The Hesse pencil and polarizations of type (1,3) on Abelian surfaces

2022/12/30 by Fabrizio Catanese, Catanese, Fabrizio, Edoardo Sernesi +1
Mathematics · #14D06 #14D15 #14J10 #14J29 #14J60 #14L30 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #Complex Variables (math.CV) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2212.14877

openalex publication_date 2022/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this short note we prove two theorems, the first one is a sharpening of a result of Lange and Sernesi: the discriminant curve W of a general Abelian surface A endowed with an irreducible polarization D of type (1,3) is an irreducible curve of degree 18 whose singularities are exactly 36 nodes and 72 cusps. Moreover, we analyze the degeneration of the discriminant curve W and its singularities as A tends to the product of two equal elliptic curves. The second theorem, using the first one in order to prove a transversality assertion, shows that the general element of a family of surfaces constructed by Alessandro and Catanese is a smooth surface, thereby proving the existence of a new family of minimal surfaces of general type with p_g=q=2, K²=6 and Albanese map of degree 3.

Related

• Genus two curves on abelian surfaces
• Severi varieties and branch curves of abelian surfaces of type (1,3)
• Severi Varieties and Brill-Noether theory of curves on abelian surfaces
• Non-simple polarised abelian surfaces and genus 3 curves with completely decomposable Jacobians
• Degenerations of (1,7)-polarized abelian surfaces