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 D0, 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 D0 for trigonal pointed curves, namely with total ramification at P.