Smoothness results for the schemes of special divisors on general k-gonal curves

TL;DR

This paper investigates the smoothness properties of Brill-Noether schemes and their degeneracy loci on general k-gonal curves, providing criteria for smoothness and describing the structure of these schemes.

Contribution

It introduces natural open subsets of degeneracy schemes that ensure smoothness and characterizes the smoothness of Brill-Noether schemes at certain points, extending previous understanding.

Findings

Certain degeneracy schemes are smooth along specific open subsets.

W^r_d(C) is smooth at points in these open subsets unless they lie in multiple components.

The singular locus of W^r_d(C) differs from the union of W^{r+1}_d(C) and component intersections.

Abstract

For a general $k$-gonal curve $C$ with a morphism $f: C \rightarrow \mathbb{P}^1$ of degree $k$, we consider the refinement of the Brill-Noether schemes $W^r_d(C)$ by means of the Brill-Noether degeneracy schemes $\overline{\Sigma}_{\overrightarrow {e}}(C,f)$. The schemes $\overline{\Sigma}_{\overrightarrow {e}}(C,f)$ as sets are closures of subsets $\Sigma_{\overrightarrow {e}}(C,f)$ of $\Pic (C)$ and as a scheme $\Sigma_{\overrightarrow {e}}(C,f)$ is a smooth open subscheme of $\overline{\Sigma}_{\overrightarrow {e}}(C,f)$. In this paper we describe naturally defined open subsets of $\overline{\Sigma}_{\overrightarrow {e}}(C,f)$ in general strictly containing $\Sigma_{\overrightarrow {e}}(C,f)$ such that $\overline{\Sigma}_{\overrightarrow {e}}(C,f)$ is smooth along them. As an application we describe all invertible sheaves $L$ on $C$ having an injective Petri map. Some of those sets $\overline{\Sigma}_{\overrightarrow {e}}(C,f)$ are the irreducible components of $W^r_d(C)$. In those cases we prove $W^r_d(C)$ is smooth at a point $L$ of those larger open subsets of $\overline{\Sigma}_{\overrightarrow {e}}(C,f)$ unless $L$ belongs to at least two irreducible components of $W^r_d(C)$ (such points exist). On the other hand in general the singular locus of the schemes $W^r_d(C)$ is not equal to the complement of the union of $W^{r+1}_d(C)$ and the intersections of two different components of $W^r_d(C)$.