Pfaffian Subschemes

Abstract

A subscheme $X\subset {\mathbb{P}}^{n+3}$ of codimension $3$ is {\em Pfaffian} if it is the degeneracy locus of a skew-symmetric map $f:\cal{E}\spcheck(-t) @>>> \cal{E}$ with $\mathcal{E}$ a locally free sheaf of odd rank on ${\mathbb{P}}^{n+3}$. It is shown that a codimension $3$ subscheme $X\subset {\mathbb{P}}^{n+3}$ is Pfaffian if and only if it is locally Gorenstein, subcanonical (i.e.\ ${\omega }_X\cong {\mathcal{O}}_{\mathcal{X}}(l)$ for some integer $l$), and the following parity condition holds: if $n\equiv 0(\mathrm{mod}4)$ and $l$ is even, then $\chi ({\mathcal{O}}_{\mathcal{X}}(l/2))$ is also even. The paper includes a modern version of the Horrocks correspondence, stated in the language of derived categories. A local analogue of the main theorem is also proved.

26 pages, AMS-LaTeX

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