Pfaffian Subschemes

1994/06/17 by Charles Walter, Charles H. Walter, Walter, Charles H. · 16 citations
Mathematics · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Biology #Codimension #Combinatorics #Degeneracy (biology) #Discrete mathematics #Homotopy and Cohomology in Algebraic Topology #Integer (computer science) #Locus (genetics) #Mathematics #Omega #Pfaffian #Physics #Pure mathematics #Sheaf #Skew #alg-geom #math.AG

paper · pdf · doi:10.48550/arxiv.alg-geom/9406005

published in arXiv (Cornell University) (Cornell University) · 26 pages, AMS-LaTeX

arxiv created 1994/06/17 · openalex publication_date 1994/06/17 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

A subscheme X⊂ \Bbb Pⁿ⁺³ of codimension 3 is \em Pfaffian if it is the degeneracy locus of a skew-symmetric map f:\calE\spcheck(-t) @>>> \calE with \calE a locally free sheaf of odd rank on \Bbb Pⁿ⁺³. It is shown that a codimension 3 subscheme X⊂\Bbb Pⁿ⁺³ is Pfaffian if and only if it is locally Gorenstein, subcanonical (i.e. ω_X≅\cal O_X(l) for some integer l), and the following parity condition holds: if n≡ 0\pmod4 and l is even, then χ(\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.

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