Theta Surfaces

A theta surface in affine 3-space is the zero set of a Riemann theta function in genus 3. This includes surfaces arising from special plane quartics that are singular or reducible. Lie and Poincaré showed that any analytic surface that is the Minkowski sum of two space curves in two different ways i...

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Bibliographic Details
Published in:Vietnam journal of mathematics 2021-06, Vol.49 (2), p.319-347
Main Authors: Agostini, Daniele, Çelik, Türkü Özlüm, Struwe, Julia, Sturmfels, Bernd
Format: Article
Language:English
Subjects:
Algebra
Curves
Mathematics
Mathematics and Statistics
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Summary:A theta surface in affine 3-space is the zero set of a Riemann theta function in genus 3. This includes surfaces arising from special plane quartics that are singular or reducible. Lie and Poincaré showed that any analytic surface that is the Minkowski sum of two space curves in two different ways is a theta surface. The four space curves that generate such a double translation structure are parametrized by abelian integrals, so they are usually not algebraic. This paper offers a new view on this classical topic through the lens of computation. We present practical tools for passing between quartic curves and their theta surfaces, and we develop the numerical algebraic geometry of degenerations of theta functions.
ISSN:2305-221X
2305-2228
DOI:10.1007/s10013-020-00443-x