The Riemann constant for a non-symmetric Weierstrass semigroup

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.