The intersection of four quadrics in \(P^ 6\), Abelian surfaces and their moduli
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.
intersection of four quadrics in projective 6-space
algebraic completely integrable geodesic flow
scientific article; zbMATH DE number 3981349
scientific article