Schur quadrics, cubic surfaces and rank 2 vector bundles over the projective plane

Abstract

A cubic surface in $P^3$ is known to contain 27 lines, out of which one can form 36 Schlafli double - sixes i.e., collections $l_1,...,l_6,l_1^{\prime },...,l_6^{\prime }\}$ of 12 lines such that each $l_i$ meets only $l_j^{\prime },j\ne i$ and does not meet $l_j,j\ne i$. In 1881 F. Schur proved that any double - six gives rise to a certain quadric $Q$ , called Schur quadric which is characterized as follows: for any $i$ the lines $l_i$ and $l_i^{\prime }$ are orthogonal with respect to (the quadratic form defining) $Q$. The aim of the paper is to relate Schur's construction to the theory of vector bundles on $P^2$ and to generalize this construction along the lines of the said theory.

arXiv comment27 pages, plain TEX

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