On the computation of the minimal resolution of smooth parametric varieties
- Autores: Forruccio Orecchia
- Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 52, Fasc. 1, 2001, págs. 21-34
- Idioma: inglés
- Títulos paralelos:
- Cálculo de la resolución mínima de variedades paramétricas regulares
- Enlaces
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Resumen
In this paper we give some results on the computation of the minimal resolutionof the idealI(V)of a smooth parametric varietyV⊂Pnkof dimensionmrepresented by homogeneous polynomials of the same degreerand without basepoints. We show that the Castelnuovo-Mumford regularity ofVisreg(V)=min{d≥m−mr|HV(d)=dr+mm}+1, whereHV(d)is the Hilbertfunction ofV.IfVhas maximal rank and the minimal degree of a generator ofI(V)isα≥m−mrthenreg(V)≤α+1. In this case the shifts of the freemodulesFiof a minimal free resolution ofI(V)are at most two and we show thatthe Betti numbers are determined by computing the linear part of the resolution.In particular, ifVis minimally resolved theFihave one shift, for all but onei.We show that if(a1, ..., aq)∈kqare the coefficients of the polynomials thatrepresentVthere is an open subsetUofAqksuch that, if(a1, ..., aq)∈U,Visminimally resolved and that it is possible to check thatUis non-empty for fixedm, n, rby computer.
