Genus two curves on abelian surfaces

2019/01/22 by Knutsen, Andreas Leopold, Lelli-Chiesa, Margherita · 1 citation
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1901.07603

Abstract

This paper deals with singularities of genus 2 curves on a general (d₁,d₂)-polarized abelian surface (S,L). In analogy with Chen's results concerning rational curves on K3 surfaces [Ch1,Ch2], it is natural to ask whether all such curves are nodal. We prove that this holds true if and only if d₂ is not divisible by 4. In the cases where d₂ is a multiple of 4, we exhibit genus 2 curves in |L| that have a triple, 4-tuple or 6-tuple point. We show that these are the only possible types of unnodal singularities of a genus 2 curve in |L|. Furthermore, with no assumption on d₁ and d₂, we prove the existence of at least a nodal curve in |L|. As a corollary, we obtain nonemptiness of all Severi varieties on general abelian surfaces and hence generalize [KLM, Thm 1.1] to nonprimitive polarizations.

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