The intersection of four quadrics in \(P^ 6\), Abelian surfaces and their moduli
type: Item
instance of: scholarly article
title: The intersection of four quadrics in \(P^ 6\), Abelian surfaces and their moduli
zbMATH Open document ID: 0607.14030
DOI: 10.1007/BF01456191
author: Mark Adler
author: Pierre van Moerbeke
published in: Mathematische Annalen
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164307
review text: When do four quadrics in \({\mathbb{P}}^ 6\) intersect in the affine part of an Abelian surface? It is shown that \(4\quad quadrics\) of the form \(\sum ^{3}_{i=1}(a_ ix^ 2_ i+a_{i+3}x^ 2_{i+3}+a_{i,i+3}x_ ix_{i+3})-cx^ 2_ 0,\quad whose\) linear span contains a nondegenerate curve of \(rank\quad 4\) quadrics intersect in the affine part of an Abelian surface; the latter is either principally polarized or it has polarization (1,2). According to the case, we provide a very simple normal form for the \(4\quad quadrics\) and a set of commuting vector fields defined on the affine part. This normal form provide an explicit set of moduli, on the one hand, for the intersection of \(4\quad quadrics\) with the properties above and, on the other hand, for the Abelian surfaces of principal polarization and polarization (1,2). When the linear span contains a degenerate curve of \(rank\quad 4\) quadrics, the intersection completes into a K3 surface. It is also shown that the curve of \(rank\quad 4\) quadrics ties up very closely with the curve which completes the affine into an Abelian surface. Moreover, there is a natural generalization of these ideas to quartics in \({\mathbb{P}}^ 6\). Finally the situations obtained lead to algebraic completely integrable geodesic flow on SO(4) for certain left-invariant metrics.
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 14K99
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 53D25
zbMATH DE Number: 3981349
zbMATH Keywords: intersection of four quadrics in projective 6-space
zbMATH Keywords: algebraic completely integrable geodesic flow
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3981349