Abstract
In this article we study the asymptotic behavior of the regularity of symbolic powers of ideals of points in a weighted projective plane. By a result of Cutkosky, Ein, and Lazarsfeld, regularity of such powers behaves asymptotically like a linear function, which is deeply related to the Seshadri constant of a blowup. We study the difference between regularity of such powers and this linear function. Under some conditions, we prove that this difference is bounded or eventually periodic.
As a corollary, we show that if there exists a negative curve, then the regularity of symbolic powers of a monomial space curve is eventually a periodic linear function. We give a criterion for the validity of Nagata’s conjecture in terms of the lack of existence of negative curves.
Citation
Steven Dale Cutkosky. Kazuhiko Kurano. "Asymptotic regularity of powers of ideals of points in a weighted projective plane." Kyoto J. Math. 51 (1) 25 - 45, Spring 2011. https://doi.org/10.1215/0023608X-2010-019
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