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

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 $W^r_d(C)$.