On Rao's Theorems and the Lazarsfeld-Rao Property

Abstract

Let $X$ be an integral projective scheme satisfying the condition $S_3$ of Serre and $H^1({\mathcal O}_X(n)) = 0$ for all $n \in {\mathbb Z}$. We generalize Rao's theorem by showing that biliaison equivalence classes of codimension two subschemes without embedded components are in one-to-one correspondence with pseudo-isomorphism classes of coherent sheaves on $X$ satisfying certain depth conditions. We give a new proof and generalization of Strano's strengthening of the Lazarsfeld--Rao property, showing that if a codimension two subscheme is not minimal in its biliaison class, then it admits a strictly descending elementary biliaison. For a three-dimensional arithmetically Gorenstein scheme $X$, we show that biliaison equivalence classes of curves are in one-to-one correspondence with triples $(M,P,\alpha)$, up to shift, where $M$ is the Rao module, $P$ is a maximal Cohen--Macaulay module on the homogeneous coordinate ring of $X$, and $\alpha: P^{\vee} \to M^* \to 0$ is a surjective map of the duals.