LAUSR

Abstract Let X = m P 1 + ⋯ + m P n + k be a fat point subscheme of P n , where S u p p ( X ) consists of n + k distinct points which generate P n . 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 S u p p ( X ) on a linear subspace Λ ≃ P 3 . We assume S u p p ( 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 S u p p ( 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 S u p p ( 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 ≃ P 3 . Finally we prove that, in general, 2 m − 1 ≤ τ ( X ) ≤ 2 m 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.