It was conjectured in [R] (and will not be proved in the present paper) what (for every d and n) the minimal free resolution of a general finite set Sc P" with card(S)=d should be (for a full discussion of the conjecture, see [L]). In particular it was conjectured in [R], [GO] (and will not be proved in the present paper) that such a set S should have a homogeneous ideal, I, with the minimal possible number of generators (compatible with the dimensions of the graded components of I). To state more precisely what this means and the results proven in this paper we introduce a few notations; before that, we stress the conjectures raised in Section 3 of this paper related to the minimal free resolution for suitable projective curves (one of the main aims of this note).
We fix an algebraically closed field K. Every scheme will be of finite type over K. For all integers a, b with a/> b/> 0 we write {a; b} for the binomial coefficients, i.e., {a;b} :=a!/((a-b)!b!). Fix an integer n~>2 and an integer d with, say d~>n+ 1. There is a unique integer k>~2 such that {n+k-1;n} ~<d< {n+k;n}; k is called the critical value for the pair (n, d). Set P := P". For a subscheme Y of the scheme X, let Ir.x be the ideal scheme of Y in X; we will write I r instead of 1~. e. Fix a general subset S of P with card(S)=d and let I=E)It, l,:=H~ ls(t)), be its homogeneous ideal. By the generality of S and the choice of k, we have It= {0}if t <k. A theorem of Mumford's (see [EG], proposition stated in the Introduction]) states in particular that I is generated by Ik~Ik+l. Clearly, to generate I we need Ik. The question is: how much of Ik+l do we need to generate I? Equivalently, if we choose homogeneous coordinates Xo ..... x,,, what is the codimension of the submodule
xolk+ ;.. Jrxnl k of Ik+l? Define integers a(n,k) and r(n,k) by the relation .~(., k) + r(., k) = (. + l){. + k; ,, } -{. + k-l;. }, 0 ~< r(n, k) < n. 206