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

Bounds for the regularity of monomial ideals

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Einstieg in die Ingenieurmathematik aus der Berufspraxis - Unterstützung in Mathematik und fachadäquaten Lernstrategien

Das Projekt „Einstieg in die Ingenieurmathematik aus der Berufspraxis“ wurde im Wintersemester 2014/2015 an der Leibniz Universität Hannover pilotiert und richtet sich an Studierende, die nach längerer Zeit der Berufspraxis ihr Studium ohne bzw. mit länger zurückliegender Allgemeiner Hochschulreife aufnehmen. Für diese Gruppe von Studierenden stellt die Veranstaltung Mathematik für Ingenieure I in der Regel ein großes Hindernis für den erfolgreichen Einstieg ins Studium dar

An Explicit Non-smoothable Component of the Compactified Jacobian

This paper studies the components of the moduli space of rank 1, torsion-free sheaves, or compactified Jacobian, of a non-Gorenstein curve. We exhibit a generically reduced component of dimension equal to the arithmetic genus and prove that this is the only non-smoothable component when the curve has a unique singularity that is of finite representation type. Analogous results are proven for the Hilbert scheme of points and the Quot scheme parameterizing quotients of the dualizing sheaf.Comment: 23 pages; expository changes, error in table corrected, notation for singularities changed; to appear in Journal of Algebr

A normal form algorithm for the Brieskorn lattice

This article describes a normal form algorithm for the Brieskorn lattice of an isolated hypersurface singularity. It is the basis of efficient algorithms to compute the Bernstein-Sato polynomial, the complex monodromy, and Hodge-theoretic invariants of the singularity such as the spectral pairs and good bases of the Brieskorn lattice. The algorithm is a variant of Buchberger's normal form algorithm for power series rings using the idea of partial standard bases and adic convergence replacing termination.Comment: 23 pages, 1 figure, 4 table