81 research outputs found
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
-modules, Bernstein-Sato polynomials and -invariants of direct summands
We study the structure of -modules over a ring which is a direct
summand of a polynomial or a power series ring with coefficients over a
field. We relate properties of -modules over to -modules over . We
show that the localization and the local cohomology module
have finite length as -modules over . Furthermore, we show the existence
of the Bernstein-Sato polynomial for elements in . In positive
characteristic, we use this relation between -modules over and to
show that the set of -jumping numbers of an ideal is
contained in the set of -jumping numbers of its extension in . As a
consequence, the -jumping numbers of in form a discrete set of
rational numbers. We also relate the Bernstein-Sato polynomial in with the
-thresholds and the -jumping numbers in .Comment: 24 pages. Comments welcome
Algorithms for computing multiplier ideals
We give algorithms for computing multiplier ideals using Gr\"obner bases in
Weyl algebras. The algorithms are based on a newly introduced notion which is a
variant of Budur--Musta\c{t}\v{a}--Saito's (generalized) Bernstein--Sato
polynomial. We present several examples computed by our algorithms.Comment: 23 pages, title changed, Theorem 4.5 added, typos corrected, and some
minor revisions (some notation changed, and Definition 2.1, Proposition 2.11,
Observation 3.1, and Observation 4.2 added
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