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. Mustaţă (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 Mustaţăʼs conjecture (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 there exists a -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.