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Santiago Encinas

Publications and source records attributed to Santiago Encinas.

10 recordsLinked to original sources
arXiv2025

Sequences of point blow-ups over perfect fields from a combinatorial point of view

We associate a combinatorial object to sequences of point blow-ups over perfect fields, the weighted directed graph, and another one to the composition of all blow-ups, which we call associated sequential morphisms, the $d-$ary intersection form. Then, in order to consider different fields extensions, we introduce the concepts of algebraically and combinatorially compatible partitions of the exceptional divisor for both sequences of point blow-ups and sequential morphisms, which lead us to define the corresponding algebraic and combinatorial equivalence classes. We prove that there exists a bijection between the respective combinatorial equivalence classes of sequences of point blow-ups and the associated sequential morphisms, and moreover, we also give a proof of the existence of a suitable bijection between the respective algebraic equivalence classes.

math.AG↗
arXiv2017

A procedure for computing the log canonical threshold of a binomial ideal

We present a procedure for computing the log-canonical threshold of an arbitrary ideal generated by binomials and monomials. The computation of the log canonical threshold is reduced to the problem of computing the minimum of a function, which is defined in terms of the generators of the ideal. The minimum of this function is attained at some ray of a fan which only depends on the exponents of the monomials appearing in the generators of the ideal.

math.AG↗
arXiv2012

Lojasiewicz exponent of families of ideals, Rees mixed multiplicities and Newton filtrations

We give an expression for the Łojasiewicz exponent of a wide class of n-tuples of ideals $(I_1,..., I_n)$ in $Ø_n$ using the information given by a fixed Newton filtration. In order to obtain this expression we consider a reformulation of Łojasiewicz exponents in terms of Rees mixed multiplicities. As a consequence, we obtain a wide class of semi-weighted homogeneous functions $(\mathbb{C}^n,0)\to (\mathbb{C},0)$ for which the Łojasiewicz of its gradient map $\nabla f$ attains the maximum possible value.

math.AG↗
arXiv2011

Some natural properties of constructive resolution of singularities

These expository notes, addressed to non-experts, are intended to present some of Hironaka's ideas on his theorem of resolution of singularities. We focus particularly on those aspects which have played a central role in the constructive proof of this theorem. In fact, algorithmic proofs of the theorem of resolution grow, to a large extend, from the so called Hironaka's fundamental invariant. Here we underline the influence of this invariant in the proofs of the natural properties of constructive resolution, such as: equivariance, compatibility with open restrictions, with pull-backs by smooth morphisms, with changes of the base field, independence of the embedding, etc.

math.AG↗
arXiv2010

Coefficient and elimination algebras in Resolution of Singularities

We compare some algebras appeared in the recent attempts to prove resolution of singularities in positive characteristic. We also construct an algebra which encodes the same information and it is equivalent, up to integral closure, to the previous structures. In the case of characteristic zero one may use these structures to obtain a resolution of singularities.

math.AG↗
arXiv2009

The Łojasiewicz exponent of a set of weighted homogeneous ideals

We give an expression for the Łojasiewicz exponent of a set of ideals which are pieces of a weighted homogeneous filtration. We also study the application of this formula to the computation of the Łojasiewicz exponent of the gradient of a semi-weighted homogeneous function $(\C^n,0)\to (\C,0)$ with an isolated singularity at the origin.

math.AG↗
arXiv2009

Lojasiewicz exponents and resolution of singularities

We show an effective method to compute the Łojasiewicz exponent of an arbitrary sheaf of ideals of $\OO_X$, where $X$ is a non-singular scheme. This method is based on the algorithm of resolution of singularities.

math.AG↗
arXiv2009

Embedded desingularization of toric varieties

We present a new method to achieve an embedded desingularization of a toric variety. Let $W$ be a regular toric variety defined by a fan $Σ$ and $X\subset W$ be a toric embedding. We construct a finite sequence of combinatorial blowing-ups such that the final strict transforms $X'\subset W'$ are regular and $X'$ has normal crossing with the exceptional divisor.

math.AG↗
arXiv2007

Rees algebras and resolution of singularities

Embedded principalization of ideals in smooth schemes, also known as Log-resolutions of ideals, play a central role in algebraic geometry. If two sheaves of ideals, say $I_1$ and $I_2$, over a smooth scheme $V$ have the same integral closure, it is well known that Log-resolution of one of them induces a Log-resolution of the other. On the other hand, in case $V$ is smooth over a field of characteristic zero, an algorithm of desingularization provides, for each sheaf of ideals, a unique Log-resolution. In this paper we show that algorithms of desingularization define the same Log-resolution for two ideals having the same integral closure. We prove this result here by using the form of induction introduced by Włodarczyk. We extend the notion of Log-resolution of ideals over a smooth scheme $V$, to that of Rees algebras over $V$; and then we show that two Rees algebras with the same integral closure undergo the same constructive resolution. The key point is the interplay of integral closure with differential operators.

math.AG↗