On moduli of vector bundles on surfaces with negative Kodaira dimension

Content Provider	Semantic Scholar
Author	Ballico, Edoardo
Copyright Year	1982
Abstract	SuntoSiaX una superficie algebrica liscia completa con dimensione di Kodaira negativa e definita su un campo algebricamente chiusoK; fissiamoH eL∈Pic (X),t∈Z; siaq l’irregolarità diX e τ≔8(1−q)−KXEmphasis>2; siaM(X, 2,H, L, t) to schema dei moduli dei fibrati vettorialiH-stabili di rango 2 suX con determinateL ec2=t. Si dimostra che esiste una costantew che dipende solo daq, da τ e dalla classe numerica diH e diL (ma non da char (K) o dalla classe di isomorphismo diX) tale che per ognit≥w il numero, la dimensione e «la struttura birazionale» delle componenti irriducibili diM(X, 2,H, L, t)red non dipende dalla scelta di char (K),K eX ma solo daq, τ e dalle classi diH eL inNS(X). Inoltre la «struttura birazionale» di queste componenti irriducibili può essere grossolanamente descritta in termini delle componenti di opportuni spazi di moduliM(S, 2,H’, L’, t’) (doveS è un modello minimale diX).SummaryHere 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.
Starting Page	33
Ending Page	40
Page Count	8
File Format	PDF HTM / HTML
DOI	10.1007/BF02827081
Alternate Webpage(s)	https://page-one.springer.com/pdf/preview/10.1007/BF02827081
Alternate Webpage(s)	https://doi.org/10.1007/BF02827081
Volume Number	38
Language	English
Access Restriction	Open
Content Type	Text
Resource Type	Article