2003/02/07 by Robin Hartshorne, Hartshorne, Robin
Mathematics · #13C40 #14M06 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #math.AG #msc:13C40 #msc:14M06
paper · pdf · doi:10.48550/arxiv.math/0302078
17 pages
arxiv created 2003/02/07 · openalex publication_date 2003/02/07 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X be an integral projective scheme satisfying the condition S3 of Serre and H1(\mathcal OX(n)) = 0 for all n ∈ \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,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\vee → M^* → 0 is a surjective map of the duals.