A subscheme $X\subset \Bbb P^{ n+3}$ 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 $\cal{ E}$ a locally free sheaf of odd rank on $\Bbb P^{ n+3}$. It is shown that a codimension $3$ subscheme $X\subset\Bbb P^{ n+3}$ is Pfaffian if and only if it is locally Gorenstein, subcanonical (i.e.\ $\omega_X\cong\cal O_X(l)$ for some integer $l$), and the following parity condition holds: if $n\equiv 0\pmod{ 4}$ and $l$ is even, then $\chi (\cal O_X (l/2))$ 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.
Categories: alg-geom math.AG