arXiv subjects
Publications and source records attributed to Enrique Chávez-Martínez.
We study the problem of resolving singularities via the blow-up of the module of derivations. Our main results are a positive answer for the case of curves and log-canonical surface singularities, i.e., a finite sequence of blow-ups along the module of derivations resolves the singularities of such varieties.
We study exceptional loci of F-blowups of normal toric varieties. In the $\Q$-factorial case, this study amounts to studying the exceptional loci of $G$-Hilbert schemes. We give a formula for the dimension of the center of a prime divisor on the F-blowup in terms of combinatorial data, together with an algorithm for computing it. Moreover, we study the relation between F-blowups and essential divisors for three-dimensional terminal singularities and canonical singularities. Finally, we give a simple condition ensuring that a prime divisor over the given toric variety has a positive-dimensional center on the F-blowup.
We describe the semigroup of the Lipschitz saturation of a complex analytic toric singularity in arbitrary dimension. We give a necessary and sufficient condition for a monomial in the normalization to belong to the Lipschitz saturation, in terms of Newton polyhedra and lattice conditions, and deduce a finite algorithm to compute it. We also show that, in dimension greater than two, Campillo's notion of presaturation differs from the Lipschitz saturation, even for complex singularities.
We investigate the relation between essential divisors and F-blowups, in particular, address the problem whether all essential divisors appear on the $e$-th F-blowup for large enough $e$. Focusing on the case of normal affine toric varieties, we establish a simple sufficient condition for a divisor over the given toric variety to appear on the normalized limit F-blowup as a prime divisor. As a corollary, we show that if a normal toric variety has a crepant resolution, then the above problem has a positive answer, provided that we use the notion of essential divisors in the sense of Bouvier and Gonzalez-Sprinberg. We also provide an example of toric threefold singularities for which a non-essential divisor appears on an F-blowup.
We show that the normalization of the Nash blow-up of order n of the toric surface singularity An can be factorized by the minimal resolution of An. The result is obtained using the combinatorial description of these objects.