The Riemann constant for a non-symmetric Weierstrass semigroup

Abstract

The zero divisor of the theta function of a compact Riemann surface X of genus g is the canonical theta divisor of Pic${^{(g-1)}}$ up to translation by the Riemann constant ${\Delta}$ for a base point P of X. The complement of the Weierstrass gaps at the base point P gives a numerical semigroup, called the Weierstrass semigroup. It is classically known that the Riemann constant ${\Delta}$ is a half period, namely an element of ${\frac{1}{2} \Gamma_\tau}$ , for the Jacobi variety ${\mathcal{J}(X)=\mathbb{C}^{g}/\Gamma_\tau}$ of X if and only if the Weierstrass semigroup at P is symmetric. In this article, we analyze the non-symmetric case. Using a semi-canonical divisor D ₀, we express the relation between the Riemann constant ${\Delta}$ and a half period in the non-symmetric case. We point out an application to an algebraic expression for the Jacobi inversion problem. We also identify the semi-canonical divisor D ₀ for trigonal pointed curves, namely with total ramification at P.