On the class of caustic on the moduli space of odd spin curves

TL;DR

This paper investigates specific divisors in the moduli space of odd spin curves related to the behavior of linear systems, expanding their descriptions in the Picard group to understand their geometric properties.

Contribution

It provides a detailed expansion of the Base Point and Caustic divisors in the rational Picard group of the moduli space of odd spin curves, building on prior work on the non-reduced divisor.

Findings

Identified the structure of the Caustic divisor in the moduli space.

Expanded the classes of the Base Point and Caustic divisors in the Picard group.

Connected divisor properties to the geometry of odd spin curves.

Abstract

Let $C$ be a smooth projective curve of genus $g\geq 3$ and let $\eta$ be an odd theta characteristic on it such that $h^0(C,\eta) = 1$. Pick a point $p$ from the support of $\eta$ and consider the one-dimensional linear system $|\eta + p|$. In general this linear system is base-point free and all its ramification points (i.e. ramification points of the corresponding branched cover $C\to\mathbb P^1\simeq \mathbb PH^0(C,\eta+p)$) are simple. We study the locus in the moduli space of odd spin curves where the linear system $|\eta + p|$ fails to have this general behavior. This locus splits into a union of three divisors: the first divisor corresponds to the case when $|\eta+p|$ has a base point, the second one corresponds to theta characteristics which are not reduced at $p$ (and therefore $|\eta + p|$ must have a triple point at $p$) and the third one corresponds to the case when $|\eta + p|$ has a triple point different from $p$. The second divisor was studied by G. Farkas and A. Verra in\cite{FARo} where its expansion in the rational Picard group was used to prove that the moduli space of odd spin curves is of general type for genus at least $12$. We call the first divisor a Base Point divisor and the third one a Caustic divisor (following Arnold terminology for Hurwitz spases). The objective of this paper is to expand these two divisors via the set of standard generators in the rational Picard group of the moduli space of odd spin curves.