Rationality of moduli spaces of torsion free sheaves over rational surfaces
type: Item
instance of: scholarly article
title: Rationality of moduli spaces of torsion free sheaves over rational surfaces
zbMATH Open document ID: 0870.14003
DOI: 10.1007/BF02567513
published in: Manuscripta Mathematica
publication date: 1996-06-05T00:00:00Z
full work available at URL: https://eudml.org/doc/156158
review text: Let \(S\) be a smooth rational ruled surface with \(F\) a general fibre. Let \(H\) be an ample divisor with \((K_S+F)\cdot H<0\). Then the author shows that in most cases, the moduli space \(M_H^s(r,c_1,c_2)\) of \(H\)-Gieseker semistable sheaves of rank \(r\), first Chern class \(c_1\) and \(c_1\cdot F=0\) and given (large enough, with some additional condition) \(c_2\), is rational. \textit{A. Hirschowitz} and \textit{Y. Laszlo} [Math. Ann. 297, No. 1, 85-102 (1993; Zbl 0808.14007)] introduced the notion of priority sheaves over \(\mathbb{P}^2\) and thus proved that the moduli space as above is irreducible. \textit{C. Walter} generalised this notion to arbitrary birationally ruled rational surfaces, in his Nice preprint (1993), which the reviewer has not seen in print. A torsion free sheaf \(E\) on a rational ruled surface \(S\) is called prioritary, if \(\Hom(E,E(K_S+ F))=0\). (In the Hirschowitz-Laszlo definition, \(\text{Ext}^2 (E,E(-1))=0\), because they are over \(\mathbb{P}^2\).) Walter proved that the appropriate moduli stack is irreducible and smooth. Now, the condition \((K_S+F)\cdot H<0\) becomes relevant, since under that condition, every \(H\)-Gieseker semistable sheaf is prioritary. It is also true that such an \(H\) exists.
reviewed by: N. Mohan Kumar
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 14M20
Mathematics Subject Classification ID: 14J26
zbMATH DE Number: 866131
zbMATH Keywords: rationality of moduli space
zbMATH Keywords: semistable sheaves
zbMATH Keywords: rational ruled surface
zbMATH Keywords: prioritary sheaf
author: Lothar Göttsche
MaRDI profile type: Publication
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OpenAlex ID: W1990572520
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description: scientific article; zbMATH DE number 866131