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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Plane models for Riemann surfaces admitting certain half-canonical linear series. II


Author: Robert D. M. Accola
Journal: Trans. Amer. Math. Soc. 263 (1981), 243-259
MSC: Primary 14H15; Secondary 30F20, 32G15
DOI: https://doi.org/10.1090/S0002-9947-1981-0590422-3
MathSciNet review: 590422
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Abstract: For $ r \geqslant 2$, closed Riemann surfaces of genus $ 3r + 2$ admitting two simple half-canonical linear series $ g_{3r + 1}^r,h_{3r + 1}^r$ are characterized by the existence of certain plane models of degree $ 2r + 3$ where the linear series are apparent. The plane curves have $ r - 2$ $ 3$-fold singularities, one $ (2r - 1)$-fold singularity $ Q$, and two other double points (typically tacnodes) whose tangents pass through $ Q$. The lines through $ Q$ cut out a $ g_4^1$ which is unique. The case where the $ g_4^1$ is the set of orbits of a noncyclic group of automorphisms of order four is characterized by the existence of $ 3r + 3$ pairs of half-canonical linear series of dimension $ r - 1$, where the sum of the two linear series in any pair is linearly equivalent to $ g_{3r + 1}^r + h_{3r + 1}^r$.


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DOI: https://doi.org/10.1090/S0002-9947-1981-0590422-3
Keywords: Riemann surface, linear series, algebraic curve, automorphism
Article copyright: © Copyright 1981 American Mathematical Society