Covering gonality of symmetric products of curves and Cayley-Bacharach condition on Grassmannians

TL;DR

This paper investigates the covering gonality of symmetric products of algebraic curves, establishing new results for the 3-fold and 4-fold cases, and explores the Cayley-Bacharach condition on Grassmannians to support these findings.

Contribution

It extends the understanding of covering gonality for symmetric products of curves, proving it equals the curve's gonality for higher symmetric powers, and analyzes Cayley-Bacharach conditions on Grassmannians.

Findings

Covering gonality of the second symmetric product equals the gonality of the curve.

Proved the same for the third and fourth symmetric products.

Described the geometry of linear subspaces satisfying Cayley-Bacharach condition.

Abstract

Given an irreducible projective variety $X$, the covering gonality of $X$ is the least gonality of an irreducible curve $E\subset X$ passing through a general point of $X$. In this paper we study the covering gonality of the $k$-fold symmetric product $C^{(k)}$ of a smooth complex projective curve $C$ of genus $g\geq k+1$. It follows from a previous work of the first author that the covering gonality of the second symmetric product of $C$ equals the gonality of $C$. Using a similar approach, we prove the same for the $3$-fold and the $4$-fold symmetric product of $C$. A crucial point in the proof is the study of Cayley-Bacharach condition on Grassmannians. In particular, we describe the geometry of linear subspaces of $\mathbb{P}^n$ satisfying this condition and we prove a result bounding the dimension of their linear span.