A long-standing problem in Algebraic Geometry and Commutative Algebra is to determine the minimal graded free resolution of a 0- dimensional scheme Z in Pn or in an arbitrary projective variety X. In [18], M. Musta ¸ta (1998) predicted the graded Betti numbers of ˘ the minimal free resolution of a general set of distinct points Z in X. In this paper, we state a refined version of Musta ¸ta’s con- ˘ jecture (MRC) and we predict the existence of a non-empty open subset U ⊂ Hilbs (X) such that any [Z] ∈ U has a minimal graded free resolution without ghost terms (WMRC). In this paper, we are going to prove: (1) for any s d+3 3 − 1 there exists a d+2 2 - dimensional family of irreducible generically smooth surfaces of degree d in P3 satisfying the WMRC for s; and (2) any smooth cubic surface satisfies the MRC for any s 19

Minimal free resolution for points on surfaces / Miró-Roig, Rosa M.; Pons-Llopis, Juan Francisco. - In: JOURNAL OF ALGEBRA. - ISSN 0021-8693. - 357:(2012), pp. 304-318. [10.1016/j.jalgebra.2012.01.034]

Minimal free resolution for points on surfaces

Pons-Llopis, Juan Francisco
2012

Abstract

A long-standing problem in Algebraic Geometry and Commutative Algebra is to determine the minimal graded free resolution of a 0- dimensional scheme Z in Pn or in an arbitrary projective variety X. In [18], M. Musta ¸ta (1998) predicted the graded Betti numbers of ˘ the minimal free resolution of a general set of distinct points Z in X. In this paper, we state a refined version of Musta ¸ta’s con- ˘ jecture (MRC) and we predict the existence of a non-empty open subset U ⊂ Hilbs (X) such that any [Z] ∈ U has a minimal graded free resolution without ghost terms (WMRC). In this paper, we are going to prove: (1) for any s d+3 3 − 1 there exists a d+2 2 - dimensional family of irreducible generically smooth surfaces of degree d in P3 satisfying the WMRC for s; and (2) any smooth cubic surface satisfies the MRC for any s 19
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11583/2859414