_p$ and $T_R|_p$ of $T_P|_p$ .

Let us restrict the exact sequence in equation (2.9) to X. We see that the composite $T_X \to T_P|_X \to N_{Z/P}|_X$ is zero, and hence the map $T_P|_X \to N_{Z/P}|_X$ factors as

$$\begin{align*}T_P|_X \to N_{X/P} \to N_{Z/P}|_X.\end{align*}$$

As a result, equation (2.9) gives an exact sequence

(2.10)

The sequence in equation (2.10) exhibits $N_{Z/P}|_X$ as a degree 1 inflation of $N_{X/P}$ at p.

Let us understand the defining quotient of the inflation in equation (2.10). We simply carry out the computation of equation (2.4) in the current context. We have the locally free sheaves

$$\begin{align*}\begin{array}{cc} E^+ = N_{Z/P}|_X, &\qquad {E^+}^\vee = I_{Z/P}|_X, \\ E = N_{X/P}, &\qquad E^\vee = I_{X/P}|_X, \end{array} \end{align*}$$

and the skyscraper sheaves

$$ \begin{align*} B &= N_{p/X} \otimes N_{p/R},\\ A &= I_{p/X}|_p \otimes I_{p/R}|_p \otimes N_{p/X}|_p, \\ &= I_{p/R}|_p. \end{align*} $$

In the last equation, we have contracted the first and the third factors using the natural pairing.

Proposition 2.9. The defining quotient of the inflation in equation (2.10) is the map

$$ \begin{align*} q {\colon} I_{X/P}|_p \to I_{p/R}|_p \end{align*} $$

induced by the restriction map ${\mathcal O}_P \to {\mathcal O}_R$ .

Proof. Let $g \in I_{X/P}|_p$ be given. We recall from equation (2.4) the description of $q(g)$ as a functional

$$\begin{align*}q(g) {\colon} N_{p/X} \otimes N_{p/R} \otimes I_{p/X}|_p \to k. \end{align*}$$

Starting with $\lambda \otimes \mu \otimes t$ , we first lift $\lambda \otimes \mu $ to a section $e^+$ of $N_{Z/P}|_X$ and t to a section $\widetilde t$ of $I_{p/X}$ . We then let e be the section of $N_{X/P}$ that maps to $\widetilde t \cdot e^+$ . The functional $q(g)$ is then given by

(2.11) $$ \begin{align} q(g) \colon \lambda \otimes \mu \otimes t \mapsto \langle g, e|_p\rangle. \end{align} $$

To compute $\langle g, e|_p\rangle $ , let us choose a lift $\widetilde g \in I_{X/P}|_X$ of $g \in I_{X/P}|_p$ . Then we have

(2.12) $$ \begin{align} \langle g, e|_p \rangle = \langle \widetilde g, e \rangle|_p, \end{align} $$

where $\langle -,- \rangle $ on the right denotes the natural pairing

$$\begin{align*}\langle -, - \rangle {\colon} I_{X/P}|_X \otimes N_{X/P} \to {\mathcal O}_X.\end{align*}$$

By construction, the image of $e \in N_{X/P}|_X$ in $N_{Z/P}|_X$ is $\widetilde t \cdot e^+$ . We claim that this means

(2.13) $$ \begin{align} \langle h, e \rangle = \langle \widetilde t \cdot h, e^+ \rangle \end{align} $$

for any $h \in I_{X/P}|_X$ . (In equation (2.13), on the left, we have the natural pairing between $I_{X/P}|_X$ and $N_{X/P}$ , and on the right, the natural pairing between $I_{Z/P}|_X$ and $N_{Z/P}|_X$ .) Indeed, if h lies in the subsheaf $I_{Z/P}|_X \subset I_{X/P}|_X$ , then equation (2.13) is true by the definition of $\widetilde t \cdot e^+$ . But $I_{Z/P}|_X$ and $I_{X/P}|_X$ are locally free of the same rank, so equation (2.13) must be true for all $h \in I_{X/P}|_X$ . Applying equation (2.13) to $h = \widetilde g$ yields

(2.14) $$ \begin{align} \langle \widetilde g, e \rangle|_p &= \langle \widetilde t \cdot \widetilde g, e^+ \rangle|_p\nonumber\\ &= \langle \widetilde t \cdot \widetilde g \big|_p, e^+|_p \rangle. \end{align} $$

Let $\overline g \in I_{p/R}|_p$ be the image of $g \in I_{X/P}|_p$ under the map $I_{X/P} \to I_{p/R}$ induced by the restriction ${\mathcal O}_P \to {\mathcal O}_R$ . Recall that $\widetilde t \in I_{p/X}$ is a lift of $t \in I_{p/X}|_p$ . Observe that the element $\widetilde t \cdot \widetilde g|_p \in I_{Z/P}|_p$ is the image of $t \otimes \overline g$ under the map

$$\begin{align*}a {\colon} I_{p/X} \otimes I_{p/R} \to I_{Z/P}|_p \end{align*}$$

described in equation (2.7). Since $e^+ \in N_{Z/P}|_X$ is a lift of $\lambda \otimes \mu \in N_{p/X} \otimes N_{p/R}$ under the map

$$\begin{align*}N_{Z/P} \to N_{p/X} \otimes N_{p/R},\end{align*}$$

which is dual to the map a, we have

(2.15) $$ \begin{align} \langle \widetilde t \cdot \widetilde g \big|_p, e^+|_p \rangle &= \langle t \otimes \overline g, \lambda \otimes \mu \rangle \nonumber\\[3pt] &= \langle t, \lambda \rangle \cdot \langle \overline g, \mu \rangle. \end{align} $$

By combining equations (2.11), (2.12), (2.14) and (2.15), we get

(2.16) $$ \begin{align} q(g) {\colon} N_{p/X} \otimes N_{p/R} \otimes I_{p/X}|_p &\to k \nonumber\\[3pt] \lambda \otimes \mu \otimes t &\mapsto \langle t, \lambda \rangle \cdot \langle \overline g, \mu \rangle. \end{align} $$

If we contract the first and the third factors of $N_{p/X} \otimes N_{p/R} \otimes I_{p/X}|_p$ using the natural pairing, then the functional $q(g)$ is equal to

$$ \begin{align*} q(g) {\colon} N_{p/R} &\to k \nonumber\\ \mu &\mapsto \langle \overline g, \mu \rangle. \end{align*} $$

Thus, as an element of the dual space $I_{p/R}|_p$ , the element $q(g)$ is equal to the image $\overline g$ of $g \in I_{X/P}|_p$ under the restriction map $I_{X/P} \to I_{p/R}$ .

Restricting the exact sequence in equation (2.9) to R yields an analogous picture. We may write the two inflations obtained in this way together as

(2.17)

By construction, the maps $\alpha $ and $\beta $ are induced from the same map $N_{Z/P} \to N_{p/X} \otimes N_{p/R}$ .

Finally, note that the discussion above extends naturally to the case of two smooth curves attached nodally at a finite set of points instead of a single point.

2.4 Isotrivial degenerations

We say that a bundle E isotrivially degenerates to a bundle $E_0$ if there exists a pointed smooth curve $(\Delta , 0)$ and a bundle ${\mathcal E}$ on $Y \times \Delta $ such that ${\mathcal E}_{Y \times \{0\}} \cong E_0$ and ${\mathcal E}\big |_{Y \times \{t\}} \cong E$ for every $t \in \Delta \setminus \{0\}$ .

Proposition 2.10. Let E be a vector bundle on Y, and let N be a non-negative integer. Then E isotrivially degenerates to a vector bundle $E_0$ of the form

$$\begin{align*}E_0 = L_1 \oplus \dots \oplus L_r,\end{align*}$$

where the $L_i$ are line bundles and $\deg L_i + N \leq \deg L_{i+1}$ for all $i = 1, \dots , r-1$ .

For the proof of Proposition 2.10, we need a lemma.

Lemma 2.11. There exists a filtration

$$\begin{align*}E = F_{0} \supset F_{1} \supset \dots \supset F_{r-1} \supset F_{r} = 0\end{align*}$$

satisfying the following properties:

1. 1. For every $i \in \{0, \dots , r-1\}$ , the sub-quotient $F_{i}/F_{i+1}$ is a line bundle.

2. 2. Set $L_i = F_{i}/F_{i+1}$ for $i \in \{1, \dots , r-1\}$ and $L_r = F_0/F_1$ . For every $i \in \{1, \dots , r-1\}$ , we have

   $$\begin{align*}\deg L_i + N \leq \deg L_{i+1}.\end{align*}$$

Proof. The statement is vacuous for $r = 0$ and $1$ . So assume $r \geq 2$ . Note that if $F_{\bullet }$ is a filtration of E satisfying the two conditions, and if L is a line bundle, then $F_\bullet \otimes L$ is such a filtration of $E \otimes L$ . Therefore, by twisting by a line bundle of large degree if necessary, we may assume that $\deg E \geq 0$ .

Let us construct the filtration from right to left. Let $L_{r-1} \subset E$ be a line bundle with $\deg L_{r-1} \leq -N$ and with a locally free quotient. Set $F_{r-1} = L_{r-1}$ . Next, let $L_{r-2} \subset E / F_{r-1}$ be a line bundle with $\deg L_{r-2} \leq \deg L_{r-1} - N$ and with a locally free quotient. Let $F_{r-2} \subset E$ be the preimage of $L_{r-2}$ . Continue in this way. More precisely, suppose that we have constructed

$$\begin{align*}F_j \supset F_{j+1} \supset \dots \supset F_{r-1} \supset F_r = 0\end{align*}$$

such that $L_i = F_i / F_{i+1}$ satisfy

$$\begin{align*}\deg L_{i} \leq \deg L_{i+1} - N,\end{align*}$$

and suppose $j \geq 2$ . Then let $L_{j-1} \subset E/F_j$ be a line bundle with $\deg L_{j-1} \leq \deg L_j - N$ with a locally free quotient. Let $F_{j-1} \subset E$ be the preimage of $L_{j-1}$ . Finally, set $F_0 = E$ .

Condition 1 is true by design. Condition 2 is true by design for $i \in \{1, \dots , r-2\}$ . For $i = r-1$ , note that $\deg L_{r-1} \leq -N$ by construction. On the other hand, we must have $\deg L_r \geq 0$ . Indeed, we have $\deg E \geq 0$ , but every sub-quotient of $F_\bullet $ except $F_0 / F_1$ has negative degree. Therefore, condition 2 holds for $i = r-1$ as well.

Proof of Proposition 2.10

Let $F_\bullet $ be a filtration of E satisfying the conclusions of Lemma 2.11. It is standard that a bundle degenerates isotrivially to the associated graded of a filtration. The construction of the degeneration goes as follows. Choose an affine cover $\{U_\alpha \}$ of Y on which $F_0, \dots , F_r$ are free. Suppose $U_\alpha = \operatorname {Spec} A_\alpha $ , and let $F_{i,\alpha } = F_i|_{U_\alpha }$ and $E_\alpha = E|_{U_\alpha }$ . Let $e_{0,\alpha }, \dots , e_{r-1,\alpha }$ be an $A_\alpha $ -basis of $E_\alpha $ adapted to the filtration $F_{\bullet , \alpha }$ : that is, such that

$$\begin{align*}F_{i,\alpha} = \langle e_{i,\alpha}, \dots, e_{r-1,\alpha} \rangle.\end{align*}$$

Let $U_{\alpha ,\beta } = U_\alpha \cap U_\beta $ . With the trivialization of $E_\alpha $ given by such a basis, the transition maps

$$\begin{align*}\phi_{\alpha, \beta} {\colon} U_{\alpha,\beta} \to \operatorname{GL}_r\end{align*}$$

lie in the subgroup $B \subset \operatorname {GL}_r$ of lower triangular matrices. Let $\phi ^{i,j}_{\alpha ,\beta } \in {\mathcal O}(U_{\alpha ,\beta })$ be the $(i,j)$ th entry of $\phi _{\alpha ,\beta }$ ; then $\phi ^{i,j}_{\alpha ,\beta } = 0$ if $i < j$ . Let $\mathbb A^1 = \operatorname {Spec} k[t]$ . Consider the open cover $\{U_\alpha \times \mathbb A^1\}$ of $Y \times \mathbb A^1$ . Let $\mathcal E$ be the locally free sheaf on $Y \times \mathbb A^1$ whose transition functions

$$\begin{align*}\psi_{\alpha,\beta} {\colon} U_{\alpha,\beta} \times \mathbb A^1 \to \operatorname{GL}_r\end{align*}$$

are defined by $\psi _{\alpha ,\beta }^{i,j} = t^{i-j}\phi _{\alpha ,\beta }^{i,j}$ . Then, for any $t \neq 0$ , we have $\mathcal E|_{Y \times \{t\}} \cong E$ ; and for $t = 0$ , we have $\mathcal E|_{Y \times \{0\}} \cong F_0/F_1 \oplus \dots \oplus F_{r-1}/F_r$ .

2.5 The canonical affine embedding

We end the section with a basic construction that relates finite covers and their Tschirnhausen bundles. Let d be a positive integer, and assume that $\operatorname {char} k = 0$ or $\operatorname {char} k> d$ .

Let X be a curve of arithmetic genus $g_X$ ; let $\phi {\colon } X \to Y$ be a finite flat morphism of degree d, and let E be the associated Tschirnhausen bundle. Then we have a decomposition $\phi _* {\mathcal O}_X = {\mathcal O}_Y \oplus E^\vee $ . The map $E^\vee \to \phi _* {\mathcal O}_X$ induces a surjection $\operatorname {Sym}^* E^\vee \to \phi _* {\mathcal O}_X$ . Taking the relative spectrum gives an embedding of X in the total space $\operatorname {Tot}(E)$ of the vector bundle associated to E; we often denote $\operatorname {Tot}(E)$ by E if no confusion is likely. We call $X \subset E$ the canonical affine embedding. Note that the degree of E is half of degree of the branch divisor of $\phi $ , namely

$$\begin{align*}\deg E = g_X - 1 - d(g_Y-1).\end{align*}$$

For all $y \in Y$ , the subscheme $X_y \subset E_y$ is in affine general position (not contained in a translate of a strict linear subspace of $E_y$ ).

The canonical affine embedding is characterized by the properties above.

Proposition 2.12. Retain the notation above. Let F be a vector bundle on Y of the same rank and degree as E, and let $\iota {\colon } X \to F$ be an embedding over Y such that for a general $y \in Y$ , the scheme $\iota (X_y) \subset F_y \cong \mathbf {A}^{d-1}$ is in affine general position. Then we have $F \cong E$ , and up to an affine linear automorphism of $F / Y$ , the embedding $\iota $ is the canonical affine embedding.

Proof. The restriction map $\operatorname {Sym}^* F^\vee \to \phi _*{\mathcal O}_X = {\mathcal O}_Y \oplus E^\vee $ induces a map

$$\begin{align*}\lambda {\colon} F^\vee \to E^\vee.\end{align*}$$

Since a general fibre $X_y \subset F_y$ is in affine general position, the map $\lambda $ is an injective map of sheaves. But the source and the target are locally free of the same degree and rank. Therefore, $\lambda $ is an isomorphism.

Recall that the affine canonical embedding is induced by the map

$$\begin{align*}(0, \operatorname{id}) {\colon} E^\vee \to {\mathcal O}_Y \oplus E^\vee = \phi_* {\mathcal O}_X.\end{align*}$$

Suppose $\iota $ induces the map

$$\begin{align*}(\alpha, \lambda) {\colon} F^\vee \to {\mathcal O}_Y \oplus E^\vee.\end{align*}$$

Compose $\iota $ with the affine linear isomorphism of $T_\alpha {\colon } \operatorname {Tot}(F) \to \operatorname {Tot}(F)$ over Y defined by the map $\operatorname {Sym}^*F^\vee \to \operatorname {Sym}^*F^\vee $ induced by

$$\begin{align*}(-\alpha, \operatorname{id}) {\colon} F^\vee \to {\mathcal O}_Y \oplus F^\vee.\end{align*}$$

Then $T_\alpha \circ \iota {\colon } X \to F$ is the affine canonical embedding, as desired.

3 Proof of the main theorem

Let d be a positive integer, and assume that $\operatorname {char} k = 0$ or $\operatorname {char} k> d$ . Throughout, Y is a smooth, projective, connected curve over k.

3.1 The split case with singular covers

As a first step, we treat the case of a suitable direct sum of line bundles and allow the source curve X to be singular.

Proposition 3.1. Let $E = L_1 \oplus \dots \oplus L_{d-1}$ , where the $L_i$ are line bundles on Y with $\deg L_1 \geq 2g_Y-1$ and $\deg L_{i+1} \geq \deg L_i + (2g_Y-1)$ for $i \in \{1, \dots , d-2\}$ . There exists a nodal curve X and a finite flat map $\phi {\colon } X \to Y$ of degree d such that $E_\phi \cong E$ .

The proof is inductive, based on the following ‘pinching’ construction. Let $\psi {\colon } Z \to Y$ be a finite cover of degree r. Let X be the reducible nodal curve $Z \cup Y$ , where Z and Y are attached nodally at distinct points (see Figure 1). More explicitly, let $y_i \in Y$ and $z_i \in Z$ be points such that $\psi (z_i) = y_i$ . Define R as the kernel of the map

$$\begin{align*}\psi_* {\mathcal O}_Z \oplus {\mathcal O}_Y \to \bigoplus_i k_{y_i},\end{align*}$$

defined around $y_i$ by

$$\begin{align*}(f,g) \mapsto f(z_i) - g(y_i).\end{align*}$$

Then $R \subset \psi _* {\mathcal O}_Z \oplus {\mathcal O}_Y$ is an ${\mathcal O}_Y$ -subalgebra and $X := \operatorname {Spec}_Y R$ is a nodal curve. Let $\phi {\colon } X \to Y$ be the natural finite flat map. Set $D = \sum y_i$ .

Figure 1 The pinching construction, in which pairs of points indicated by dotted lines are identified to form nodes

Lemma 3.2. In the setup above, we have an exact sequence

$$\begin{align*}0 \to E_\psi \to E_\phi \to {\mathcal O}_Y(D) \to 0.\end{align*}$$

Proof. The closed embedding $Z \to X$ gives a surjection

$$\begin{align*}\phi_* {\mathcal O}_X \to \psi_* {\mathcal O}_Z \end{align*}$$

whose kernel is ${\mathcal O}_Y(-D)$ . Factoring out the ${\mathcal O}_Y$ summand from both sides and taking duals yields the claimed exact sequence.

Proof of Proposition 3.1

We use induction on d, starting with the base case $d = 1$ , which is vacuous.

By the inductive hypothesis, we may assume that there exists a nodal curve Z and a finite cover $\psi {\colon } Z \to Y$ of degree $(d-1)$ such that $E_\psi \cong L_2 \oplus \dots \oplus L_{d-1}$ . Let $X = Z \cup Y \to Y$ be a cover of degree d obtained from $Z \to Y$ by a pinching construction such that ${\mathcal O}_Y(D) = L_1$ . By Lemma 3.2, we get an exact sequence

(3.1) $$ \begin{align} 0 \to L_2 \oplus \dots \oplus L_{d-1} \to E_\phi \to L_1 \to 0. \end{align} $$

But we have $\operatorname {Ext}^1(L_1, L_i) = H^1(L_i \otimes L_{1}^{\vee }) = 0$ since $\deg (L_i \otimes L_{1}^{\vee }) \geq 2g_Y-1$ . Therefore, the sequence in equation (3.1) is split, and we get $E_\phi = L_1 \oplus \dots \oplus L_{d-1}$ . The induction step is then complete.

3.2 Attaching rational curves

We now describe a procedure to make a finite cover more flexible so that it can be deformed easily. The procedure is local on Y, so we zoom in to a cleaner local situation.

Let Y be a smooth curve and $0 \in Y$ an arbitrary point. Set $P = \mathbf {P}^{d-1} \times Y$ , and let $H \subset P$ be a divisor, flat over Y, which restricts to hyperplanes on the fibres. Let $X \subset P$ be finite and étale of degree d over Y and disjoint from H. Assume that the fibres of $X \to Y$ are in linear general position in $\mathbf {P}^{d-1}$ . Use the subscript $0$ to denote the fibre over $0$ .

Let $R \subset P_0$ be a rational normal curve that contains $X_0$ and is transverse to $H_0$ . Let $\widetilde P \to P$ be the blow-up along $H_0$ . We use the same notation to denote the proper transforms of R, H, X and $P_0$ in $\widetilde P$ (these are isomorphic copies). Denote by D the exceptional divisor of the blow-up. Let Z be the nodal curve $Z = X \cup R$ . Set $\delta = X \cap R = X_0$ . See Figure 2 for a sketch of the setup. Denote by $\phi $ the projection to Y.

Figure 2 Attaching rational normal curves to X to make the normal bundle positive

Our goal is to establish $Z \subset \widetilde P$ as a more flexible replacement of $X \subset P$ . (The reason for the blow-up is to keep the curve away from the divisor H.) For this goal, we must relate the normal bundle $N_{Z/\widetilde P}$ with $N_{X/P}$ . Establishing this relationship takes some effort; the upshot is Corollary 3.7, which is the only statement we use in the later sections.

The following identifies the restriction of $N_{Z/\widetilde P}$ to R.

Lemma 3.3. We have the following diagram with exact rows and columns

where the maps in the first row and middle column are standard, and the others are induced from them. The sheaf F is canonically isomorphic to $\phi ^*N_{0/Y} \otimes _{{\mathcal O}_{\widetilde P}} {\mathcal O}_R(\delta - D)$ and thus isomorphic to ${\mathcal O}_R(1)$ .

Proof. The first row is standard. Augment it by considering the natural map $N_{R/\widetilde P} \to N_{Z/\widetilde P}\big |_R$ recalled in Section 2.3. Since $R \subset P_0 = \mathbf {P}^{d-1}$ is a rational normal curve, we have an isomorphism $N_{R/P_0} \cong {\mathcal O}(d+1)^{d-2}$ (see [34, II] or [36, Example 4.6.6]). A local calculation shows that the map $N_{R/P_0} \to N_{Z/\widetilde {P}}\big |_{R}$ remains an injection when restricted to any point of R, and hence its cokernel F is locally free and, plainly, of rank 1. From Section 2.3, we know that the cokernel of the natural map $N_{R/\widetilde P} \to N_{Z/\widetilde P}\big |_R$ is $N_{\delta /X} \otimes N_{\delta /R}$ . By the snake lemma, the cokernel of the map $N_{P_0/\widetilde P}\big |_R \to F$ is the same. Since both $N_{P_0/\widetilde P}\big |_R$ and F are line bundles, and the map between them degenerates exactly at $\delta $ , we obtain an isomorphism

$$\begin{align*}F = N_{P_0/\widetilde P} \big|_R(\delta).\end{align*}$$

Combined with the isomorphism

$$\begin{align*}N_{P_0/\widetilde P} = N_{P_0/P}(-D)\end{align*}$$

and the isomorphism $N_{P_0/P} = \phi ^*N_{0/Y}$ , we get the canonical isomorphism $F = \phi ^*N_{0/Y}\otimes {\mathcal O}_R(\delta -D)$ , as claimed. Since the degree of $\delta $ is d and that of $D|_R$ is $d-1$ , we see that $F \cong {\mathcal O}_R(1)$ .

Let us describe the three maps in equation (3.3) involving the bundle F. For all three, it is easier to describe the duals.

The ${\mathcal O}_R$ -dual of $N_{P_0/\widetilde P}|_R \to F$ is the map

(3.2) $$ \begin{align} \phi^*I_{0/Y} \otimes_{{\mathcal O}_{\widetilde P}} {\mathcal O}_R(D - \delta) \to I_{P_0/\widetilde P}|_R \end{align} $$

given as follows. Let $U \subset \widetilde P$ be an open set. Let t be a section of $I_{0/Y}$ , and let f be a section of ${\mathcal O}_R(D-\delta )$ on $U \cap R$ , interpreted as a rational function on R vanishing along $\delta $ with at most simple poles along D. Let $\widetilde f$ be any section of ${\mathcal O}_{\widetilde P}(D)$ on U that restricts to f on $U \cap R$ . Then the map in equation (3.2) sends the section represented by $t \otimes f$ to the section represented by $t \cdot \widetilde f$ . Observe that the possible poles of $\widetilde f$ along D are cancelled by the vanishing of t along D. Since t vanishes on $P_0$ , so does the product $t \cdot \widetilde f$ , which represents a section of $I_{P_0/\widetilde P}$ . It is easy to check that its image in $I_{P_0/\widetilde P}|_R$ depends only on f and not on the lift $\widetilde f$ .

The ${\mathcal O}_R$ -dual of $N_{Z/\widetilde P}|_R \to F$ is the map

(3.3) $$ \begin{align} \phi^*I_{0/Y} \otimes_{{\mathcal O}_{\widetilde P}} {\mathcal O}_R(D - \delta) \to I_{Z/\widetilde P}|_R \end{align} $$

given as follows. Let $U \subset \widetilde P$ be an open set. Let t be a section of $I_{0/Y}$ , and let f be a section of ${\mathcal O}_R(D-\delta )$ on $U \cap R$ . Let $\widetilde f$ be any section of $I_{X/\widetilde P} \otimes {\mathcal O}_{\widetilde P}(D)$ on U that restricts to f on $U \cap R$ . Then the map in equation (3.3) sends the section represented by $t \otimes f$ to the section represented by $t \cdot \widetilde f$ . As before, observe that $t \cdot \widetilde f$ has no poles, it vanishes along Z, and its image in $I_{Z/\widetilde P}|_R$ does not depend on the lift $\widetilde f$ .

The ${\mathcal O}_\delta $ -dual of $F|_\delta \to N_{\delta /X} \otimes N_{\delta /R}$ is the map

(3.4) $$ \begin{align} I_{\delta/X}\big|_\delta \otimes_{{\mathcal O}_{\widetilde P}} I_{\delta/R}\big|_\delta = \phi^*I_{0/Y} \otimes_{{\mathcal O}_{\widetilde P}} I_{\delta/R}\big|_\delta\to \phi^*I_{0/Y} \otimes_{{\mathcal O}_{\widetilde P}}{\mathcal O}_R(D - \delta)\big|_\delta, \end{align} $$

given as follows. Let t be a section of $I_{0/Y}$ and f a section of $I_{\delta /R}$ on some open subset of $\widetilde P$ . On this open set, lift f to a section $\widehat f$ of ${\mathcal O}_R(D-\delta )$ . Then the map in equation (3.4) sends the element represented by $t \otimes f$ to the element represented by $t \otimes \widehat f$ . Dually, the map

(3.5) $$ \begin{align} F = \phi^*N_{0/Y} \otimes_{{\mathcal O}_{\widetilde P}} {\mathcal O}_R(\delta-D) \to N_{\delta/X} \otimes N_{\delta/R} = \phi^*N_{0/Y} \otimes N_{\delta/R} \end{align} $$

is simply $\operatorname {id} \otimes \mathrm {res}$ , where

$$\begin{align*}\mathrm{res} {\colon} {\mathcal O}_R(\delta-D) \to {\mathcal O}_R(\delta-D)|_\delta = N_{\delta/R} \end{align*}$$

is the restriction map.

To see why the map in equation (3.4) is as claimed above, consider the following diagram, obtained by restricting the bottom-right square in the diagram in Lemma 3.3 to $\delta $ and taking ${\mathcal O}_\delta $ -duals:

Let $t \in I_{0/Y}$ , $f \in I_{\delta /R}$ and $\widehat f \in {\mathcal O}_R(D-\delta )$ be as in the definition of equation (3.5). To see that c indeed maps $t \otimes f$ to $t \otimes \widehat f$ , it suffices to observe that

$$\begin{align*}b( t \otimes \widehat f) = a (t \otimes f).\end{align*}$$

To compute the left-hand side, we use the description of b from equation (3.3). We choose a lift $\widetilde f \in I_{X/\widetilde P} \otimes {\mathcal O}_{\widetilde P}(D)$ of $\widehat f \in {\mathcal O}_R(D-\delta )$ . Then $b(t \otimes \widehat f) \in I_{Z/\widetilde P}|_\delta $ is the element represented by $t \widetilde f$ . To compute the right-hand side, we use the description of a from equation (2.8). We choose the lift $t \in I_{R/\widetilde P}$ of $t \in I_{\delta /X}$ and $\widetilde f \in I_{X/\widetilde P}(D)$ of $f \in I_{\delta /R}$ , observing that $\widetilde f$ is indeed a section of $I_{X/\widetilde P}$ in a neighbourhood of $\delta $ . Then $a(t \otimes f) \in I_{Z/\widetilde P}|_\delta $ is also the element represented by $t \widetilde f$ .

Lemma 3.3 and the discussion following it give us a good understanding of the relationship between $N_{Z/\widetilde P}$ and $N_{R/P}$ . Next, we must relate $N_{Z/\widetilde P}$ and $N_{X/P}$ . For this purpose, we need an auxiliary bundle M, which we now define. Set $N^+ = N_{Z/\widetilde P}$ and $N = N_{X/P}$ . Consider the diagram

Define M by the sequence

(3.6)

The following explains how M is related to $N^+=N_{Z/\widetilde P}$ .

Lemma 3.4. We have $R^1\phi _* N^+ = 0$ and an isomorphism of sheaves on Y

$$\begin{align*}M = \phi_*N^+ / \text{torsion}.\end{align*}$$

Proof. By Lemma 3.3, we see that $H^1(R, N^+|_R) = 0$ . Let K be the kernel of the restriction map $N^+ \to N^+|_R$ . Apply $R^i\phi _*$ to the sequence

$$\begin{align*}0 \to K \to N^+ \to N^+|_R \to 0.\end{align*}$$

The support of K is contained in X, which is finite over Y; hence $R^i\phi _*K = 0$ for $i> 0$ . Since the map $\phi {\colon } R \to Y$ contracts R to a point and $H^1(R, N^+|_R) = 0$ , we get $R^1\phi _* \left ( N^+|_R \right )= 0$ . As a result, we get that $R^1\phi _* N^+ = 0$ .

To see the isomorphism $M = \phi _*N^+/\text {torsion}$ , consider the sequence on Z

$$\begin{align*}0 \to N^+ \to N^+|_X \oplus N^+|_R \to N^+|_\delta \to 0,\end{align*}$$

where the last map sends $(f,g)$ to $f|_\delta - g|_\delta $ . Its push-forward to Y and the defining sequence of M fit in the diagram

where the middle vertical map is the projection on the first coordinate. (In the diagram, the dashed arrow is induced from the others.) By the snake lemma, we see that the map $\phi _*N^+ \to M$ is surjective. Since M is torsion free and of the same generic rank as $\phi _*N^+$ , we conclude that

$$\begin{align*}M = \phi_*N^+/\text{torsion}.\\[-36pt]\end{align*}$$

Having related M and $N^+ = N_{Z/\widetilde P}$ , we now relate M and $N = N_{X/P}$ .

Lemma 3.5. The bundle M is an inflation of $\phi _*N$ of degree 2 at $0 \in Y$ . More precisely, we have an exact sequence

$$\begin{align*}0 \to \phi_* N \to M \to \phi_*F \to 0.\end{align*}$$

Proof. The proof involves some standard diagram chases. From Lemma 3.3, we obtain the diagram

From the isomorphism $N_{R/P_0} \cong {\mathcal O}(d+1)^{d-2}$ in Lemma 3.3, we get that the map $N_{R/P_0} \to N_{R/P_0}\big |_\delta $ is surjective on global sections. By the snake lemma, we get

$$\begin{align*}\operatorname{coker}\left(\phi_*\left(N^+|_R\right) \to \phi_*\left(N^+|_\delta\right)\right) = \operatorname{coker} \left( \phi_* F \xrightarrow{e} \phi_*\left(N_{\delta/X} \otimes N_{\delta/R}\right)\right). \end{align*}$$

Substituting in the defining sequence in equation (3.6) of M, we obtain

(3.7)

Recall from equation (2.18) in Section 2.3 the sequence of sheaves on X

$$\begin{align*}0 \to N \to N^+|_X \to N_{\delta/X} \otimes N_{\delta/R} \to 0. \end{align*}$$

Since $X \to Y$ is finite, the sequence remains exact after applying $\phi _*$ . We thus have the diagram

(3.8)

where the dashed arrow is induced from the others. By the snake lemma, we get

(3.9) $$ \begin{align} 0 \to \phi_* N \to M \to \phi_*F \to 0, \end{align} $$

as asserted.

Let us explicitly compute the defining quotient of the inflation $\phi _*N \to M$ . We dualize—apply $\operatorname {Hom}_Y(-,{\mathcal O}_Y)$ —to the diagram in equation (3.8). The first two columns consist of locally free sheaves on Y. The last column consists of skyscraper sheaves supported at $0 \in Y$ , which contribute only $\operatorname {\mathscr {E}xt}^1$ terms. Recall from equation (2.3) that for a vector space A considered as a sheaf supported at $0$ , we have

(3.10) $$ \begin{align} \operatorname{\mathscr{E}xt}^1_{{\mathcal O}_Y}(A, {\mathcal O}_Y) = \operatorname{\mathscr{E}xt}_{{\mathcal O}_Y}^1(k, {\mathcal O}_Y) \otimes A^\vee = N_{0/Y} \otimes A^\vee. \end{align} $$

By applying equation (3.10) to $A = \phi _* F = N_{0/Y} \otimes \phi _* {\mathcal O}_R(\delta -D)$ , we get

$$\begin{align*}\operatorname{\mathscr{E}xt}^1_{{\mathcal O}_Y}(\phi_*F, {\mathcal O}_Y) = H^0({\mathcal O}_R(\delta-D))^\vee \text{, supported at}\ 0 \in Y. \end{align*}$$

By applying equation (3.10) to $A = \phi _*\left ( N_{\delta /X} \otimes N_{\delta /R}\right ) = N_{0/Y} \otimes \phi _*N_{\delta /R}$ , we get

$$\begin{align*}\operatorname{\mathscr{E}xt}^1_{{\mathcal O}_Y}(\phi_*\left( N_{\delta/X} \otimes N_{\delta/R} \right), {\mathcal O}_Y) = H^0(N_{\delta/R})^\vee \text{, supported at}\ 0 \in Y. \end{align*}$$

Thus, the dual of equation (3.8) gives the following diagram (some details suppressed)

(3.11)

In this diagram, the vector spaces in the leftmost column should be thought of as skyscraper sheaves at $0 \in Y$ . The defining quotient we seek is thus the composite

(3.12) $$ \begin{align} (\phi_*N)^\vee|_0 \to H^0\left(N_{\delta/R}\right)^\vee \xrightarrow{e^\vee} H^0({\mathcal O}_R(\delta-H))^\vee. \end{align} $$

The first term in equation (3.12) is equal to $H^0(N_{X_0/P_0})^\vee = H^0(N_{\delta /P_0})^\vee $ , since $X_0 = \delta $ . The first map in equation (3.12), namely the map $H^0(N_{\delta /P_0})^\vee \to H^0(N_{\delta /R})^\vee $ , is just $H^0$ applied to the defining quotient of the inflation $N \to N^+|_X$ . Proposition 2.9 describes this map. To recall, write

$$ \begin{align*} H^0(N_{\delta/P_0})^\vee &= H^0\left(I_{\delta/P_0}\big|_{\delta}\right), \text{ and }\\ H^0(N_{\delta/R})^\vee &= H^0\left(I_{\delta/R}\big|_\delta\right). \end{align*} $$

By Proposition 2.9, the first map in equation (3.12) is $H^0$ applied to the restriction map

$$\begin{align*}I_{X/\widetilde P}|_\delta = I_{\delta/P_0}\big|_\delta \to I_{\delta/R}\big|_\delta.\end{align*}$$

The second map in equation (3.12) is dual to the map

$$\begin{align*}e {\colon} H^0(F) \to H^0(N_{\delta/X} \otimes N_{\delta/R}) \end{align*}$$

obtained by applying $H^0$ to the map $F \to N_{\delta /X} \otimes N_{\delta /R}$ , which we identified as $\operatorname {id} \otimes \mathrm {res}$ in equation (3.5). Hence, we can explicitly describe the composite map in equation (3.12)

(3.13) $$ \begin{align} \alpha {\colon} H^0(I_{\delta/P_0}|_\delta) \to H^0({\mathcal O}_R(\delta - D))^\vee \end{align} $$

as follows. Consider an element $g \in H^0(I_{\delta /P_0}|_\delta )$ . In a neighbourhood of $\delta $ , choose a lift $\widetilde g \in I_{\delta /P_0}$ of g. Then $\alpha (g)$ is the function $H^0({\mathcal O}_R(\delta - D)) \to k$ defined by

$$\begin{align*}\alpha(g) {\colon} f \mapsto \sum_{x \in \delta} (f \cdot \widetilde g)|_x.\end{align*}$$

Here $(f \cdot \widetilde g)|_x$ is the evaluation at x of the function $(f \cdot \widetilde g)$ . Observe that $f \cdot \widetilde g$ is a regular function on R in a neighbourhood of $\delta $ , and its evaluation at $x \in \delta $ depends only on g, not on the lift $\widetilde g$ .

The following proposition shows that a general R gives an M that contains a general degree 1 inflation of $\phi _*N$ . Recall that by our convention, projectivizations parameterise one-dimensional quotients.

Proposition 3.6. Let $Q \subset \mathbf {P} H^0\left (I_{\delta /P_0}|_\delta \right )$ be the set consisting of projections $q {\colon } H^0\left (I_{\delta /P_0}|_\delta \right ) \to k$ that factor through the map $\alpha {\colon } H^0\left (I_{\delta /P_0}|_\delta \right ) \to H^0({\mathcal O}_R(\delta - D))^\vee $ for some rational normal curve $R \subset P_0$ containing $\delta $ . Then Q spans $\mathbf {P} H^0\left (I_{\delta /P_0}|_\delta \right )$ .

Proof. We begin by explicitly writing the curves R. Without loss of generality, $\delta \subset P_0 = \mathbf {P}^{d-1}$ consists of the d coordinate points, and the hyperplane $H_0 \subset \mathbf { P}^{d-1}$ is cut out by the equation $\sum X_i = 0$ . We can write rational curves $R \subset P_0$ that contain $\delta $ as follows. Let $b_1, \dots , b_d \in k^\times $ and $a_1, \dots , a_d \in k$ be arbitrary constants with $a_i \neq a_j$ for $i \neq j$ . Let x be a variable, and let $\Pi $ be the product $(x-a_1)\cdots (x-a_d)$ . Consider the map $\mathbf {A}^1 \to P_0 = \mathbf {P}^{d-1}$ defined by

(3.14) $$ \begin{align} x \mapsto \left[\frac{b_1 \Pi} {x-a_1}: \dots : \frac{b_d \Pi}{x-a_d}\right]. \end{align} $$

Let $R \subset P_0$ be the closure of the image; this is a rational normal curve. To see that R contains $\delta $ , simply observe that the map in equation (3.14) sends $a_i$ to the ith coordinate point. The divisor $D \subset R$ is cut out by $\sum X_i$ , which pulls back under equation (3.14) to the polynomial

$$\begin{align*}\Gamma = \sum b_i \frac{\Pi}{x-a_i}. \end{align*}$$

We now choose a basis of $H^0\left (I_{\delta /P_0}|_\delta \right )$ . Let $Y_1, \dots , Y_d$ be homogeneous coordinates on $P_0 = \mathbf {P}^{d-1}$ , and let $\delta _j \in \delta $ be the jth coordinate point (where only $Y_j \neq 0$ ). For $i,j \in \{1, \dots , d\}$ with $i \neq j$ , define $g(i,j) \in H^0\left (I_{\delta /P_0}|_\delta \right )$ by

$$\begin{align*}g(i,j)\big|_{\delta_\ell} = \begin{cases} Y_i / Y_j & \text{if}\ \ell = j, \\ 0 &\text{if}\ \ell \neq j.\end{cases} \end{align*}$$

Plainly, $\langle g(i,j) \rangle $ forms a basis of $H^0\left (I_{\delta /P_0}|_\delta \right )$ .

The parameterisation of R in equation (3.14) gives a basis of $H^0({\mathcal O}_R(\delta -D))$ as follows. Identifying $H^0({\mathcal O}_R(\delta -D))$ with the set of rational functions with zeros along $D = H_0 \cap R$ and possible poles along $\delta $ , we get a basis of $H^0({\mathcal O}_R(\delta -D))$ given by $\langle \Gamma /\Pi , x\Gamma /\Pi \rangle $ . Let $\langle u,v \rangle $ be the dual basis.

Let us use the description of $\alpha $ after equation (3.13) to compute the map

$$\begin{align*}\alpha {\colon} \langle g(i,j) \rangle \to \langle u, v \rangle. \end{align*}$$

We see that $\alpha (g(i,j))$ is the functional on $\langle \Gamma /\Pi , x\Gamma /\Pi \rangle $ given by

$$ \begin{align*} \frac{\Gamma}{\Pi} &\mapsto \left(\frac{Y_i}{Y_j} \cdot \frac{\Gamma}{\Pi}\right) \big|_{a_j}\\ &= \left( \frac{b_i(x-a_j)}{b_j(x-a_i)}\cdot \sum_\ell \frac{b_\ell}{x-a_\ell} \right)\big|_{a_j} \\ &= \frac{b_i}{a_j-a_i}, \text{ and }\\ \frac{x\Gamma}{\Pi} &\mapsto \left(\frac{Y_i}{Y_j} \cdot \frac{x\Gamma}{\Pi}\right) \big|_{a_j}\\ &= \frac{a_jb_i}{a_j-a_i}. \end{align*} $$

Thus, the map $\alpha $ is

(3.15) $$ \begin{align} g(i,j) \mapsto \frac{b_i}{a_j-a_i} \cdot u + \frac{a_jb_i}{a_j-a_i} \cdot v. \end{align} $$

The maps $q {\colon } \langle g(i,j) \rangle \to k$ in Q are precisely the maps in equation (3.15) with u and v replaced by arbitrary elements of k. It is easy to verify that, as rational functions in the variables $a_1,\dots , a_d$ , $b_1, \dots , b_d$ , u and v, the $d(d-1)$ functions $\frac {b_i(u+a_jv)}{a_j-a_i}$ are k-linearly independent. In other words, there is no k-linear equation that is satisfied by the maps q for all values of the a’s, the b’s, u and v. The proposition follows.

Corollary 3.7. We have $R^1\phi _*N_{Z/\widetilde P} = 0$ , and if R is general, then the sheaf $\phi _*N_{Z/\widetilde P}$ contains an inflation of $\phi _*N_{X/P}$ with a general defining quotient (away from any prescribed linear subspace).

Proof. The vanishing of $R^1\phi _*N_{Z/\widetilde P}$ is from Lemma 3.4. Let $\Lambda \subset \mathbf {P} H^0(N_{X_0/P_0}) = \mathbf {P} H^0(I_{\delta /P_0}|_\delta )$ be any proper linear subspace. Choose $q {\colon } H^0(I_{\delta /P_0}|_\delta ) \to k$ that is not contained in $\Lambda $ and that factors through the map $\alpha {\colon } H^0(I_{\delta /P_0}|_\delta ) \to H^0({\mathcal O}_R(\delta -D))^\vee $ for some rational normal curve $R \subset P_0$ containing $X_0$ . Such a q exists by Proposition 3.6. Let $N^\dagger $ be the inflation of $\phi _*N_{X/P}$ with defining quotient q, namely the one defined by

$$\begin{align*}\left(N^\dagger\right)^\vee = \ker \left(\left(\phi_*N_{X/P}\right)^\vee \xrightarrow{q} k\right),\end{align*}$$

where the k is supported at $0$ . By the exact sequence in Lemma 3.5, we see that $N^\dagger $ is a subsheaf of $\phi _*N_{Z/\widetilde P}/\mathrm {torsion}$ and hence of $\phi _*N_{Z/\widetilde P}$ .

3.3 Smoothing out

Proposition 3.8 Key

Let X be a nodal curve with a finite map $\phi {\colon } X \to Y$ of degree d. Let E be the Tschirnhausen bundle of $\phi $ . There exists a finite set $S \subset Y$ , a smooth curve $X'$ and a finite map $X' \to Y$ of degree d such that the following hold:

1. 1. The Tschirnhausen bundle of $X' \to Y$ is $E' = E \otimes {\mathcal O}_Y(S)$ .

2. 2. Consider $X'$ embedded in (the total space of) $E'$ by the canonical affine embedding. Then we have $H^1(X', N_{X'/ E'}) = 0$ .

Furthermore, if n is large enough (determined by $X \to Y$ ), we may take S to have size n and an arbitrary divisor class of degree n.

Proof. Consider X embedded in the total space of E by the canonical affine embedding. Compactify the total space of E to the projective bundle $P = \mathbf {P}\left ({\mathcal O}_Y \oplus E^\vee \right )$ , and let $H \subset \mathbf {P}$ be the hyperplane at infinity. Then we have $X \subset P$ , disjoint from H and $N_{X/E} = N_{X/P}$ .

Set $N = \max \{h^1(N_{X/P}(-y)), y \in Y\}$ , and let $n \geq 2N$ . Choose a general $S \subset Y$ of size n, and over every $y \in S$ , perform the surgery described in Section 3.2. Explicitly, let $\widetilde P \to P$ be the blow-up at $\sqcup _{y \in S} H_y \subset P$ , and let $R_y$ be a rational normal curve in the proper transform of $P_y$ passing through $X_y$ . Let $Z \subset \widetilde P$ be the curve

$$\begin{align*}Z = X \bigcup \cup_{y \in S} R_y.\end{align*}$$

Note that Z is a connected nodal curve with arithmetic genus

$$\begin{align*}p_a(Z) = p_a(X) + (d-1)n.\end{align*}$$

Thanks to Corollary 3.7, if we choose the rational curves $R_y$ generically, then $\phi _*N_{Z/\widetilde P}$ contains a degree n inflation of $\phi _*N_{X/P}$ at S with general defining quotients. By Corollary 2.7, we conclude that $H^1(N_{Z/\widetilde P}) = 0$ and $N_{Z/\widetilde P}$ is globally generated.

Consider the Hilbert scheme of curves in $\widetilde P$ . Since $H^1(N_{Z/\widetilde P}) = 0$ , the Hilbert scheme is smooth at $[Z \subset \widetilde P]$ (see [36, Theorem 3.2.12]). Furthermore, since $N_{Z/\widetilde P}$ is globally generated, for every node $z \in Z$ , the surjective map

$$\begin{align*}N_{Z/\widetilde P} \to \operatorname{\mathscr{E}xt}^1_{{\mathcal O}_Z}(\Omega_Z, {\mathcal O}_Z)\big|_z\end{align*}$$

is also surjective on global sections. As a result, Z is the flat limit of a family of smooth curves in $\widetilde P$ (see [20, Proposition 1.1]). Let $X' \subset \widetilde P$ be a general member of such a family. By semi-continuity, the vanishing of $H^1$ and global generation continue to hold for $N_{X'/\widetilde P}$ .

Let $\pi {\colon } \widetilde P \to P'$ be the blow-down of all the $P_y \subset \widetilde P$ (it is helpful to refer to Figure 2 again). We now check that $X' \subset \widetilde P$ maps isomorphically to its image in $P'$ . Indeed, see that $P_y \cdot Z = 1$ , and hence $P_y \cdot X' = 1$ . Since $X'$ is smooth and connected, $P_y \cap X'$ consists of a single (reduced) point, and hence, the blow-down of $P_y$ does not change $X'$ .

Let $H' \subset P'$ be the proper transform of $H \subset P$ . Plainly, $X' \subset P'$ stays away from $H'$ .

We claim that $P' \setminus H'$ is the total space of $E' = E \otimes {\mathcal O}_Y(S)$ . Granting this claim, it is easy to see that $X' \subset E'$ is the canonical affine embedding and $H^1(N_{X'/E'}) = 0$ . Indeed, $E'$ has the correct degree

$$ \begin{align*} \deg E' &= \deg E + n(d-1) \\ &= p_a(X) - 1 - d(g_Y-1) + n(d-1)\\ &= p_a(X') - 1 - d(g_Y-1), \end{align*} $$

and a general fibre of $X' \to Y$ is in affine general position in $E'$ , so Proposition 2.12 applies. To see the vanishing of $H^1$ , observe that we have an injection

$$\begin{align*}N_{X'/\widetilde P} \xrightarrow{d\pi} N_{X'/P'}.\end{align*}$$

Since $H^1(N_{X'/\widetilde P}) = 0$ , we get $H^1(N_{X'/P'}) = H^1(N_{X'/E'}) = 0$ .

Let us prove the remaining claim that $P'\setminus H'$ is the total space of the bundle $E' = E \otimes {\mathcal O}_Y(S)$ . Choose a point $y \in S$ , and consider the rational map

(3.16) $$ \begin{align} \mathbf{P}({\mathcal O}_Y \oplus E^\vee) \dashrightarrow \mathbf{P}({\mathcal O}_Y \oplus E^\vee(-y)) \end{align} $$

induced by the natural map

$$\begin{align*}{\mathcal O}_Y \oplus E^\vee(-y) \to {\mathcal O}_Y \oplus E^\vee. \end{align*}$$

Let t be a local coordinate on Y at y. Let x ₁, …, x _d−1 be generators of $E^\vee $ around y. Then $tx_1, \dots , tx_{d-1}$ are generators of $E^\vee (-y)$ around y. Together with $1 \in {\mathcal O}_Y$ , we get local generators for ${\mathcal O}_Y \oplus E^\vee $ and ${\mathcal O}_Y \oplus E^\vee (-y)$ , respectively, and hence homogeneous coordinates on their projectivizations. In these coordinates, the rational map in equation (3.16) is given by

$$ \begin{align*} \begin{split} (t,[x_0:x_1:\dots:x_{d-1}]) &\mapsto (t, [x_0:tx_1:\dots:tx_{d-1}]).\\ \end{split} \end{align*} $$

It is easy to check explicitly that this map is the composite of the blow-up at $H_y \subset P = \mathbf {P}({\mathcal O}_Y \oplus E)$ and the blow-down of the proper transform of $P_y$ ; see [19] for the case of $d = 2$ . It is also easy to see that this map transforms the hyperplane at infinity in the domain to the hyperplane at infinity in the codomain. By performing this operation at every $y \in S$ , we see that $P'$ is $\mathbf {P}({\mathcal O}_Y \oplus E^\vee (-S))$ , and $H' \subset P'$ is the hyperplane at infinity. In particular, $P' \setminus H'$ is the total space of $E' = E \otimes {\mathcal O}_Y(S)$ .

Finally, instead of $n \geq 2N$ , if we take $n \geq 2N + g(Y)$ , then the additional freedom to choose the $g(Y)$ points allows us to put S in any prescribed divisor class of degree n.

3.4 The general case

We now use the results of Section 3.1 and Section 3.3 to deduce the main theorem. Recall that Y is a connected, projective and smooth curve over k, an algebraically closed field with $\operatorname {char} k = 0$ or $\operatorname {char} k> d$ .

Theorem 3.9. Let E be a vector bundle on Y of rank $(d-1)$ . There exists an n (depending on E) such that for any line bundle L of degree at least n, there exists a smooth curve X and a finite flat morphism $\phi {\colon } X \to Y$ of degree d such that $E_\phi \cong E \otimes L$ . Furthermore, we have $H^1(X, N_{X/ E \otimes L}) = 0$ , where $X \subset E \otimes L$ is the canonical affine embedding.

Proof. Choose an isotrivial degeneration $E_0$ of E of the form

$$\begin{align*}E_0 = L_1 \oplus \dots \oplus L_{d-1},\end{align*}$$

where the $L_i$ ’s are line bundles with $\deg L_i + (2g_Y-1) \leq \deg L_{i+1}$ . That is, let $(\Delta , 0)$ be a pointed curve and $\mathcal E$ a vector bundle on $Y \times \Delta $ such that $\mathcal E|_0 = E_0$ and $\mathcal E|_t \cong E$ for all $t \in \Delta \setminus \{0\}$ . Such a degeneration exists by Proposition 2.10. Let $\pi {\colon } Y \times \Delta \to Y$ be the first projection. After replacing $\mathcal E$ by $\mathcal E \otimes \pi ^*\lambda $ for a line bundle $\lambda $ on Y of large degree, we may also assume that $\deg L_1 \geq 2g_Y-1$ .

By Proposition 3.1, there exists a nodal curve W and a finite flat morphism $W \to Y$ with Tschirnhausen bundle $E_0$ . By the key proposition (Proposition 3.8), there exists an n such that for any line bundle L of degree at least n, we can find a smooth curve $X_0$ and a finite map $X_0 \to Y$ with Tschirnhausen bundle $E_0' = E_0 \otimes L$ . Furthermore, we can make $X_0$ satisfy $H^1(N_{X_0/E_0'}) = 0$ . Set $\mathcal E' = \mathcal E \otimes \pi ^*L$ . Let $\mathcal H$ be the connected component of the relative Hilbert scheme of $\operatorname {Tot}(\mathcal E') \to \Delta $ containing the point $[X_0 \subset E_0']$ .

Before we proceed, we clarify what we mean by the relative Hilbert scheme for a quasi-projective morphism such as $\operatorname {Tot}(\mathcal E') \to \Delta $ . Consider the functor

$$\begin{align*}\mathrm{Hilb} \colon \Delta\text{-Schemes} \to \text{Sets}\end{align*}$$

defined by

$$\begin{align*}\mathrm{Hilb}(S) = \{Z \subset \operatorname{Tot}(\mathcal E') \times_\Delta S \mid Z \to S \text{ is proper and flat}.\}\end{align*}$$

The functor $\mathrm {Hilb}$ is represented by a disjoint union of quasi-projective $\Delta $ -schemes. (The union is indexed by the Hilbert polynomial with respect to a chosen projective embedding.) By the relative Hilbert scheme, we mean this disjoint union. We refer the reader to [2, Section 2], more precisely [2, Corollary 2.8], for a proof of the representability of $\mathrm {Hilb}$ . The infinitesimal study of the Hilbert scheme is equivalent to the deformation theory of closed subschemes, for which we refer to [26, I.2].

Recall that $\mathcal H$ is the connected component of the relative Hilbert scheme of $\operatorname {Tot}(\mathcal E') \to \Delta $ containing the point $[X_0 \subset E_0']$ . We use the infinitesimal lifting criterion for smoothness to show that the map $\mathcal H \to \Delta $ is smooth at $[X_0 \subset E^{\prime }_0]$ . Let $\iota \colon B \to A$ be a surjection of local Artin k-algebras such that the kernel of $\iota $ is isomorphic to k as a B-module. Suppose we are given a map $\operatorname {Spec} B \to \Delta $ and a subscheme $\mathcal X_A \subset \operatorname {Tot}(\mathcal E') \times _\Delta \operatorname {Spec} A$ , flat over $\operatorname {Spec} A$ with special fibre $X_0 \subset E^{\prime }_0$ . The obstruction to extending $\mathcal X_A$ to a subscheme $\mathcal X_B \subset \operatorname {Tot}(\mathcal E') \times _\Delta \operatorname {Spec} B$ , flat over $\operatorname {Spec} B$ , lies a priori in $\operatorname {Ext}^1(I_{X_0/E^{\prime }_0}, {\mathcal O}_{X_0})$ by [26, Proposition 2.5]. Since $X_0$ and $E_0'$ are smooth, the inclusion $X_0 \subset E^{\prime }_0$ is a local complete intersection. As a result, $X_0 \subset E^{\prime }_0$ is locally unobstructed by [26, Lemma 2.12], and hence the obstruction lies in $H^1(X_0, N_{X_0/E^{\prime }_0})$ by [26, Proposition 2.14]. We know that $H^1(X_0, N_{X_0/E^{\prime }_0}) = 0$ , and hence we conclude that $\mathcal X_B$ exists. By the lifting criterion for smoothness, the map $\mathcal H \to \Delta $ is smooth at $[X_0 \subset E^{\prime }_0]$ .

Since $\mathcal H \to \Delta $ is smooth at a point, it is dominant. As a result, there exists a point $[X \subset \mathcal E^{\prime }_t] \in \mathcal H$ , where X is smooth and $t \in \Delta $ is generic. By the choice of $\mathcal E$ , we have $\mathcal E^{\prime }_t = E \otimes L$ . Since $H^1(N_{X_0/E_0 \otimes L}) = 0$ , we get that $H^1(N_{X/E \otimes L}) = 0$ by semi-continuity. Let $\phi {\colon } X \to Y$ be the projection. Since $X_0 \subset E_0 \otimes L$ is the canonical affine embedding, Proposition 2.12 implies that $X \subset E \otimes L$ is also the canonical affine embedding. The proof is now complete.

Remark 3.10. Theorem 3.9 can be stated in terms of moduli stacks of covers and bundles in the following way. Denote by $\mathcal H_d(Y)$ the stack whose S points are finite flat degree d morphisms $\phi {\colon } C \to Y \times S$ , where $C \to S$ is a smooth curve. Let $\operatorname {Vec}_{d-1}(Y)$ be the stack whose S points are vector bundles of rank $(d-1)$ on $Y \times S$ . Both $\mathcal H_d(Y)$ and $\operatorname {Vec}_{d-1}(Y)$ are algebraic stacks, locally of finite type and smooth over k. The rule

$$\begin{align*}\tau {\colon} \phi \mapsto E_\phi\end{align*}$$

defines a morphism $\tau {\colon } \mathcal H_d(Y) \to \operatorname {Vec}_{d-1}(Y)$ . Then Theorem 3.9 says that given $E \in \operatorname {Vec}_{d-1}(Y)$ and given any line bundle L on Y of large enough degree, there exists a point $[\phi {\colon } X \to Y]$ of $\mathcal H_d(Y)$ such that $\tau (\phi ) = E \otimes L$ and furthermore such that the map $\tau $ is smooth at $[\phi ]$ .

3.5 Hurwitz spaces and Maroni loci

We turn to the proof of Theorem 1.5 stated in the introduction. First we establish notation and conventions regarding the various Hurwitz spaces. Throughout Section 3.5, take the base field $k = \mathbf {C}$ .

Let $\mathcal H_{d,g}^{\mathrm {all}}(Y)$ be the stack whose objects over S are S-morphisms $\phi {\colon } C \to Y \times S$ , where $C \to S$ is a smooth, proper, connected curve of genus g, and $\phi $ is a finite morphism of degree d. Observe that $\mathcal H_{d,g}^{\mathrm {all}}(Y)$ is an open substack of the Kontsevich stack of stable maps $\overline {\mathcal M}_g(Y, d[Y])$ constructed, for example, in [17] or in [6]. As a result, $\mathcal H_{d,g}^{\mathrm {all}}(Y)$ is a separated Deligne–Mumford stack of finite type over k. Using the deformation theory of maps [36, Example 3.4.14], it follows that $\mathcal H_{d,g}^{\mathrm {all}}(Y)$ is smooth and equidimensional of dimension $b = (2g-2)-d(2g_Y-2)$ . Denote by $\mathcal H_{d,g}^{\mathrm {simple}}(Y) \subset \mathcal H_{d,g}^{\mathrm {all}}(Y)$ the open substack of simply branched maps, namely the substack whose S-points correspond to maps $\phi {\colon } C \to Y \times S$ whose branch divisor $\operatorname {br} \phi \subset Y \times S$ is étale over S (the branch divisor is defined as the subscheme whose ideal is generated by the discriminant [40, Tag 0BVH]). The transformation $\phi \mapsto \operatorname {br}\phi $ gives a morphism

$$\begin{align*}\mathcal H_{d,g}^{\mathrm{all}}(Y) \to \operatorname{Sym}^{b} Y\end{align*}$$

with finite fibres. Since the source is equidimensional of the same dimension as the target and the map is quasi-finite, each component of $\mathcal H_{d,g}^{\mathrm {all}}(Y)$ maps dominantly on $\operatorname {Sym}^{b}(Y)$ . In particular, $\mathcal H_{d,g}^{\mathrm {simple}}(Y)$ is dense in $\mathcal H_{d,g}^{\mathrm {all}}(Y)$ . By a celebrated theorem of Clebsch [13], if $g_Y = 0$ , then $\mathcal H_{d,g}^{\mathrm {simple}}(Y)$ is connected (equivalently, irreducible). More generally, by [18, Theorem 9.2], the connected ( $=$ irreducible) components of $\mathcal H_{d,g}^{\mathrm {all}}(Y)$ are classified by the subgroup $\phi _* \pi _1(C)$ of $\pi _1(Y)$ . Recall that $\phi $ is called primitive if $\phi _* \pi _1(C) = \pi _1(Y)$ , or equivalently, if $\phi $ does not factor through an étale covering $\widetilde Y \to Y$ . Denote by $\mathcal H_{d,g}^{\mathrm {primitive}}(Y) \subset \mathcal H_{d,g}^{\mathrm {all}}(Y)$ the connected ( $=$ irreducible) component whose points correspond to primitive covers.

The connection between primitive and simply branched covers is the following. By [7, Proposition 2.5], if $\phi {\colon } C \to Y$ is a simply branched covering, then $\phi $ is primitive if and only if the monodromy map

$$\begin{align*}\pi_1(Y \setminus \operatorname{br} \phi) \to S_d \end{align*}$$

is surjective. Therefore, we can view $\mathcal H_{d,g}^{\mathrm {primitive}}(Y)$ as a partial compactification of the stack of simply branched covers of Y with full monodromy group $S_d$ . By convention, $\mathcal H_{d,g}(Y)$ (without any superscript) denotes the component $\mathcal H_{d,g}^{\mathrm {primitive}}(Y)$ of $\mathcal H_{d,g}^{\mathrm {all}}(Y)$ .

Being open substacks of the Kontsevich stack, the Hurwitz stacks described above admit quasi-projective coarse moduli spaces, which we denote by the roman equivalent $H_{d,g}$ of $\mathcal H_{d,g}$ .

Denote by $M_{r,k}(Y)$ the moduli space of semi-stable vector bundles of rank r and degree k on Y. This is the space $U(r,k)$ constructed by Seshadri in [38, Chapter 7.III]. It contains the coarse moduli space of stable vector bundles as an open subset, and in general, its points correspond to S-equivalence classes of semi-stable vector bundles. It satisfies an evident universal property: a vector bundle on $Y \times T$ whose restriction to every fibre $Y \times \{t\}$ is semi-stable of rank r and degree k induces a morphism $T \to M_{r,k}(Y)$ .

Let $\mathcal U \subset \mathcal H_{d,g}(Y)$ be the (possibly empty) open substack consisting of points $[\phi ] \in \mathcal H_{d,g}(Y)$ such that $E_\phi $ is semi-stable. Let $e = b/2 = g - 1 - d(g_Y-1)$ . We have a morphism $\mathcal U \to M_{d-1,e}(Y)$ defined functorially as follows. An object $\phi {\colon } C \to Y \times S$ of $\mathcal U$ maps to the unique morphism $S \to M_{d-1,e}(Y)$ induced by the bundle $E_\phi $ on $Y \times S$ . Let $U \subset H_{d,g}(Y)$ be the coarse space of $\mathcal U$ . By the universal property of coarse spaces, the morphism $\mathcal U \to M_{d-1,e}(Y)$ descends to a morphism $U \to M_{d-1,e}(Y)$ . If U is non-empty, then we can think of $U \to M_{d-1,e}(Y)$ as a rational map $H_{d,g}(Y) \dashrightarrow M_{d-1,e}(Y)$ .

Recall that Y is a smooth, projective, connected curve over $\mathbf {C}$ .

Theorem 3.11. Let $g_Y \geq 2$ . If g is sufficiently large (depending on Y and d), then the Tschirnhausen bundle associated to a general point of $H_{d,g}(Y)$ is stable. Moreover, the rational map

$$\begin{align*}H_{d,g}(Y) \dashrightarrow M_{d-1,e}(Y)\end{align*}$$

given by $[\phi ] \mapsto E_\phi $ is dominant, where $e = g-1-d(g_Y-1)$ .

The same statement holds for $g_Y = 1$ with ‘stable’ replaced by ‘regular poly-stable’.

Proof. Let $g_Y \geq 2$ ; the proof for $g_Y = 1$ is identical with ‘stable’ replaced by ‘regular poly-stable’.

Let $\phi _0 {\colon } X_0 \to Y$ be an element of the primitive Hurwitz space $H_{d,g_0}(Y)$ with Tschirnhausen bundle $E_0$ . By Theorem 3.9, for every sufficiently large n, there exists a line bundle L of degree n and $\phi {\colon } X \to Y$ with X smooth and with Tschirnhausen bundle $E = E_0 \otimes L$ with $H^1(N_{X/E}) = 0$ . Note that the genus of X is $g = g_0 + (d-1)n$ . From the proof of Theorem 3.9, we know that $X \to Y$ is obtained as a deformation of the singular curve formed by attaching vertical rational curves to $X_0$ . Recall that in a deformation, the $\pi _1$ of a general fibre surjects onto the $\pi _1$ of the special fibre. Hence, since $\pi _1(X_0) \to \pi _1(Y)$ is surjective, so is $\pi _1(X) \to \pi _1(Y)$ . That is, $X \to Y$ is primitive.

We know that the moduli stack of vector bundles on Y is irreducible [22, Appendix A], and therefore, the locus of stable bundles forms a dense open substack. So, we can find a pointed curve $(\Delta , 0)$ and a vector bundle $\mathcal E$ on $Y \times \Delta $ such that $\mathcal E_{Y \times \{0\}} = E$ and $\mathcal E_{Y \times \{t\}}$ is stable for $t \in \Delta \setminus \{0\}$ . As $H^1(N_{X/E}) = 0$ , the curve $X \subset E$ deforms to the generic fibre of $\mathcal E \to \Delta $ , by the same relative Hilbert scheme argument as used in the proof of Theorem 3.9. Let $X_t \subset \mathcal E_t$ be such a deformation. Then $X_t \to Y$ is a primitive cover with a stable Tschirnhausen bundle. We conclude that for sufficiently large g congruent to $g_0\mod {(d-1)}$ , the Tschirnhausen bundle of a general element of $H_{d,g}(Y)$ is stable. By varying $g_0$ over all congruence classes, we see that the same is true for all sufficiently large g.

Let $\phi {\colon } X \to Y$ be an element of $H_{d,g}(Y)$ with stable Tschirnhausen bundle E such that we have $H^1(N_{X/E}) = 0$ . The above argument shows that such coverings exist if g is sufficiently large. Let S be a versal deformation space for E and $\mathcal E$ a versal vector bundle on $Y \times S$ . See [29, Lemma 2.1] for a construction of S in the analytic category. In the algebraic category, we can take S to be a suitable Quot scheme (see, for example, [22, Proposition A.1]). Let $\mathcal H$ be the component of the relative Hilbert scheme of $\operatorname {Tot}(\mathcal E) / S$ containing the point $[X \subset E]$ , and let $\mathcal H^{\mathrm {sm}} \subset \mathcal H$ be the open subset parameterising $[X_t \subset \mathcal E_t]$ with smooth $X_t$ . Since $H^1(N_{X/E}) = 0$ , the map $\mathcal H^{\mathrm {sm}} \to S$ is smooth at $[X \subset E]$ by [26, Proposition 2.5]. In particular, it is dominant. By Proposition 2.12, we know that for $[X_t \subset \mathcal E_t] \in \mathcal H^{\mathrm {sm}}$ , the bundle $\mathcal E_t$ is indeed the Tschirnhausen bundle of $X_t \to Y$ . We conclude that the map $H_{d,g}(Y) \dashrightarrow M_{d-1,e}(Y)$ is dominant.

Remark 3.12. It is natural to ask for an effective lower bound on g in Theorem 3.11. By studying our proof, we get lower bounds of order $d^3g_Y$ . It may be interesting to obtain sharper results.

Recall that the Maroni locus $\operatorname {Mar}(E)$ is the locally closed subset of $H_{d,g}(Y)$ defined by

$$\begin{align*}\operatorname{Mar}(E) = \left\{[\phi] \in H_{d,g}(Y) \mid E_\phi \cong E\right\}.\end{align*}$$

Theorem 3.13. Let Y be a smooth projective curve over $\mathbf {C}$ of genus $g_Y$ . Let E be a vector bundle on Y of rank $(d-1)$ . There exists an n (depending on E) such that for every line bundle L on Y of degree at least n, the Maroni locus $\operatorname {Mar}(E \otimes L) \subset H_{d,g}(Y)$ contains an irreducible component having the expected codimension $h^{1}(\operatorname {End} E)$ . Here, g is related to E and L by the formula

$$\begin{align*}\deg E + (d-1) \deg L = g - 1 - d(g_Y-1).\end{align*}$$

Proof. By Theorem 3.9, there exists an n such that for every line bundle L on Y of degree at least n, there exists a finite cover $\phi {\colon } X \to Y$ with Tschirnhausen bundle $E' = E \otimes L$ with X smooth. Furthermore, by the same theorem, we can also arrange $H^1(N_{X/E'}) = 0$ , where $X \subset E'$ is the canonical affine embedding. Let g be the genus of X, which is related to E and L by the formula stated in the theorem.

Let $H^{\mathrm {sm}}$ be the open subset of the Hilbert scheme of curves in $\operatorname {Tot}(E')$ parameterising $[X \subset E']$ with X smooth of genus g, finite of degree d over Y and such that for all $y \in Y$ , the scheme $X_y \subset E^{\prime }_y$ is in affine general position. By Proposition 2.12, the Tschirnhausen bundle of $X \to Y$ is $E'$ . Therefore, we have a map

$$\begin{align*}\tau {\colon} H^{\mathrm{sm}} \to \operatorname{Mar}(E') \subset H_{d,g}(Y)\end{align*}$$

that sends $[X \subset E']$ to $[X \to Y]$ . Using the canonical affine embedding (Section 2.5), we see that $\tau $ is surjective. By Proposition 2.12, the fibres of $\tau $ are orbits under the group A of affine linear transformations of $E'$ over Y. Plainly, the action of the group is faithful.

As in the beginning of the proof, let $[X \subset E'] \in H^{\mathrm {sm}}$ be such that $H^1(N_{X/E'}) = 0$ . We now do a dimension count. Note that $N_{X/E'}$ is a vector bundle on X of rank $(d-1)$ and degree $(d+2)e'$ , where $e' = g_X-1 - d(g_Y-1)$ . Then the dimension of $H^{\mathrm {sm}}$ at $[X \subset E']$ is given by

$$ \begin{align*} \dim_{[X]}H^{\mathrm{sm}} &= \chi(N_{X/E'}) \\ &= (d+2)e' - (g_X-1)(d-1) \\ &= 3e' - d(d-1)(g_Y-1). \end{align*} $$

The dimension of the fibre of $\tau $ is given by

$$ \begin{align*} \dim A &= \dim \operatorname{Hom}(E^{\prime\vee}, {\mathcal O}_Y \oplus E^{\prime\vee}) \\ &= e' - d(d-1)(g_Y-1) + h^1(\operatorname{End} E). \end{align*} $$

As a result, the dimension of $\operatorname {Mar}(E')$ at $[\phi ]$ is given by

$$ \begin{align*} \dim_{[\phi]} \operatorname{Mar}(E') &= \dim_{[X]}H^{\mathrm{sm}} - \dim A \\ &= 2e' - h^1(\operatorname{End} E). \end{align*} $$

Since $\dim H_{d,g}(Y) = 2e'$ , the proof is complete.

4 Higher dimensions

In this section, we discuss the possibility of having an analogue of Theorem 1.1 for higher-dimensional Y. For simplicity, take $k = \mathbf {C}$ .

Let us begin with the following question.

Question 4.1. Let Y be a smooth projective variety, L an ample line bundle on Y and E a vector bundle of rank $(d-1)$ on Y. Is $E \otimes L^{n}$ a Tschirnhausen bundle for all sufficiently large n?

The answer to Question 4.1 is ‘No’, at least without additional hypotheses.

Example 4.2. Take $Y = \mathbf {P}^{4}$ and $E = {\mathcal O}(a) \oplus {\mathcal O}(b)$ . Then a sufficiently positive twist $E'$ of E cannot be the Tschirnhausen bundle of a smooth branched cover X.

To see this, recall that the data of a Gorenstein triple cover $X \to Y$ with Tschirnhausen bundle $E'$ is equivalent to the data of a nowhere vanishing global section of $\operatorname {Sym}^{3}E' \otimes (\det E')^{\vee }$ (see [28] or [12]). For $E' = E \otimes L^n$ with large n, the rank $4$ vector bundle $\operatorname {Sym}^{3}E' \otimes (\det E')^{\vee }$ is very ample. Thus, its fourth Chern class is nonzero. Therefore, a general global section must vanish at some points.

In fact, it is easy to see by direct calculation that the fourth Chern class of $\operatorname {Sym}^{3}E \otimes (\det E)^{\vee }$ can vanish if and only if $E = {\mathcal O}(a) \oplus {\mathcal O}(b)$ , where $b = 2a$ . Conversely, $E = {\mathcal O}(a) \oplus {\mathcal O}(2a)$ is the Tschirnhausen bundle of a cyclic triple cover of $\mathbf {P}^4$ . Thus, $E = {\mathcal O}(a) \oplus {\mathcal O}(b)$ can be a Tschirnhausen bundle of a smooth triple cover of $\mathbf {P}^4$ if and only if $b=2a$ .

Example 4.2 illustrating the failure of Theorem 1.1 can be generalized to all degrees $\geq 3$ , provided the base Y is allowed to be high-dimensional.

Proposition 4.3. Let $d \geq 3$ . The answer to Question 4.1 is ‘No’ for all Y of dimension at least $d {d \choose 2}$ .

Proof. Let $\phi {\colon } X \to Y$ be a finite, flat, degree d map. Then the sheaf $\phi _{*}{\mathcal O}_{X}$ is a sheaf of ${\mathcal O}_{Y}$ -algebras, and it splits as $\phi _{*} = {\mathcal O}_{Y}\oplus E^{\vee }$ .

Suppose over some point $y \in Y$ , the multiplication map

$$\begin{align*}m : \operatorname{Sym}^{2}E^{\vee} \to \phi_{*}{\mathcal O}_{X}\end{align*}$$

is identically zero. Then, we have a k-algebra isomorphism

$$\begin{align*}(\phi_{*}{\mathcal O}_{X})|_y \cong k[x_{1}, \dots, x_{d-1}]/(x_{1}, \dots, x_{d-1})^{2}.\end{align*}$$

That is, $\phi ^{-1}(y)$ is isomorphic to the length d ‘fat point’, defined by the square of the maximal ideal of the origin in an affine space. When $d \geq 3$ , these fat points are not Gorenstein. Since Y is smooth, this implies X can not even be Gorenstein, let alone smooth.

Now, if E is a vector bundle on Y and L is a sufficiently positive line bundle, then the bundle

$$\begin{align*}M := \operatorname{Hom}(\operatorname{Sym}^{2}(E \otimes L)^{\vee}, {\mathcal O}_{Y} \oplus (E \otimes L)^{\vee})\end{align*}$$

is very ample. A general global section $m \in H^{0}(Y, M)$ will vanish identically at some points $y \in Y$ provided

$$\begin{align*}\dim Y \geq \operatorname{rk} M = d {d \choose 2}.\end{align*}$$

We conclude that if $\dim Y \geq d {d \choose 2}$ , then Question 4.1 has a negative answer.

Observe that Proposition 4.3 remains true even if we relax the requirement that X be smooth to X be Gorenstein.

The following result due to Lazarsfeld suggests the possibility that Proposition 4.3 may be true with a much better lower bound than $d { d \choose 2}$ .

Proposition 4.4. Let E be a vector bundle of rank $(d-1)$ on $\mathbf {P}^{r}$ , where $r \geq d+1$ . Then $E(n)$ is not a Tschirnhausen bundle of a smooth, connected cover for sufficiently large n.

Proof. The proof relies on [27, Proposition 3.1], which states that for a branched cover $\phi {\colon } X \to \mathbf {P}^{r}$ of degree $d \leq r-1$ with X smooth and connected, the pullback map

$$\begin{align*}\phi^{*} {\colon} \operatorname{Pic}(\mathbf{P}^{r}) \to \operatorname{Pic} X\end{align*}$$

is an isomorphism. In particular, the dualizing sheaf $\omega _\phi $ is isomorphic to $\phi ^* {\mathcal O}(l)$ for some l. Note that $\omega _\phi $ is represented by an effective divisor (the ramification divisor), so $ l> 0$ . Therefore, we get

$$\begin{align*}{\mathcal O}_{\mathbf{P}^r} \oplus E = \phi_* \omega_\phi \cong \phi_* {\mathcal O}(l) = {\mathcal O}_{\mathbf{P}^r}(l) \oplus E^\vee(l).\end{align*}$$

Since X is connected, $E^\vee $ has no global sections. Using this, it is easy to conclude from the above sequence that ${\mathcal O}_{\mathbf {P}^r}(l)$ is a summand of E.

Suppose $E(n)$ is a Tschirnhausen bundle of a smooth connected cover for infinitely many n. Applying the reasoning above with E replaced by $E(n)$ shows that E must have line bundle summands of infinitely many degrees. Since this is impossible, the proposition follows.

The reasoning in Example 4.2 implies the following.

Proposition 4.5. For degree 3, Question 4.1 has an affirmative answer if and only if $\dim Y < 4$ .

Proof. Let $\phi {\colon } X \to Y$ be a Gorenstein finite covering of degree 3 with Tschirnhausen bundle E. Then by the structure theorem of triple covers in [28] or [12], we get an embedding $X \subset \mathbf {P} E$ as a divisor of class ${\mathcal O}_{\mathbf {P} E}(3)$ . Thus, X is given by a global section on $\mathbf {P} E$ of ${\mathcal O}_{\mathbf {P} E}(3)$ , or equivalently a global section on Y of $\operatorname {Sym}^3 E \otimes \det E^\vee $ . Note that since $X \to Y$ is flat, the global section of $\operatorname {Sym}^3 E \otimes \det E^\vee $ is nowhere vanishing.

Suppose we are given an arbitrary rank $2$ vector bundle E on Y. Set $D = {\mathcal O}_{\mathbf {P} E}(3)$ and $V = \operatorname {Sym}^3 E \otimes \det E^\vee $ . If we twist E by $L^n$ , then $\mathbf {P} E$ is unchanged but D changes to $D + 3nL$ and V changes to $V \otimes L^n$ . For sufficiently large n, the bundle $V \otimes L^n$ is ample. If $\dim Y < 4$ , then a general section of $V \otimes L^n$ is nowhere zero on Y. Furthermore, the divisor $X \subset \mathbf {P} E$ cut out by the corresponding section of ${\mathcal O}(D+3nL)$ is smooth by Bertini’s theorem. By construction, the resulting $X \to Y$ has Tschirnhausen bundle $E \otimes L^n$ .

On the other hand, if $\dim Y \geq 4$ , then every global section of $V \otimes L^n$ must vanish at some point in Y. Thus, $E \otimes L^n$ cannot arise as a Tschirnhausen bundle.

4.1 Modifications of the original question

Following the discussion in the previous section, natural modified versions of Question 4.1 emerge. The first obvious question is the following.

Question 4.6. Is the analogue of Theorem 1.1 true for all Y with $\dim Y \leq d$ ?

We can also relax the finiteness assumption on $\phi $ .

Question 4.7. Let Y be a smooth projective variety and E a vector bundle in Y. Is E isomorphic to $(\phi _* {\mathcal O}_X/{\mathcal O}_Y)^\vee $ , up to a twist, for a generically finite map $\phi {\colon } X \to Y$ with smooth X?

Remark 4.8. A similar question is addressed in work of Hirschowitz and Narasimhan [21], where it is shown that any vector bundle on Y is the direct image of some line bundle on a smooth variety X under a generically finite morphism.

Alternatively, we can keep the finiteness requirement on $\phi $ in exchange for the smoothness of X. We end the paper with the following open-ended question.

Question 4.9. What singularity assumptions on X (or the fibres of $\phi $ ) yield a positive answer to Question 4.1?