Asymptotic regularity of powers of ideals of points in a weighted projective plane
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.
This paper considers the problem of tuning natural frequencies of a linear system by a memoryless controller. Using algebro-geometric methods it is shown how it is possible to improve current sufficiency conditions.The main result is an exact combinatorial characterization of the nilpotency index of the $\bmod 2$ cohomology ring of the real Grassmannian. Using this characterization, new sufficiency results for generic pole assignment for the linear system with m-inputs, p-outputs, and McMillan degree n are given. Among other results it is shown that \[2.25 \cdot \max (m,p) + \min (m,p) - 3 \geq n\] is a sufficient condition for generic real pole placement, provided $\min (m,p) \geq 4$.