Calabi-Yau threefolds and moduli of Abelian surfaces. I.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1063.14051
DOI: 10.1023/A:1012076503121
author: Sorin Popescu
author: Mark Gross
published in: Compositio Mathematica
publication date: 2001-01-01T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0001089
Mathematics Subject Classification ID: 14K10
Mathematics Subject Classification ID: 14J32
Mathematics Subject Classification ID: 14M20
Mathematics Subject Classification ID: 14M12
zbMATH DE Number: 1652611
zbMATH Keywords: moduli spaces of Abelian surfaces
zbMATH Keywords: Calabi-Yau 3-folds
MaRDI profile type: Publication
arXiv ID: math/0001089
OpenAlex ID: W3161037083
title: Calabi-Yau threefolds and moduli of Abelian surfaces. I.
review text: The paper describes birational models of the moduli spaces \({\mathcal A}_d\) of polarized abelian surfaces of type \((1, d)\) and their coverings \({\mathcal A}^{\text{lev}}_d\) with canonical level structure, for some small numbers \(d\). It is shown in particular that \({\mathcal A}^{\text{lev}}_6\) is birational to a nonsingular quadric hypersurface in \(\mathbb{P}^4\), \({\mathcal A}^{\text{lev}}_8\) is birational to a rational conic bundle over \(\mathbb{P}^2\), and \({\mathcal A}^{\text{lev}}_{10}\) is birational to a singular prime Fano 3-fold of genus 9, index 1 in \(\mathbb{P}^{10}\). In all cases the abelian surfaces are related to some Calabi-Yau 3-folds. Here I will point out only the case \(d= 8\): If \(A\subset\mathbb{P}^7\) is a general abelian surface of type \((1, 8)\), its homogeneous ideal is generated by 4 quadrics and 16 cubics. The 4 quadrics cut out a complete intersection \(X\) in \(\mathbb{P}^7\), whose small resolution is a Calabi Yau 3-fold. \(X\) contains a pencil of \((1,8)\)-polarized abelian surfaces, and this gives a description of \({\mathcal A}^{\text{lev}}_9\) as a \(\mathbb{P}^1\)-bundle over an open subset of \(\mathbb{P}^2\). Some of the Calabi-Yau 3-folds were not known before. There is a nice analogy betweeen the Calabi-Yau's described in the paper and the del Pezzo surfaces which are double covers of \(\mathbb{P}^2\).
reviewed by: Herbert Lange
description: scientific article; zbMATH DE number 1652611