Deformation of rank 2 quasi-bundles and some strange dualities for rational surfaces
type: Item
instance of: scholarly article
title: Deformation of rank 2 quasi-bundles and some strange dualities for rational surfaces
zbMATH Open document ID: 1208.14034
DOI: 10.1215/00127094-2010-063
author: Takeshi Abe
published in: Duke Mathematical Journal
publication date: 2011-01-10T00:00:00Z
review text: Let \({\mathcal M}_n\) (resp., \({\mathcal N}_d\)) be the coarse moduli space of rank 2 torsion free (resp., pure dimension 1) Gieseker-Maruyama semistable sheaves on the projective plane \({\mathbb P}^2\) with trivial determinant and \(c_2 = n \geq 0\) (resp., with determinant \({\mathcal O}_{{\mathbb P}^2}(d)\), \(d \geq 1\), and \(\chi = 0\)). Let \(\rho : {\mathcal N}_d \rightarrow | \, {\mathcal O}_{{\mathbb P}^2}(d)\, |\) be the morphism associating to \([{\mathcal G}] \in {\mathcal N}_d\) its scheme theoretic support (defined by the determinant of a locally free presentation of \(\mathcal G\)). Using the simple fact asserting that if \(f : X \rightarrow T\) is a projective morphism and \(\mathcal F\) is a coherent sheaf on \(X\), flat over \(T\), with \(\text{Supp}\, {\mathcal F}\) of relative dimension 1 over \(T\), and with \(\chi ({\mathcal F}_t) = 0\), \(\forall t \in T\), then \([\text{det}(\text{R}f_{\ast}{\mathcal F})]^{-1}\) has a global section vanishing exactly on \(\{ t\in T \, | \, \text{H}^0({\mathcal F}_t) \neq 0 \}\), one can show that there exists a line bundle \({\mathcal D}_d\) (resp., \({\mathcal R}_n\)) on \({\mathcal M}_n\) (resp., \({\mathcal N}_d\)) and a global section of the line bundle \({\mathcal D}_d \boxtimes {\mathcal R}_n\) on \({\mathcal M}_n \times {\mathcal N}_d\) whose zero locus is \(\{ ([{\mathcal F}],[{\mathcal G}])\, | \, \text{H}^0({\mathbb P}^2,{\mathcal F} \otimes {\mathcal G}) \neq 0 \}\). This section gives rise to the \textit{strange-duality} map: \[ (\text{SD})_{n,d} : \text{H}^0({\mathcal N}_d,{\mathcal R}_n)^{\vee} \longrightarrow \text{H}^0({\mathcal M}_n,{\mathcal D}_d)\, . \] J. Le Potier conjectured that \((\text{SD})_{n,d}\) is an isomorphism. Now, for \(d = 1\) or 2, \(\rho\) is an isomorphism and \({\mathcal R}_n \simeq {\rho}^{\ast}{\mathcal O}(n)\) (for example, the points of \({\mathcal N}_1\) are of the form \([{\mathcal O}_L(-1)]\) with \(L\) line in \({\mathbb P}^2\)). \textit{G. Danila} [Ann. Inst.…
reviewed by: Iustin Coandă
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 14D23
Mathematics Subject Classification ID: 14B10
Mathematics Subject Classification ID: 14J26
zbMATH DE Number: 5834476
zbMATH Keywords: moduli stack
zbMATH Keywords: semistable sheaf
zbMATH Keywords: Donaldson determinant line bundle
zbMATH Keywords: strange duality
zbMATH Keywords: rational surface
zbMATH Keywords: quasi-bundle
zbMATH Keywords: infinitesimal deformation
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/00127094-2010-063
OpenAlex ID: W2065877345
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description: scientific article; zbMATH DE number 5834476