  In this paper 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. 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.
