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The sigma function for Weierstrass semigroups \(\langle 3,7,8\rangle\) and \(\langle 6,13,14,15,16\rangle\). (English) Zbl 1284.14045
This paper proposes a construction of \(\sigma\)-functions based on the nature of the Weierstrass semigroup at one point of the Riemann surface as a generalization of the construction for plane affine models of the Riemann surface. After an introduction, the authors give in Section 2 a brief review of Weierstrass semigroups. In Sections 3 and 4, they proceed as follows: they define monomial curves with given Weierstrass semigroups, with motivation from the Norton numbers; when the semigroup falls outside the verifiably smoothable case, they give Komeda’s proof that one smooth curve with such Weierstrass semigroup exists; Pinkham’s calculation of the expected dimension yields a positive number, therefore they can conclude that the monomial curve is smoothable. In Section 5, the authors use the local coordinates given by the monomial presentation of the curve, to manufacture a local section of certain meromorphic differentials, and they construct an abelian function on the curve, the \(\sigma\)-function, by integrating those differentials. In [the first two authors, “Sigma functions for a space curve of type \((3, 4, 5)\)”, J. Geom. Symm. Phys. 30, 75–91 (2013), arXiv:1112.4137v2], the original idea was implemented for the semigroup (3, 4, 5), including the \(\sigma\)-function construction and natural extensions of the ones previously developed for \((n, s)\) curves. The \(\sigma\)-function provides a stratification of the Jacobian in Section 6; in Section 7, the authors make some observations on the possible links with the representation theory of the Monster, starting with the above-mentioned numerical observation connecting the Norton numbers and the differentials they constructed.

MSC:
14H55 Riemann surfaces; Weierstrass points; gap sequences
14H50 Plane and space curves
14K20 Analytic theory of abelian varieties; abelian integrals and differentials
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