SMF

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

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

I. DOLGECHEV, M. KAPRANOV
  • Année : 1993
  • Tome : 218
  • Format : Papier, Électronique
  • Langue de l'ouvrage :
    Anglais
  • Class. Math. : 14J60
  • Pages : 111-144
  • DOI : 10.24033/ast.242

Let $\Sigma \subset \mathbb {P}^3$ be a smooth cubic surface. It is known that $\Sigma $ contains 27 lines. Out of these lines one can form 36 Schläfli double-sixes, i.e., collections $\{l_1,\ldots ,l_6\}$, $\{l'_l,\ldots ,l'_6\}$ of 12 lines such that each $l_i$ meets only $l'_j$, $j\neq i$ and does not meet $l_j$, $j\neq i$. In 1881 F. Schur proved that any double-six gives rise to a certain unique quadric Q, the Schur quadric, characterized as follows : for any $i$ the lines $l_i$ and $l'_i$ are orthogonal with respect to Q. The aim of the paper is to relate Schur's construction to the theory of vector bundles on $\mathbb {P}^2$. In fact, we show that the whole theory of Hulek of rank 2 vector bundles on $\mathbb {P}^2$ with odd $c_1$ can be given a “geometric” interpretation involving some natural generalizations of cubic surfaces, double-sixes and Schur quadrics.

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