650 research outputs found
Expected resurgences and symbolic powers of ideals
We give explicit criteria that imply the resurgence of a self-radical ideal
in a regular ring is strictly smaller than its codimension, which in turn
implies that the stable version of Harbourne's conjecture holds for such
ideals. This criterion is used to give several explicit families of such
ideals, including the defining ideals of space monomial curves. Other results
generalize known theorems concerning when the third symbolic power is in the
square of an ideal, and a strong resurgence bound for some classes of space
monomial curves.Comment: Final version to appear in the Journal of the London Mathematical
Societ
Bounding symbolic powers via asymptotic multiplier ideals
We revisit a bound on symbolic powers found by Ein-Lazarsfeld-Smith and
subsequently improved by Takagi-Yoshida. We show that the original argument of
Ein-Lazarsfeld-Smith actually gives the same improvement. On the other hand, we
show by examples that any further improvement based on the same technique
appears unlikely. This is primarily an exposition; only some examples and
remarks might be new.Comment: 10 pages. Primarily exposition. Originally written as appendix to
lecture notes by Brian Harbourne. v2: Minor changes. v3: Final version,
appeared in Ann. Univ. Pedagog. Crac. Stud. Mat
Segre Class Computation and Practical Applications
Let be closed (possibly singular) subschemes of a smooth
projective toric variety . We show how to compute the Segre class
as a class in the Chow group of . Building on this, we give effective
methods to compute intersection products in projective varieties, to determine
algebraic multiplicity without working in local rings, and to test pairwise
containment of subvarieties of . Our methods may be implemented without
using Groebner bases; in particular any algorithm to compute the number of
solutions of a zero-dimensional polynomial system may be used
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