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The Riemann constant for a non-symmetric Weierstrass semigroup. (English) Zbl 1360.14090
The Riemann constant is an invariant that is associated to a pointed curve \((X, P)\), whose Abel map is normalized at \(P\). In this paper, the authors work over the complex numbers, assume that \(X\) is a compact Riemann surface (a curve) of genus \(g>1\) and use standard convention. They state the algebraic-transcendental correspondence for the Riemann constant on a general pointed curve and its consequences for the Jacobi inversion problem. The zero divisor of the theta function of a compact Riemann surface \(X\) of genus \(g\) is the canonical theta divisor of \(\mathrm{Pic}^{(g-1)}\) 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 \(J(X)=\mathbb{C}^g/\Gamma_\tau\) of \(X\) if and only if the Weierstrass semigroup at \(P\) is symmetric. The aim of this paper is to analyze the non-symmetric case. Using a semi-canonical divisor \(D_0\), the authors express the relation between the Riemann constant \(\Delta\) and a half period in the non-symmetric case. They point out an application to an algebraic expression for the Jacobi inversion problem. They also identify the semi-canonical divisor \(D_0\) for trigonal pointed curves, namely with total ramification at \(P\).

MSC:
14H55 Riemann surfaces; Weierstrass points; gap sequences
14H50 Plane and space curves
14K25 Theta functions and abelian varieties
14H40 Jacobians, Prym varieties
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