Sharpness of the Segre’s Upper Bound for the Regularity Index of Fat Points

In this paper we show some results about estimating the regularity index of fat points and study when the Segre’s upper bound is sharp for arbitrary fat points in 𝕡<sup><i>n</i></sup>. We show that the Segre’s upper bound is sharp for fat points where the points are constrained by geometric conditions in 𝕡<sup><i>n</i></sup> (Corollary 2.1 and Proposition 2.1). We show that if <i>s</i> ≤ 4, the Segre’s upper bound is sharp for <i>s</i> arbitrary fat points in 𝕡<sup><i>n</i></sup> (Theorem 3.1), and the Segre’s upper bound is sharp for 5 equimultiple fat points in 𝕡<sup><i>n</i></sup> (Theorem 3.2). We also show that if <i>s</i> ≥ 6 and <i>n</i> ≥ 2, then there exists always a set of <i>s</i> fat points in 𝕡<sup><i>n</i></sup> whose the Segre’s upper bound is not sharp (Proposition 3.1). We predict that Segre’s upper bound is sharp for 5 non-equimultiple fat points, but we can not prove this prediction nor we can find an example to show that the prediction is incorrect.