Abelian surfaces of type \((1,4)\)
type: Item
instance of: scholarly article
title: Abelian surfaces of type \((1,4)\)
zbMATH Open document ID: 0714.14028
DOI: 10.1007/BF01452051
author: Christina Birkenhake
author: Herbert Lange
author: Duco Van Straten
published in: Mathematische Annalen
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164621
review text: Let \(A\) be an abelian surface over \({\mathbb{C}}\) and \(L\) a (symmetric) polarization on \(A\) of type \((1,4)\). Let \(K\) be the kernel of the natural map of \(A\) to the dual of \(A\) and \(K\) decomposes into a direct sum of two isotropic cyclic subgroups of order \(4\cdot K=K_ 1\oplus K_ 2\). The authors study the rational map (regular in fact) determined by the linear system of the polarization. This was the only remaining case to be studied for polarized abelian surfaces. - The main results are the following: (i) (Theorem I (i)) The map \(\phi_ L: A\to \bar A\subset {\mathbb{P}}_ 3\) is birational onto a singular octic \(\bar A\) in \({\mathbb{P}}_ 3\) iff the polarization divisor admits an involution which commutes with the translation with \(K_ 2\), and whose quotient is an elliptic curve. (ii) (Theorem I (ii)) In the exceptional case \(\phi_ L\) is a double covering of a singular quartic \(\bar A\) which is birational to an elliptic scroll. (iii) (Theorem II) One can write down the octic equation of \(\bar A\) explicitly with 4 homogeneous parameters and this determines a finite rational map of \({\mathbb{P}}_ 3\) to the moduli space of polarized abelian surface of type \((1,4)\) of degree 48.
Mathematics Subject Classification ID: 14K05
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14K10
zbMATH DE Number: 4175145
zbMATH Keywords: polarization divisor
zbMATH Keywords: moduli space of polarized abelian surface
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4175145