2007/05/15 by Fabrizio Catanese, Catanese, Fabrizio, Fabio Tonoli +1
Mathematics · #13D02 #14F05 #14J60 #14M99 #14N25 #14Q10 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #math.AG #msc:13D02 #msc:14F05 #msc:14J60 #msc:14M99 #msc:14N25 #msc:14Q10
paper · pdf · doi:10.48550/arxiv.0705.2184
39 pages, to appear in "Vector bundles and low codimensional subvarieties: state of the art and recent developments" in the Series "Quaderni di Matematica" della Seconda Universita' di Napoli
arxiv created 2007/05/15 · openalex publication_date 2007/05/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the moduli space \fMs(6;3,6,4) of simple rank 6 vector bundles \E on \PP3 with Chern polynomial 1+3t+6t2+4t3 and properties of these bundles, especially we prove some partial results concerning their stability. We first recall how these bundles are related to the construction of sextic nodal surfaces in \PP3 having an even set of 56 nodes (cf. \citeCaTo). We prove that there is an open set, corresponding to the simple bundles with minimal cohomology, which is irreducible of dimension 19 and bimeromorphic to an open set \fA0 of the G.I.T. quotient space of the projective space \fB:=\B∈ \PP(U^\vee⊗ W⊗ V^\vee)\ of triple tensors of type (3,3,4) by the natural action of SL(W)× SL(U). We give several constructions for these bundles, which relate them to cubic surfaces in 3-space \PP3 and to cubic surfaces in the dual space (\PP3)\vee. One of these constructions, suggested by Igor Dolgachev, generalizes to other types of tensors. Moreover, we relate the socalled \em cross-product involution for (3,3,4)-tensors, introduced in \citeCaTo, with the Schur quadric associated to a cubic surface in \PP3 and study further properties of this involution.