On moduli of vector bundles on surfaces with negative Kodaira dimension

Abstract

Here we prove the following result. Fix integersq, τ,a’, b’, a’i, 1≤i≤τ,a’, b’, a’i, 1≤i≤τ; then there is an integerew such that for every integert≥w, for every algebraically closed fieldK for every smooth complete surfaceX with negative Kodaira dimension, irregularityq andKX2=8(1−q)−τ, the following condition holds; ifX→S is a sequence fo τ blowing-downs which gives a relatively minimal model with ruling ρ:S→C, take as basis of the Neron Severi groupNS(X) a smooth rational curve which is the total transform of a fiber ofC, the total transform of a minimal section of ρ and the total transformDi, 1≤i≤τ, of the exceptional curver; then for everyH andL∈Pic (X) withH ample,H (resp.L) represented by the integersa’, b’, a’i, (resp.a’, b’, a’i), 1≤i≤τ, in the chosen basis ofNS(X) the moduli spaceM(ZX, 2,H, L, t) of rank 2H-stable vector bundles onX with determinantL andc2=t is generically smooth and the number, dimension and «birational structure» of the irreducible components ofM(X, 2,H, L, t)red do not depend on the choice ofK andX. Furthermore the birational structure of these irreducible components can be loosely described in terms of the birational structure of the components of suitableM(S, 2,H’, L’, t’)red withS a relatively minimal model ofX.