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

Let X=mP1+⋯+mPn+k be a fat point subscheme of Pn, where Supp(X) consists of n+k distinct points which generate Pn. We study the regularity index τ(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 τ(X) holds if n≥4 and there are k+3 points of Supp(X) on a linear subspace Λ≃P3. We assume Supp(X) is not in general position and call d the least integer for which there exists a linear subspace Λ 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≥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 subspace Λ≃P3. Finally we prove that, in general, 2m−1≤τ(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.