On Rao's Theorems and the Lazarsfeld-Rao Property

الأصل · EN

Let X be an integral projective scheme satisfying the condition S₃ of Serre and H¹(Oₓ(n)) = 0 for all n ∈ 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,α), up to shift, where M is the Rao module, P is a maximal Cohen--Macaulay module on the homogeneous coordinate ring of X, and α: P → M* → 0 is a surjective map of the duals.