Severi varieties and branch curves of abelian surfaces of type (1,3)

2001/04/24 by Herbert Lange, H. Lange, Lange, H. +3 · 1 citation
Computer Science · Mathematics · #14K05 #14K10 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic #FOS: Mathematics #Polynomial and algebraic computation #math.AG #msc:14K05 #msc:14K10

paper · pdf · doi:10.48550/arxiv.math/0104223

17 pages

arxiv created 2001/04/24 · openalex publication_date 2001/04/24 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (A,L) be a principally polarized abelian surface of type (1,3). The linear system |L| defines a 6:1 covering of A onto P2, branched along a curve B of degree 18 in P2. The main result of the paper is that for general (A,L) the curve B irreducible, admits 72 cusps, 36 nodes or tacnodes, each tacnode counting as two nodes, 72 flexes and 36 bitangents. The main idea of the proof is to use the fact that for a general (A,L) of type (1,3) the closure of the Severi variety V in |L| is dual to the curve B in the sense of projective geometry. We investigate V and B via degeneration to a special abelian surface.

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