Minimal free resolution for points on surfaces

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