June 18, 2024
In this survey, we examine the properties of the top dimensional singular set $\text{sing}_{*}(u)$, including several theorems, geometric properties, and relation to homogeneous degree zero minimizers.
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May 31, 2023
We survey recent applications of topology and singularity theory in the study of the algebraic complexity of concrete optimization problems in applied algebraic geometry and algebraic statistics.
October 14, 2015
We list combinatorial criteria of some singularities, which appear in the Minimal Model Program or in the study of (singular) Fano varieties, for spherical varieties. Most of the results of this paper are already known or are quite easy corollary of known results. We collect these results, we precise some proofs and add few results to get a coherent and complete survey.
March 29, 2019
Effective methods are introduced for testing zero-dimensionality of varieties at a point. The motivation of this paper is to compute and analyze deformations of isolated hypersurface singularities. As an application, methods for computing local dimensions are also described. For the case where a given ideal contains parameters, the proposed algorithms can output in particular a decomposition of a parameter space into strata according to the local dimension at a point of the a...
November 9, 2004
We correct a mistake in an earlier paper and give a complete classification of singular varieties having an extremal secant line.
September 13, 2006
The paper introduces a number of new techniques to handle minimal hyersurface singularities. In particular, they allow to extend the obstruction theory for postive scalr curvature to any dimension.
December 14, 2010
This brief survey of some singularity invariants related to Milnor fibers should serve as a quick guide to references. We attempt to place things into a wide geometric context while leaving technicalities aside. We focus on relations among different invariants and on the practical aspect of computing them.
April 21, 2003
I will discuss recent progress by many people in the program of extending natural topological invariants from manifolds to singular spaces. Intersection homology theory and mixed Hodge theory are model examples of such invariants. The past 20 years have seen a series of new invariants, partly inspired by string theory, such as motivic integration and the elliptic genus of a singular variety. These theories are not defined in a topological way, but there are intriguing hints o...
November 1, 2016
In this article, we consider the singularity of an arbitrary homogeneous polynomial with complex coefficients $f(x_0,\dots,x_n)$ at the origin of $\mathbb C^{n+1}$, via the study of the monodromy characteristic polynomials $\Delta_l(t)$, and the relation between the monodromy zeta function and the Hodge spectrum of the singularity. We go further with $\Delta_1(t)$ in the case $n=2$. This work is based on knowledge of multiplier ideals and local systems.
February 7, 2011
This is a survey on recent results regarding singularities that occur on higher dimensional stable varieties.
November 10, 2011
We consider a classification problem of ideals by codimension in case rings are the local rings of irreducible curve singularities. In this paper, we introduce a systematic method to solve this problem.