Even Linkage Classes

Abstract

In this paper we generalize the E \mathcal {E} and N \mathcal {N} -type resolutions used by Martin-Deschamps and Perrin for curves in P 3 \mathbb {P}^{3} to subschemes of pure codimension in projective space, and shows that these resolutions are interchanged by the mapping cone procedure under a simple linkage. Via these resolutions, Rao’s correspondence is extended to give a bijection between even linkage classes of subschemes of pure codimension two and stable equivalence classes of reflexive sheaves E \mathcal {E} satisfying H ∗ 1 ( E ) = 0 H^{1}_{*}( \mathcal {E})=0 and E x t 1 ( E ∨ , O ) = 0 \mathcal {E}xt^{1}( \mathcal {E}^{\vee }, \mathcal {O})=0 . Further, these resolutions are used to extend the work of Martin-Deschamps and Perrin for Cohen-Macaulay curves in P 3 \mathbb {P}^{3} to subschemes of pure codimension two in P n \mathbb {P}^{n} . In particular, even linkage classes of such subschemes satisfy the Lazarsfeld-Rao property and any minimal subscheme for an even linkage class links directly to a minimal subscheme for the dual class.