Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Scrollar Invariants of Curves Mapping to a Hirzebruch Surface

Riccardo Redigolo

TL;DR

This work analyzes the scrollar invariants of normalisations of nodal curves on the $m$-th Hirzebruch surface $\mathbb F_m$ by relating them to the Tschirnhausen bundle of the induced $k:1$ covers to $\mathbb P^1$. It develops interpolation techniques on $\mathbb F_m$ to compute the splitting type $\mathcal T\!sch(f) \cong \bigoplus_i \mathcal O_{\mathbb P^1}(e_i)$ of the pushforward of $\mathcal O_C$, yielding explicit piecewise formulas for the invariants $e_i$ in terms of the class $(k,a)$, the node count $\delta$, and the geometry of $\mathbb F_m$. The paper also proves existence results for curves in prescribed nodal configurations, extending Coppens’ and Ballico’s results to $m>0$ and linking these constructions to Brill–Noether theory and the Vakil–Vemulapalli polytope. These findings provide concrete realizations of scrollar invariants beyond the general case and offer a route to understanding the Brill–Noether geometry of $k$-gonal curves via normalisations of nodal curves on ruled surfaces.

Abstract

In this note we analyse the scrollar invariants of $k:1$ covers of $\mathbb P^1$ that factor through the normalisation of a nodal curve in the $m$-th Hirzebruch surface $\mathbb F_m$. We then give an existence theorem for nodal curves in $\mathbb F_m$ having fixed class and singular locus.

The Scrollar Invariants of Curves Mapping to a Hirzebruch Surface

TL;DR

This work analyzes the scrollar invariants of normalisations of nodal curves on the -th Hirzebruch surface by relating them to the Tschirnhausen bundle of the induced covers to . It develops interpolation techniques on to compute the splitting type of the pushforward of , yielding explicit piecewise formulas for the invariants in terms of the class , the node count , and the geometry of . The paper also proves existence results for curves in prescribed nodal configurations, extending Coppens’ and Ballico’s results to and linking these constructions to Brill–Noether theory and the Vakil–Vemulapalli polytope. These findings provide concrete realizations of scrollar invariants beyond the general case and offer a route to understanding the Brill–Noether geometry of -gonal curves via normalisations of nodal curves on ruled surfaces.

Abstract

In this note we analyse the scrollar invariants of covers of that factor through the normalisation of a nodal curve in the -th Hirzebruch surface . We then give an existence theorem for nodal curves in having fixed class and singular locus.
Paper Structure (5 sections, 16 theorems, 32 equations)

This paper contains 5 sections, 16 theorems, 32 equations.

Table of Contents

  1. INTRODUCTION
  2. BACKGROUND ON HIRZEBRUCH SURFACES
  3. INTERPOLATING POINTS IN $\mathbb F_m$
  4. COMPUTING THE SCROLLAR INVARIANTS
  5. CURVES IN $\mathbb F_m$ WITH NODES IN PRESCRIBED POSITION

Key Result

Theorem 1.2

Let $\nu:C\rightarrow \Gamma\subseteq \mathbb F_m$ be a general point of $\mathcal{M}_g(\mathbb F_m,(k,a))^{bir}$. Set $\alpha:=\pi\circ \nu$ and ${\delta:=p_a(\Gamma)-g}$. Assume that $(m,k,a)\notin \{(2,4,0),\ (1,6,0),\ (1,4,2),\ (0,4,4)\}$ and that Then, the Tschirnhausen bundle of $\alpha$ is balanced if and only if $\delta\geq \binom{k-1}{2}m$. If $\delta< \binom{k-1}{2}m$, choosing $\ell$ a

Theorems & Definitions (29)

  • Conjecture 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 3.1
  • ...and 19 more