Versal Families and the Existence of Stable Sheaves on a Del Pezzo Surface
Vol. 3 (1996), No. 3, Page 495--532.
Rudakov, Alexei N.
Versal Families and the Existence of Stable Sheaves on a Del Pezzo Surface
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Abstract:
Let r(F),c1(F),c2(F) be rank and Chern classes of an algebraic coherent sheaf F on a Del Pezzo surface X. We will call a tuple ˉc(F)=(r(F),c1(F),c2(F)) the Chern datum for the sheaf F. In the paper we write down several necessary conditions on the Chern datum of a non-exceptional stable sheaf on a Del Pezzo surface which generalize the conditions found by Drezet and Le Potier for sheaves on \PP2 and we use them to define a set \mxDX. After that we prove that if ˉc(F)∈\mxDX and F can be included in a smooth restricted versal family of sheaves on X then there are stable sheaves in the family so F can be deformed into a stable sheaf with the same Chern datum. We provide a way to construct such families and as an application we prove that for any c∈\mxDX it exists a stable sheaf F such that ˉc(F)=c provided X is P2(1) -- a Del Pezzo surface which arises by blowing up a point in \PP2.
Mathematics Subject Classification (1991): Primary 14J60; Secondary 14J10, 14J26
Mathematical Reviews Number: MR1432106
Received: 1995-03-06
