Normal Functions, Even Theta Characteristics and the Theta Divisor

Abstract

Let $[C]$ be a general point in the moduli space of curves $M_g$ with $g>1$. Let $G\subset J(C)$ be a connected compact subgroup of real dimension $1$ of the Jacobian, and let $L$ be an even theta characteristic on $C$. We prove that $\{\zeta \in G\mid H^0(C,L\otimes \zeta )\ne 0\}=\varnothing$ if and only if $L\otimes {\zeta }_G$ is an even theta characteristic on $C$, where ${\zeta }_G$ is the unique non-trivial point of $G$ of order two.