The regularity index of up to 2n−1 equimultiple fat points of P^n

Abstract. Let X = mP_1 + +mP_n+k be a fat point subscheme of P^n, where Supp(X) consists of n + k distinct points which generate P^n: We study the regularity index tau (X) of X, which is the least degree in which the Hilbert function of X equals its Hilbert polynomial. We prove that the generalized Segre's bound for tau (X) holds if n>=4 and there are k+3 points of Supp(X) on a linear 3-dimensional subspace. We assume Supp(X) is not in general position and call d the least integer for which there exists a linear subspace of dimension d containing at least d+2 points of Supp(X). We prove that the generalized Segre's bound holds for simple points when either 3<=k<=n+1 and d>k-3 or k = 4 with no restriction on d. For m greater than or equal to 2 we prove the generalized Segre's bound when Supp(X) consists of n+4 points and either there are at least 3 points on a line or at least 5 points on a plane or at least 6 points on a linear 3-dimensional subspace. Finally we prove that, in general, 2m-1<=tau(X) <= 2m when 3<=k<=n-1 and d > k-1; and we extend this result to the non-equimultiple case. We also provide cases in which the previous bound gives the generalized Segre's bound.