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.
| Published in | Pure and Applied Mathematics Journal (Volume 14, Issue 2) |
| DOI | 10.11648/j.pamj.20251402.12 |
| Page(s) | 24-28 |
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
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Copyright © The Author(s), 2025. Published by Science Publishing Group |
Segre’s Upper Bound, Fat Points, Regularity Index
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APA Style
Thien, P. V. (2025). Sharpness of the Segre’s Upper Bound for the Regularity Index of Fat Points. Pure and Applied Mathematics Journal, 14(2), 24-28. https://doi.org/10.11648/j.pamj.20251402.12
ACS Style
Thien, P. V. Sharpness of the Segre’s Upper Bound for the Regularity Index of Fat Points. Pure Appl. Math. J. 2025, 14(2), 24-28. doi: 10.11648/j.pamj.20251402.12
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Download@article{10.11648/j.pamj.20251402.12,
author = {Phan Van Thien},
title = {Sharpness of the Segre’s Upper Bound for the Regularity Index of Fat Points
},
journal = {Pure and Applied Mathematics Journal},
volume = {14},
number = {2},
pages = {24-28},
doi = {10.11648/j.pamj.20251402.12},
url = {https://doi.org/10.11648/j.pamj.20251402.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20251402.12},
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.
},
year = {2025}
}
TY - JOUR T1 - Sharpness of the Segre’s Upper Bound for the Regularity Index of Fat Points AU - Phan Van Thien Y1 - 2025/05/06 PY - 2025 N1 - https://doi.org/10.11648/j.pamj.20251402.12 DO - 10.11648/j.pamj.20251402.12 T2 - Pure and Applied Mathematics Journal JF - Pure and Applied Mathematics Journal JO - Pure and Applied Mathematics Journal SP - 24 EP - 28 PB - Science Publishing Group SN - 2326-9812 UR - https://doi.org/10.11648/j.pamj.20251402.12 AB - 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. VL - 14 IS - 2 ER -
Department of Mathematics, University of Education, Hue University, Hue City, Vietnam
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