2.1 Moduli spaces of pointed curves with prescribed ramification

Let $ {\mathcal M}_{g,n}$ denote the moduli space of smooth n-pointed genus g curves. For $(C,x_1,\ldots , x_n) \in {\mathcal M}_{g,n}$ and $\mathbf {e}=(e_1,\ldots ,e_n)$ we let

$$\begin{align*}G^1_k(C,(x_1,e_1),\ldots,(x_n,e_n))\end{align*}$$

be the variety parametrizing $g^1_k$ s on C with ramification order $e_i$ at $x_i$ . Note that

$$\begin{align*}e \leq 2(k-1+g)+n \end{align*}$$

(recall (2.2)) by Riemann-Hurwitz. In the classical (unramified) setting, $G^1_k(C)$ denotes the variety parametrizing $g^1_k$ s on C. For a general curve C, a necessary and sufficient condition for $G^1_k(C)\neq \emptyset $ is $\rho (g,1,k)\geq 0$ , where $\rho $ is the Brill-Noether number.

Let ${\mathcal M}_{g,k,\mathbf {e}} \subset {\mathcal M}_{g,n}$ be the locus of n-pointed curves $(C,x_1,\ldots ,x_n)$ such that $G^1_k(C,(x_1,e_1),\ldots ,(x_n,e_n)) \neq \emptyset $ . A necessary and sufficient condition for such nonemptiness on a general n-pointed curve is that $\widetilde {\rho }=\widetilde {\rho }(g,1,k;e_1,\ldots ,e_n)\geq 0$ , where

(2.3) $$ \begin{align} \widetilde{\rho}=\widetilde{\rho}(g,1,k;e_1,\ldots,e_n):=\rho(g,1,k)-\sum_{i=1}^n(e_i-1)=2k-2-g-e+n \end{align} $$

is the adjusted Brill-Noether number. In other words, ${\mathcal M}_{g,k,\mathbf {e}} ={\mathcal M}_{g,n}$ if and only if $\widetilde {\rho }=\widetilde {\rho }(g,1,k;e_1,\ldots ,e_n)\geq 0$ . In contrast to the classical case (with no ramification), the ‘if statement’ relies on the computation of a nonzero intersection number of Schubert classes in the Chow ring of the Grassmannian ${\mathbb G}(d -2, d)$ provided by Ossermann, cf., [Reference Harris and Morrison23, Thm. 5.42] and [Reference Osserman35, Thms. 2.4 and 2.6].

We denote by ${\mathcal G}^1_{g,k,\mathbf {e}}$ the variety parametrizing pairs $\left ((C,x_1,\ldots ,x_n),\mathfrak {g}\right )$ such that $(C,x_1,\ldots ,x_n) \in {\mathcal M}_{g,k,\mathbf {e}}$ and $\mathfrak {g} \in G^1_k(C,(x_1,e_1),\ldots ,(x_n,e_n))$ and by

(2.4) $$ \begin{align} \kappa_{g,k,\mathbf{e}}:{\mathcal G}^1_{g,k,\mathbf{e}} \longrightarrow {\mathcal M}_{g,k,\mathbf{e}} \end{align} $$

the forgetful map, with fibre over $(C,x_1,\ldots ,x_n)$ being $G^1_k(C,(x_1,e_1),\ldots ,(x_n,e_n))$ . This is related to the Hurwitz moduli space

$$\begin{align*}{\mathcal H}_{g,k,\mathbf{e}}:={\mathcal H}_{g,k}(e_1,\ldots,e_n,2^r) \end{align*}$$

parametrizing degree k covers $X \to {\mathbb P}^1$ with X a smooth projective curve of genus g with points of ramification $e_i$ over n ordered points of the target, and simple ramification over $r:=2(k-1+g)+n-e$ ordered points of the target completing the ramification profileFootnote ¹, cf., for example, [Reference Harris and Mumford24, Reference Abramovich, Corti and Vistoli1, Reference Mochizuki31, Reference Faber and Pandharipande15]. We have a natural finiteFootnote ² forgetful map dominating each componentFootnote ³

(2.5) $$ \begin{align} \lambda_{g,k,\mathbf{e}}: {\mathcal H}_{g,k,\mathbf{e}} \longrightarrow {\mathcal G}^1_{g,k,\mathbf{e}}. \end{align} $$

By Riemann’s Existence TheoremFootnote ⁴ (cf., e.g., [Reference Miranda30, III, Cor. 4.10]),

(2.6) $$ \begin{align} {\mathcal H}_{g,k,\mathbf{e}} \;\; \text{and} \;\; {\mathcal G}^1_{g,k,\mathbf{e}} \;\; \text{are equidimensional of dimension} \;\; (3g-3+n)+\widetilde{\rho}, \text{if nonempty}. \end{align} $$

This, along with the classical result [Reference Eisenbud and Harris13, Thm. 4.5] of Eisenbud and Harris when $\widetilde {\rho } \geq 0$ , implies that all components of $G^1_k(C,(x_1,e_1),\ldots ,(x_n,e_n))$ have dimension at least $ \max \{0,\widetilde {\rho }\}$ , and if equality holds for some $(C,x_1,\ldots ,x_n)$ , then ${\mathcal M}_{g,k,\mathbf {e}}$ has a component of codimension

$$\begin{align*}\max\{0,-\widetilde{\rho}\}\end{align*}$$

in ${\mathcal M}_{g,n}$ . We call this the expected codimension of ${\mathcal M}_{g,k,\mathbf {e}}$ . One of the main results of this paper proves the existence of a component of the expected codimension under the additional condition $\widetilde {\rho } \geq -g$ :

Theorem 2.1. In the setting (2.1)-(2.2), assume that $\widetilde {\rho }(g,1,k;\mathbf {e}) \geq -g$ . Then ${\mathcal M}_{g,k,\mathbf {e}}$ has a component of the expected codimension $\max \{0,-\widetilde {\rho }\}$ in ${\mathcal M}_{g,n}$ whose general member $(C,x_1,\ldots ,x_n)$ satisfies

$$\begin{align*}\dim G^1_k(C,(x_1,e_1),\ldots,(x_n,e_n))=\max\{0,\widetilde{\rho}\}.\end{align*}$$

As a consequence, we also solve the classical problem regarding the dominance or generic finiteness of the forgetful map from the Hurwitz scheme ${\mathcal H}_{g,k,\mathbf {e}}$ to ${\mathcal M}_g$ :

(where $\lambda _{g,k,\mathbf {e}}$ and $\kappa _{g,k,\mathbf {e}}$ are as in (2.5) and (2.4), respectively, and $\phi _{g,k,\mathbf {e}}$ is the forgetful map). Since $\dim {\mathcal H}_{g,k,\mathbf {e}}= \dim {\mathcal G}^1_{g,k,\mathbf {e}} =\dim {\mathcal M}_g+(n+\widetilde {\rho })$ one may expect, as best case scenario, this map to be dominant if $n+\widetilde {\rho } \geq 0$ and generically finite (on some component) if $n+\widetilde {\rho } < 0$ , so that, in the latter case, the codimension of its image is $-(n+\widetilde {\rho })$ . We prove that, under the assumption $\widetilde {\rho } \geq -g$ , this best case scenario actually happens:

Both Theorems 2.1 and 2.3 will be proved in §3.

2.2 Gonality loci on $K3$ surfaces

Let $(S,H)$ be a primitively polarized complex $K3$ surface of genus $p \geq 2$ . Let $V_{|H|,\delta }\subseteq \vert H\vert $ be the Severi variety of curves with $0 \leq \delta \leq p$ nodes, which is known to be nonempty of dimension $g:=p-\delta $ , if $(S,H)$ is general in moduli. It has recently been proved in [Reference Bruno and Lelli-Chiesa4] that $V_{|H|,\delta }$ is irreducible for general $(S,H)$ whenever $g \geq 4$ and connected whenever $g\in \{1,2,3\}$ , but we will not make use of this.

For $C \in V_{|H|,\delta }$ , denote by $\nu :C^{\nu } \to C$ its normalization map. Then $C^{\nu }$ has genus g. We have a moduli map

$$\begin{align*}m_g: V_{|H|,\delta} \longrightarrow {\mathcal M}_g \end{align*}$$

mapping $[C]$ to $[C^{\nu }]$ . We let ${\mathcal M}^1_{g,k}$ denote the locus of curves carrying a $g^1_k$ , which is known to be irreducible of codimension $\max \{0,\rho (g,1,k)\}$ . We set

$$\begin{align*}V^k_{|H|,\delta}:=m_g^{-1}({\mathcal M}^1_{g,k}), \end{align*}$$

that is, $V^k_{|H|,\delta }\subseteq V_{|H|,\delta }$ is the subvariety of curves whose normalizations carry a $g^1_k$ . If $(S,H)$ is general, then every smooth curve in $|H|$ is Brill-Noether general by Lazarsfeld’s famous result [Reference Lazarsfeld28, Lem. 1.1], whence $V^k_{|H|,0} \neq \emptyset $ if and only if $\rho (p,1,k) \geq 0$ , that is, $p \leq 2k-2$ ; in fact $V^k_{|H|,0} = V_{|H|,0}$ when $p \leq 2k-2$ . In [Reference Ciliberto and Knutsen12, Thm. 0.1] a complete description is given in the case $\delta>0$ : if $(S,H)$ is general, then $V^k_{|H|,\delta } \neq \emptyset $ if and only if

(2.7) $$ \begin{align} \rho(p,\alpha,k\alpha+\delta) \geq 0, \text{where}\; \; \alpha:=\Big\lfloor \frac{p-\delta}{2(k-1)}\Big\rfloor, \end{align} $$

equivalently

$$\begin{align*}\delta \geq \alpha \big(p-\delta -(k-1)(\alpha+1)\big). \end{align*}$$

In particular, this provides, quite surprisingly, the existence of curves in $|H|$ whose normalizations carry pencils with negative Brill-Noether number, in contrast to the smooth case. More precisely, it is proved that $V^k_{|H|,\delta }$ is equidimensional of dimension $\min \{2(k-1),g\}$ under this condition [Reference Knutsen, Lelli-Chiesa and Mongardi27, Rem. 5.6], and has an irreducible component whose general element is an irreducible curve C such that $G^1_k(C^{\nu })$ is again as ‘nice’ as possible, that is, $\dim G^1_k(C^{\nu })=\max \{0,\rho (g,1,k)\}$ , and when $\rho (g,1,k) \leq 0$ (resp. $\rho> 0$ ), any (resp. the general) $g^1_k$ on $C^{\nu }$ has simple ramification and all nodes of C are non-neutral with respect to itFootnote ⁵.

We will study the loci of pointed curves in $V^k_{|H|,\delta }$ with prescribed ramification:

Definition 2.4. We define $V_{|H|,\delta ,n}$ to be the $(g+n)$ -dimensional scheme parametrizing pointed curves $(C,x_1,\ldots ,x_n)$ such that $C \in V_{|H|,\delta }$ and $x_1,\ldots ,x_n$ are distinct points different from the nodes of C.

Recalling (2.1)-(2.2), we define

$$\begin{align*}V^k_{|H|,\delta,\mathbf{e}}=V^k_{|H|,\delta,e_1,\ldots,e_n}\subseteq V_{|H|,\delta,n} \end{align*}$$

to be the sublocus of $V_{|H|,\delta ,n}$ such that $G^1_k(C^{\nu },(x_1,e_1),\ldots ,(x_n,e_n)) \neq \emptyset $ .

In the definition we identify $x_i \in C$ with $\nu ^{-1}(x_i) \in C^{\nu }$ . Note that we have a moduli map

(2.8) $$ \begin{align} m_{g,n}: V_{|H|,\delta,n} \longrightarrow {\mathcal M}_{g,n}, \end{align} $$

mapping $(C,x_1,\ldots ,x_n)$ to $(C^{\nu },x_1,\ldots ,x_n)$ , and we then see that

$$\begin{align*}V^k_{|H|,\delta,\mathbf{e}}=m_{g,n}^{-1}({\mathcal M}_{g,k,\mathbf{e}}).\end{align*}$$

By (2.8) and what we said in the previous subsection, the expected codimension of $V^k_{|H|,\delta ,\mathbf {e}}$ in $V_{|H|,\delta ,n}$ is $\max \{0,-\widetilde {\rho }\}$ .

A necessary condition for the nonemptiness of $V^k_{|H|,\delta ,\mathbf {e}}$ is the nonemptiness of $V^k_{|H|,\delta }$ , which as recalled above requires (2.7) to hold. Under the assumption $\widetilde {\rho } \geq -g$ , we will prove that the latter is also a sufficient condition. More precisely, we will prove the following version of Theorems 2.1 and 2.3 on $K3$ surfaces. Since the properties in Remark 2.2 are not automatic in this case, we will make use of the following:

Theorem 2.6 will be proved in §5.

The special case of $V^k_{|H|,\delta ,k,k}$ , that is, of curves carrying pencils with two points of total ramification, will find a special application in the study of cycles on $K3$ surfaces, as we will explain in the next subsection. In this case, $n=2$ and $e_1=e_2=k$ , and one checks that the condition $\widetilde {\rho } \geq -g$ is automatically satisfied; we will actually prove that $V^k_{|H|,\delta ,k,k}$ is equidimensional of dimension $2$ , cf. (2.9) and Theorem 2.8 below.

2.3 Cycles of torsion differences of points on curves on $K3$ surfaces, Beauville-Voisin points and tautological points in ${\mathcal M}_{g,2}$

Let $(S,H)$ be a primitively polarized complex $K3$ surface of genus $p \geq 2$ and $0 \leq \delta \leq p$ . As above, for any irreducible curve $C \in |H|$ , we denote by $C^{\nu }$ its normalization and $\nu :C^{\nu }\to C$ its normalization map.

The next definition generalizes [Reference Torelli39, Def. 3.1 and 3.4] to the case of nodal curves:

Definition 2.7. Fix an integer $k \geq 1$ . We define the loci

$$\begin{align*}Z^{\prime\circ}_{k,\delta}(S,H):= \left\{ (C,p,q)\in V_{|H|,\delta,2} \; |\; k\cdot[p-q]=0\in Jac(C^{\nu})\right\} \subset V_{|H|,\delta,2} \end{align*}$$

and $Z^{\circ }_{k,\delta }(S,H)$ its image in $S \times S$ by the forgetful map. We denote their closures by $Z^{\prime }_{k,\delta }(S,H)$ and $Z_{k,\delta }(S,H)$ .

With this notation, the cases studied in [Reference Torelli39, Def. 3.1 and 3.4] are $Z^{\prime }_{k,0}(S,H)$ and $Z_{k,0}(S,H)$ .

It is immediate to see that

(2.9) $$ \begin{align} Z^{\prime\circ}_{k,\delta}(S,H)=V^k_{|H|,\delta,k,k}. \end{align} $$

Since one may check that the condition $\widetilde {\rho } \geq -g$ is satisfied in this case, Theorem 2.6 yields that $Z^{\prime }_{k,\delta }(S,H)$ has a component of dimension $2$ whenever (2.7) is satisfied, and is otherwise empty. We can improve this result to give equidimensionality and show that the dimension does not drop when forgetting the curves:

If $\delta =0$ , this says that $Z^{\prime }_{k,0}(S,H)$ and $Z_{k,0}(S,H)$ are nonempty with a $2$ -dimensional component for $p\leq 2(k-1)$ and are otherwise empty, which proves Conjecture [Reference Torelli39, Conj. 1]. On the other hand, for any $k\geq 2$ there is a $\delta _0(p,k)\geq 0$ computed by (2.7) such that for any $\delta _0(p,k)\leq \delta < p$ , the varieties $Z^{\prime }_{k,\delta }(S,H)$ and $Z_{k,\delta }(S,H)$ are nonempty (and $2$ -dimensional). In other words, one can decrease k by adding nodes, as depicted in the following example for low genus p:

We next relate the loci $Z_{k,\delta }(S,H)\subset S \times S$ provided by Theorem 2.8 to the theory of Beauville-Voisin points and constant cycle curves. Recall that, by a classical result of Mumford [Reference Mumford33], the Chow group $CH_0(S)$ of $0$ -cycles on S is huge. Nonetheless, it contains a distinguished class $c_S$ of degree $1$ , called the Beauville-Voisin class, which is defined as the class of a point on a rational curve (cf. [Reference Beauville and Voisin3, Thm. 1]). A Beauville-Voisin point (BV-point in short) $p\in S$ is a point with class $c_S$ . By [Reference Maclean29, Thm. 1.2] BV-points are dense in S and the set is expected to be a union of curves. A curve whose points all define the class $c_S$ is called constant cycle curve (cf. [Reference Huybrechts25, Def. 3.1] and [Reference Voisin40]). By [Reference Huybrechts25, Thm. 11.1] and [Reference Chen and Gounelas9, Introd.], constant cycle curves are dense.

In [Reference Torelli39] $2$ -cycles Z such that $Z_*$ preserves BV-points and constant cycle curves are introduced, a notion we rephrase as:

In [Reference Torelli39] it is shown that $Z_{k,0}(S,H)_*$ preserves constant cycle curves. We can now extend this study to $\delta>0$ . Indeed, as a consequence of Theorem 2.8, we obtain:

The latter property in the corollary allows us to give an application to tautological points in ${\mathcal M}_{g,2}$ , in view of [Reference Pandharipande and Schmitt36, Thm. 1.5] relating them to BV-points on curves on $K3$ surfaces. Recall that, by [Reference Graber and Vakil22, Thm. 1.1], the degree- $0$ tautological group $R_0(\overline {\mathcal {M}_{g,n}}) \subseteq CH_0(\overline {\mathcal {M}_{g,n}})$ is always isomorphic to $\mathbb {Q}$ , even though $CH_0(\overline {\mathcal {M}_{g,n}})$ is expected to be huge except for finitely many $(g,n)$ (cf. [Reference Pandharipande and Schmitt36, Spec. 1.1]). This setting is similar to the one on K3 surfaces. In [Reference Pandharipande and Schmitt36, Thm. 1.5] it is proved that $(C,x_1,..,x_n)\in \overline {{\mathcal M}_{g,n}}$ is tautological if C is a smooth curve of genus g sitting in a K3 surface S, $x_i\in C$ is a BV-point for $1\leq i\leq n$ and $n\leq g$ . As an application of this relation and Corollary 2.11 we obtain:

Theorem 2.8 and Corollaries 2.11 and 2.12 will be proved in §6.

4.1 Limit K3 surface

We describe now the degeneration of $(S,H)\in {\mathcal K}_p$ to a pair $(S_0,H_0)$ (cf. [Reference Ciliberto and Knutsen12, §4]).

Let E be a smooth elliptic curve with two distinct general line bundles $L_1,L_2 \in \operatorname {Pic}^2(E)$ . Consider the embedding of $E \subset {\mathbb P}^{2p-1}$ as a smooth elliptic normal curve of degree $2p$ given by $L_2^ {\otimes p}$ . Let $R_1$ and $R_2$ be the rational normal scrolls of degree $2p-2$ in ${\mathbb P}^{2p-1}$ defined by $L_1$ and $L_2$ , respectively, that is, $R_i$ is the union of lines spanned by the divisors of $\vert L_i\vert $ . Then

• ○ $R_1 \cong {\mathbb P}^1 \times {\mathbb P}^1$ and $R_2 \cong \mathbb {F}_2$ ,

• ○ $R_1 \cap R_2 =E$ transversally,

• ○ E is anticanonical on each $R_i$ .

Set $R:=R_1 \cup R_2$ .

We will denote by $f:{\mathbb P}^1 \times {\mathbb P}^1 \to {\mathbb P}^1$ the projection to ${\mathbb P}^1$ defined by the pencil $|L_1|$ . Let $\mathfrak {s}_i$ and $\mathfrak {f}_i$ denote the classes of the nonpositive section and fibre, respectively, of $R_i$ , for $1\le i\le 2$ . Thus, the members of $|\mathfrak {f}_1|$ are the fibres of f. Then ${\mathcal O}_{R_1}(1) \cong {\mathcal O}_{R_1}(\mathfrak {s}_1 +(p-1)\mathfrak {f}_1)$ and ${\mathcal O}_{R_2}(1) \cong {\mathcal O}_{R_2}(\mathfrak {s}_2 +p\mathfrak {f}_2)$ . The section $\mathfrak {s}_2$ does not intersect E, hence lies in the smooth locus of R. In particular, $\mathfrak {s}_2$ is a Cartier divisor on R, so that

(4.1) $$ \begin{align} H_0: ={\mathcal O}_{R}(1)\otimes {\mathcal O}_R(-\mathfrak{s}_2) \end{align} $$

is also Cartier, with $H_0^2=2p-2$ and $\mathfrak {s}_2 \cdot H_0=p$ . By [Reference Ciliberto and Knutsen12, Lem. 4.1] the pair $(S_0:=R,H_0)$ can be flatly deformed to a member of ${\mathcal K}_p$ . More precisely, there is a flat family $\pi :{\mathcal X}\to {\mathbb D}$ , where ${\mathbb D}$ is the disc, such that the special fibre $\pi ^{-1}(0)\simeq R$ and the fibres $S_t:=\pi ^{-1}(t)$ for $t \neq 0$ are smooth $K3$ surfaces, and a line bundle ${\mathcal H}$ on ${\mathcal X}$ such that ${\mathcal H}|_{S_0}=H_0$ and, setting $H_t:={\mathcal H}|_{S_t}$ also for $t \neq 0$ , we have that $(S_t,H_t) \in {\mathcal K}_p$ is general.

Note that the total space ${\mathcal X}$ has $16$ double points on E in the central fibre, and is otherwise smooth. One may perform a small resolution of the singularities of ${\mathcal X}$ , obtaining a new family with unchanged fibres for $t \neq 0$ , but whose new central fibre is a birational modification of R. Precisely one can make sure that the new central fibre is $R_1\cup \widetilde R_2$ , where $\widetilde R_2$ is the blow-up of $R_2$ at the $16$ singular points of ${\mathcal X}$ , and $R_1$ and $\widetilde R_2$ meet transversally along $E\subset R_1$ and its strict transform on $\widetilde R_2$ . Since we can make sure that the interesting curves we work with on R will lie off the singular points of ${\mathcal X}$ , we can choose to work with ${\mathcal X}$ without caring about the singularities.

4.2 Limit Severi varieties

We now describe limit curves of $V_{|H_t|,\delta }$ , following [Reference Ciliberto and Knutsen12, §6].

Let m be a positive integer satisfying $m \leq p$ . A chain of length $2m-1$ is a sum of $2m-1$ distinct lines

$$\begin{align*}f_{2,1}+f_{1,1}+f_{2,2}+f_{1,2} + \cdots + f_{2,m-1}+f_{1,m-1}+f_{2,m}, \; f_{i,j} \in |\mathfrak{f}_i|, \end{align*}$$

where $f_{1,j}$ intersects only $f_{2,j}$ and $f_{2,j+1}$ , for $j=1, \ldots , m-1$ . The chain intersects E in $2m$ points, consisting of m divisors of $|L_2|$ , and $2m-2$ of them lie on the intersections between two lines, whereas the remaining two are on $f_{2,1}$ and $f_{2,m}$ and will be denoted by a and b, respectively. This pair of points will be called the distinguished pair of points of the chain. Figure 1 shows a chain of length $9$ and its distinguished pair of points. If the chain is contained in a member C of $|H_0|$ , and we denote by $\Gamma $ the section of the ruling on $R_1$ contained in C, the distinguished points are $a=\Gamma \cap f_{2,1}$ and $b=\Gamma \cap f_{2,m}$ . Denoting the restriction of f to E still by f (so that $f:E \to \Gamma \cong {\mathbb P}^1$ is the morphism determined by the linear series $|L_1|$ ), we note that $a+b \in f_*\left (|mL_2-(m-1)L_1|\right )$ .

Figure 1

A chain of length $9$ contained in a member of $|H_0|$ .

We denote by ${\mathcal C}_{m}$ the family of chains of length $2m-1$ , which is a locally closed subvariety of a Hilbert scheme of curves on R. In [Reference Ciliberto and Knutsen12, Lem. 4.2] it is proved that the map

(4.2) $$ \begin{align} {\mathcal C}_m \longrightarrow |mL_2-(m-1)L_{1}| \end{align} $$

sending a chain of length $2m-1$ to its pair of distinguished points on E is birational and injective.

Definition 4.2. For each tuple $(\alpha _1,\ldots , \alpha _p)$ of nonnegative integers satisfying

(4.3) $$ \begin{align} p=\displaystyle\sum_{j=1}^p j\alpha_j \end{align} $$

one defines $V(\alpha _{1}, \ldots , \alpha _{p})$ to be the locally closed subset of reduced curves in $|H_0|$ not passing through the singular locus of ${\mathcal X}$ and containing exactly $\alpha _{j}$ chains of length $2j-1$ , for $j=1,\ldots ,p$ .

By (4.2) we have a morphism

(4.4)

which is proved to be bijective in [Reference Ciliberto and Knutsen12, Proof of Prop. 5.2]. Indeed, its inverse is given in the following way: the coordinates of any member of the target uniquely define a chain, and the intersection between the union of the lines in $|\mathfrak {f}_2|$ of these chains and E is an effective divisor lying in $|(\sum j \alpha _j)L_2|=|pL_2|=|{\mathcal O}_E(1)|$ , by (4.3). This can be extended to a unique hyperplane section of R, which necessarily contains $\mathfrak {s}_2$ , so it gives a curve in $V(\alpha _{1}, \ldots , \alpha _{p})$ . As the dimension of the target in (4.4) is

(4.5) $$ \begin{align} g:=\sum_{j=1}^p\alpha_j, \end{align} $$

we see that $V(\alpha _{1}, \ldots , \alpha _{p})$ is g-dimensional.

Any curve C in $V(\alpha _{1}, \ldots , \alpha _{p})$ is nodal, and we denote as above by $\Gamma $ the section of the ruling on $R_1$ contained in C (i.e., C is the union of $\Gamma $ and of chains). We will call $\Gamma $ the sectional component of C. The curve C has

(4.6) $$ \begin{align} \delta=\delta(\alpha_{1}, \ldots, \alpha_{p}):= \displaystyle\sum_{j=1}^p (j-1) \alpha_{j} \end{align} $$

nodes lying off E, which we call the marked nodes of C, and they all lie on its sectional component $\Gamma $ . In Figure 1 they are marked with circles. We note that (4.3), (4.5) and (4.6) yield

(4.7) $$ \begin{align} g=p-\delta. \end{align} $$

It is proved in [Reference Ciliberto and Knutsen12, Prop. 5.2] that $V(\alpha _{1}, \ldots , \alpha _{p})$ is a component of the flat limit of $V_{| H_t|,\delta }$ .

4.3 Limit Severi varieties of curves whose normalization carry a $g^1_k$

We now describe limit curves of $V^k_{|H_t|,\delta }$ and compute the dimension of the component they define, following [Reference Ciliberto and Knutsen12, §6].

As in Figure 2, we denote by $\widetilde {C}$ the partial normalization of C at its $\delta $ marked nodes and by $C'$ the stable model of $\widetilde {C}$ , obtained by contracting all chains of C. We already noted that the distinguished pair of points of each chain contained in C lies on $\Gamma $ . There are g such pairs. Thus, $C'$ is equivalently the g-nodal curve obtained by identifying each distinguished pair of points on $\Gamma $ , as shown in Figure 2. We note that the natural map $\Gamma \to C'$ is the normalization, and that a degree k cover $\Gamma \to {\mathbb P}^1$ factors through $C'$ if and only if all distinguished pairs of points lie in the same fibres of $\Gamma \to {\mathbb P}^1$ . This motivates the following definition.

Figure 2

The partial normalization $\widetilde {C}$ of C and its stable model $C'$ , around a chain.

Definition 4.3. Let $C \in V(\alpha _1,\ldots ,\alpha _p)$ . We say that a $g^1_k$ on the sectional component $\Gamma $ of C is a descending $g^1_k$ if all distinguished pairs of points of each chain contained in C belong to divisors of the $g^1_{k}$ .

We denote by $G^1_k(C')$ the variety of all descending $g^1_k$ s on $\Gamma $ and by ${\mathcal G}^k(\alpha _1,\ldots ,\alpha _p)$ the variety parametrizing pairs $(C,\mathfrak {g})$ such that $C \in V(\alpha _1,\ldots ,\alpha _p)$ and $\mathfrak {g} \in G^1_k(C')$ .

We denote by $V^k(\alpha _1,\ldots ,\alpha _p)$ the image of the forgetful map

$$\begin{align*}\sigma: {\mathcal G}^k(\alpha_1,\ldots,\alpha_p) \longrightarrow V(\alpha_1,\ldots,\alpha_p) \end{align*}$$

(which has fibres $G^1_k(C')$ ).

By what we said prior to the definition, C lies in $V^k(\alpha _1,\ldots ,\alpha _p)$ if and only if $C'$ lies in $\overline {{\mathcal M}^1_{g,k}}$ (see [Reference Ciliberto and Knutsen12, Lem. 5.3]).

In [Reference Ciliberto and Knutsen12, Prop. 6.5 and Cor. 6.6] the following is proved:

Proposition 4.4. If (4.3) and

(4.8) $$ \begin{align} \alpha_{j} \leq 2(k-1) \; \; \text{for all} \; \; j \end{align} $$

hold, the variety $V^k( \alpha _1,\ldots ,\alpha _p)$ is nonempty and irreducible of dimension $\min \{2(k-1),g\}$ , and is contained in the limit of $V^k_{|H_t|,\delta }$ as t tends to $0$ .

It is also proved in [Reference Ciliberto and Knutsen12] that there are $(\alpha _1,\ldots ,\alpha _p)$ satisfying conditions (4.3) and (4.8) whenever (2.7) is satisfied (cf. [Reference Ciliberto and Knutsen12, Prop. 7.1]).

To prove Proposition 4.4, the variety $V^k( \alpha _1,\ldots ,\alpha _p)$ is explicitly constructed in the following way. Let $f: E \rightarrow {\mathbb P}^1$ be the morphism determined by $\vert L_1\vert $ and $f^{(2)}: \operatorname {Sym}^2(E) \rightarrow \operatorname {Sym}^2({\mathbb P}^1)$ the induced map. We identify $\operatorname {Sym}^2({\mathbb P}^1)$ with ${\mathbb P}^2$ as in the previous section. We denote by $\mathfrak {c}_{j}$ , for $1\le j \le n$ , the image via $f^{(2)}$ of the smooth rational curves in $\operatorname {Sym}^2(E)$ defined by the pencils $|jL_{2}-(j-1)L_{1}|$ . These are distinct conics, each intersecting the diagonal $\Delta $ in four distinct points corresponding to the ramification points of the pencils. Recall from the previous section that to each $\mathfrak {g} \in G^1_k({\mathbb P}^1)$ we associate the degree $k-1$ curve in ${\mathbb P}^ 2=\operatorname {Sym}^2({\mathbb P}^1)$ given by $ C_{\mathfrak {g}}= \{ W \in \operatorname {Sym}^2({\mathbb P}^1) \; | \; \mathfrak {g}(-W) \geq 0\}$ and that we denote the family of such curves by ${\mathcal F}_k$ (cf. Definition 3.1).

We state here a stronger version of [Reference Ciliberto and Knutsen12, Lem. 6.3], as we will need it later to prove our main results:

By (4.8) we can pick $\alpha _{j}$ of the intersection points $C_{\mathfrak {g}} \cap \mathfrak {c}_{j}$ . Each of these $\alpha _{j}$ intersection points gives rise to a chain of length $2j-1$ with distinguished pair given by the $2$ -cycle corresponding to the point itself (recall (4.2)) and the one-to-one correspondence (4.4) yields the existence of curves in $V( \alpha _1,\ldots ,\alpha _p)$ containing all the above chains, which lie in $V^k( \alpha _1,\ldots ,\alpha _p)$ by construction. In other words, we have natural surjective maps

(4.9)

where $\sigma _k$ is the forgetful map and $\gamma _k$ is finite. The fibre of $\gamma _k$ over a curve $C_{\mathfrak {g}}$ is given by all the choices of $\alpha _j$ of the intersection points in each $C_{\mathfrak {g}} \cap \mathfrak {c}_{j}$ .

When $g=\sum \alpha _i>2(k-1)$ , this is used in [Reference Ciliberto and Knutsen12] to prove that $V^k( \alpha _1,\ldots ,\alpha _p)$ is irreducible (and nonempty) of dimension $2(k-1)$ and that its general element corresponds to finitely many of the curves $C_{\mathfrak {g}}$ . When $g \leq 2(k-1)$ , one has $V^k( \alpha _1,\ldots ,\alpha _p)= V( \alpha _1,\ldots ,\alpha _p)$ and its general element corresponds to a family of dimension $2(k-1)-\sum \alpha _{j}=2(k-1)-g=\rho (g,1,k)$ of the curves $C_{\mathfrak {g}}$ . This also shows that the generic fibre dimension of $\sigma _k$ is $\max \{0,\rho (g,1,k)\}$ , in other words that $\dim G^1_k(C')=\max \{0,\rho (g,1,k)\}$ for general C in $V^k( \alpha _1,\ldots ,\alpha _p)$ .

We will also make use of the following observation, implicitly used in [Reference Ciliberto and Knutsen12, Proof of Cor. 6.6(iv)]:

Under conditions (4.3) and (4.8) we thus have a relative scheme $\pi _{\delta }: {\mathcal V}_{\delta } \to {\mathbb D}$ of dimension $\dim {\mathcal V}_{\delta }=g+1$ such that the fibre over $t \neq 0$ is a component of $V_{|H_t|,\delta }$ (of dimension $g:=p-\delta $ ) and a subscheme ${\mathcal V}^k_{\delta } \subset {\mathcal V}_{\delta }$ such that the fibre over $t \neq 0$ is a component of $V^k_{|H_t|,\delta }$ of dimension $\min \{2(k-1),g\}$ and the special fibre contains a dense open subset of $V^k( \alpha _1,\ldots ,\alpha _p)$ . By construction, the map from ${\mathcal V}^k_{\delta }$ sending a curve to the stable model of its normalization at the $\delta $ marked nodes lands in $\overline {{\mathcal M}^1_{g,k}}$ , the compactification of ${\mathcal M}^1_{g,k}$ in the Deligne-Mumford compactification $\overline {{\mathcal M}_g}$ of ${\mathcal M}_g$ ; in other words, we have moduli maps

(4.10)