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.
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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.