472 research outputs found
Embedded desingularization of toric varieties
We present a new method to achieve an embedded desingularization of a toric
variety.
Let be a regular toric variety defined by a fan and
be a toric embedding. We construct a finite sequence of combinatorial
blowing-ups such that the final strict transforms are regular
and has normal crossing with the exceptional divisor.Comment: Some comments have been corrected and references adde
A short note on the multiplier ideals of monomial space curves
Thompson (2014) exhibits a formula for the multiplier ideal with multiplier
lambda of a monomial curve C with ideal I as an intersection of a term coming
from the I-adic valuation, the multiplier ideal of the term ideal of I, and
terms coming from certain specified auxiliary valuations. This short note shows
it suffices to consider only one auxiliary valuation. This improvement is
achieved through a more intrinsic approach, reduction to the toric case.Comment: This version adds Corollary 10 and fixes several typo
Formal Desingularization of Surfaces - The Jung Method Revisited -
In this paper we propose the concept of formal desingularizations as a
substitute for the resolution of algebraic varieties. Though a usual resolution
of algebraic varieties provides more information on the structure of
singularities there is evidence that the weaker concept is enough for many
computational purposes. We give a detailed study of the Jung method and show
how it facilitates an efficient computation of formal desingularizations for
projective surfaces over a field of characteristic zero, not necessarily
algebraically closed. The paper includes a generalization of Duval's Theorem on
rational Puiseux parametrizations to the multivariate case and a detailed
description of a system for multivariate algebraic power series computations.Comment: 33 pages, 2 figure
Desingularization in Computational Applications and Experiments
After briefly recalling some computational aspects of blowing up and of
representation of resolution data common to a wide range of desingularization
algorithms (in the general case as well as in special cases like surfaces or
binomial varieties), we shall proceed to computational applications of
resolution of singularities in singularity theory and algebraic geometry, also
touching on relations to algebraic statistics and machine learning. Namely, we
explain how to compute the intersection form and dual graph of resolution for
surfaces, how to determine discrepancies, the log-canoncial threshold and the
topological Zeta-function on the basis of desingularization data. We shall also
briefly see how resolution data comes into play for Bernstein-Sato polynomials,
and we mention some settings in which desingularization algorithms can be used
for computational experiments. The latter is simply an invitation to the
readers to think themselves about experiments using existing software, whenever
it seems suitable for their own work.Comment: notes of a summer school talk; 16 pages; 1 figur
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