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 $P$ given as a numerical semigroup plays an important role, which is called the Weierstrass semigroup. It is classically known that the Riemann constant $\Delta$ is a half period $\frac{1}{2}\Gamma_\tau$ 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 show a relation between the Riemann constant $\Delta$ and a half period $\frac{1}{2}\Gamma_\tau$ of the non-symmetric case. We also identify the semi-canonical divisor $D_0$ for trigonal curves, and remark on an algebraic expression for the Jacobi inversion problem using the relation