Year: 2025
Type: article
Abstract: 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 𝕡n. We show that the Segre’s upper bound is sharp for fat points where the points are constrained by geometric conditions in 𝕡n (Corollary 2.1 and Proposition 2.1). We show that if s ≤ 4, the Segre’s upper bound is sharp for s arbitrary fat points in 𝕡n (Theorem 3.1), and the Segre’s upper bound is sharp for 5 equimultiple fat points in 𝕡n (Theorem 3.2). We also show that if s ≥ 6 and n ≥ 2, then there exists always a set of s fat points in 𝕡n 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.
Author: Phan Thien
Institutions: Hue University, Hue University of Education
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4410414396