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.

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.

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.

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.

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.

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 $I_{X/P} \to I_{p/R}$ .

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 $\mathcal E|_{Y \times \{0\}} \cong F_0/F_1 \oplus \dots \oplus F_{r-1}/F_r$ .

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.

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.

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 [Reference Sacchiero34, II] or [Reference Sernesi36, 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 $F \cong {\mathcal O}_R(1)$ .

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

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

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.

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.

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 $\phi _*N_{Z/\widetilde P}$ .

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 [Reference Sernesi36, 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 [Reference Hartshorne and Hirschowitz20, 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 [Reference Hartshorne19] 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.

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 [Reference Altman and Kleiman2, Section 2], more precisely [Reference Altman and Kleiman2, 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 [Reference Kollár26, 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 [Reference Kollár26, 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 [Reference Kollár26, Lemma 2.12], and hence the obstruction lies in $H^1(X_0, N_{X_0/E^{\prime }_0})$ by [Reference Kollár26, 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.

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 [Reference Hoffmann22, 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 [Reference Narasimhan and Seshadri29, 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, [Reference Hoffmann22, 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 [Reference Kollár26, 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.

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.

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

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?

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