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Brill-Noether loci of pencils with prescribed ramification on moduli of curves and on Severi varieties on $K3$ surfaces

Andreas Leopold Knutsen, Sara Torelli

TL;DR

The paper extends Brill-Noether theory to pencils with prescribed ramification on pointed curves and studies analogous ramification phenomena for Severi varieties on $K3$ surfaces. It proves that, when the adjusted Brill-Noether number satisfies $\widetilde{\rho}\ge -g$, the moduli loci ${\mathcal{M}}_{g,k,\mathbf{e}}$ admit components of the expected codimension and that the Hurwitz map behaves generically as dominant or finite depending on $n+\widetilde{\rho}$; it then transfers these ramified results to nodal curves on $K3$ surfaces, showing existence of non-general BN behavior and establishing equidimensional limit Severi varieties with controlled $g^1_k$-descents. The approach combines degeneration to nodal rational curves, the theory of admissible covers, and the CK construction of limit Severi varieties, yielding precise dimension counts and dominance statements. The authors apply these geometric insights to cycle theory on $K3$ surfaces, proving results about Beauville-Voisin points, constant cycle curves, and tautological points in $\mathcal{M}_{g,2}$, thereby linking ramified BN theory with moduli-cycle phenomena. Overall, the work broadens the scope of BN theory in ramified settings and provides new tools and consequences for the geometry of curves on $K3$ surfaces and their moduli spaces.

Abstract

Under the assumption that the adjusted Brill-Noether number $\widetildeρ$ is at least $-g$, we prove that the Brill-Noether loci in $\mathcal{M}_{g,n}$ of pointed curves carrying pencils with prescribed ramification at the marked points have a component of the expected codimension with pointed curves having Brill-Noether varieties of pencils of the minimal dimension. As an application, the map from the Hurwitz scheme to $\mathcal{M}_g$ is dominant if $n+\widetildeρ \geq 0$ and generically finite otherwise, settling a variation of a classical problem of Zariski. In the second part of the paper, we study the analogous loci of curves in Severi varieties on $K3$ surfaces, proving existence of curves with non-general behaviour from the point of view of Brill-Noether theory. This extends previous results of Ciliberto and the first named author to the ramified case. We apply these results to study correspondences and cycles on $K3$ surfaces in relation to Beauville-Voisin points and constant cycle curves.

Brill-Noether loci of pencils with prescribed ramification on moduli of curves and on Severi varieties on $K3$ surfaces

TL;DR

The paper extends Brill-Noether theory to pencils with prescribed ramification on pointed curves and studies analogous ramification phenomena for Severi varieties on surfaces. It proves that, when the adjusted Brill-Noether number satisfies , the moduli loci admit components of the expected codimension and that the Hurwitz map behaves generically as dominant or finite depending on ; it then transfers these ramified results to nodal curves on surfaces, showing existence of non-general BN behavior and establishing equidimensional limit Severi varieties with controlled -descents. The approach combines degeneration to nodal rational curves, the theory of admissible covers, and the CK construction of limit Severi varieties, yielding precise dimension counts and dominance statements. The authors apply these geometric insights to cycle theory on surfaces, proving results about Beauville-Voisin points, constant cycle curves, and tautological points in , thereby linking ramified BN theory with moduli-cycle phenomena. Overall, the work broadens the scope of BN theory in ramified settings and provides new tools and consequences for the geometry of curves on surfaces and their moduli spaces.

Abstract

Under the assumption that the adjusted Brill-Noether number is at least , we prove that the Brill-Noether loci in of pointed curves carrying pencils with prescribed ramification at the marked points have a component of the expected codimension with pointed curves having Brill-Noether varieties of pencils of the minimal dimension. As an application, the map from the Hurwitz scheme to is dominant if and generically finite otherwise, settling a variation of a classical problem of Zariski. In the second part of the paper, we study the analogous loci of curves in Severi varieties on surfaces, proving existence of curves with non-general behaviour from the point of view of Brill-Noether theory. This extends previous results of Ciliberto and the first named author to the ramified case. We apply these results to study correspondences and cycles on surfaces in relation to Beauville-Voisin points and constant cycle curves.

Paper Structure

This paper contains 12 sections, 15 theorems, 53 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Setting and main results of the paper
  3. Moduli spaces of pointed curves with prescribed ramification
  4. Gonality loci on $K3$ surfaces
  5. Cycles of torsion differences of points on curves on $K3$ surfaces, Beauville-Voisin points and tautological points in ${\mathcal{M}}_{g,2}$
  6. Proofs of Theorems \ref{['thm:main_mod']} and \ref{['thm:mainHur']}
  7. Building on the construction and results in CK
  8. Limit K3 surface
  9. Limit Severi varieties
  10. Limit Severi varieties of curves whose normalization carry a $g^1_k$
  11. Proof of Theorem \ref{['thm:main']}
  12. Proofs of Theorem \ref{['thm:main_cycle']} and of Corollaries \ref{['cor:BV']} and \ref{['cor:corBV']}

Key Result

Theorem 2.1

In the setting eq:setting1-eq:setting2, 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 $\operatorname{max}\{0,-\widetilde{\rho}\}$ in ${\mathcal{M}}_{g,n}$ whose general member $(C,x_1,\ldots,x_n)$ satisfies

Figures (2)

  • Figure 1: A chain of length $9$ contained in a member of $|H_0|$
  • Figure 2: The partial normalization $\widetilde{C}$ of $C$ and its stable model $C'$, around a chain

Theorems & Definitions (39)

  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Definition 2.4
  • Theorem 2.6
  • Definition 2.7
  • Theorem 2.8
  • Example 2.9
  • Definition 2.10
  • Corollary 2.11
  • ...and 29 more