Any smooth plane quartic can be reconstructed from its bitangents

Details Any smooth plane quartic can be reconstructed from its bitangents Any smooth plane quartic can be reconstructed from its bitangents

2001-11-02

arxiv.org/abs/math/0111017v3

Israel Journal of Mathematics Volume 146, Number 1 (2005) 371-379

Mathematics

Algebraic Geometry

24 pages, LaTeX 2e amsart, xypic

Scientific paper

10.1007/BF02773542

In this paper, we present two related results on curves of genus 3. The first gives a bijection between the classes of the following objects: * Smooth non-hyperelliptic curves C of genus 3, with a choice of an element a in Jac(C)[2]-{0}, such that the cover C/|K_C+a|^* does not have an intermediate factor; up to isomorphism. * Plane curves E,Q in P^2 and an element in b' in Pic(E)[2]-{0}, where E,Q are of degrees 3,2, the curve E is smooth and Q,E intersect transversally ; up to projective transformations. We discuss the degenerations of this bijection, and give an interpretation of the bijection in terms of Abelian varieties. Next, we give an application of this correspondence: An EXPLICIT proof of the reconstructability of ANY smooth plane quartic from its bitangents.

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