The scrollar invariants of k-gonal curves having a nodal model on a smooth quadric having its nodes on few lines

TL;DR

This paper computes the scrollar invariants of the normalization of certain nodal curves on a smooth quadric surface, under specific node configurations, simplifying previous proofs and extending related existence results.

Contribution

It provides a direct proof for the scrollar invariants of k-gonal curves with nodes on few lines on a smooth quadric, simplifying prior approaches and deriving new existence results.

Findings

Explicit determination of scrollar invariants for the given curves

Simplified proof using an easy lemma

Extension of Ballico's existence results for curves with prescribed invariants

Abstract

We determine the scrollar invariants of the normalization $C$ of a nodal curve $\Gamma$ of type $(k,a)$ on a smooth quadric $\mathbb{P}^1 \times \mathbb{P}^1$ associated to the $g^1_k$ defined by the pencil of lines of type $(0,1)$ in case all nodes are contained in at most $k-1$ lines of type $(1,0)$. This result is very much related to results obtained by E. Ballico, but in this paper the proof follows directly from an easy lemma. Also a result of E. Ballico on the existence of curves with prescribed scrollar invariant is a consequence of that lemma making the arguments much shorter.