# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] J.D. Fay, Theta functions on Riemann Surfaces, Lecture Notes in Mathematics, Vol. 352, Springer-Verlag, Berlin-New York, 1973. · Zbl 0281.30013 [2] J. Herzog, Generators and relations of Abelian semigroup and semigroup ring, Manuscripta Math. 3 (1970), 175-193 · Zbl 0211.33801 [3] Lewittes, J., Riemann surfaces and the theta functions, Acta Math., 111, 37-61, (1964) · Zbl 0125.31803 [4] H. M. Farkas and S. Zemel, Generalizations of Thomae’s Formula for Zn Curves (Developments in Mathematics), Springer, New York, 2010. · Zbl 1222.14001 [5] J. Komeda, S. Matsutani, and E. Previato, The sigma function for Weierstrass semigroup $${〈3,7,8〉}$$ and $${〈6,13,14,15,16〉}$$, Int. J. Math 24 (2013), 1350085 (58 pages). · Zbl 1284.14045 [6] S. Matsutani and J. Komeda, Sigma functions for a space curve of type (3, 4, 5), J. Geom. Symm. Phys. 30 (2013), 75-91. · Zbl 1311.14033 [7] D. Mumford, Tata Lectures on Theta I, Progress in Mathematics, 28, Birkhäuser, Boston, 1983. · Zbl 0509.14049 [8] D. Mumford, Tata Lectures on Theta II, Progress in Mathematics, 43, Birkhäuser, Boston, 1984. · Zbl 0549.14014 [9] Pinkham, H.C., Deformation of algebraic varieties with $$G$$_{$$m$$} action, Astérisque, 20, 1-131, (1974) · Zbl 0304.14006 [10] Shiga, H., On the representation of the Picard modular function by $${θ}$$ constants I-II, Publ. RIMS, Kyoto Univ., 24, 311-360, (1988) · Zbl 0678.10020 [11] Stöhr, K.-O., On the moduli spaces of Gorenstein curves with symmetric Weierstrass semigroups, J. Reine Angew. Math., 441, 189-213, (1993) · Zbl 0771.14009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.