Papers by Santiago Encinas
Contemporary Mathematics, 2003
Journal of Pure and Applied Algebra, 2011
We give an expression for the {\L}ojasiewicz exponent of a set of ideals which are pieces of a we... more We give an expression for the {\L}ojasiewicz 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 {\L}ojasiewicz exponent of the gradient of a semi-weighted homogeneous function $(\C^n,0)\to (\C,0)$ with an isolated singularity at the origin.
Eprint Arxiv 1103 1731, Mar 9, 2011
We give an expression for the {\L}ojasiewicz exponent of a wide class of n-tuples of ideals $(I_1... more We give an expression for the {\L}ojasiewicz exponent of a wide class of n-tuples of ideals $(I_1,..., I_n)$ in $\O_n$ using the information given by a fixed Newton filtration. In order to obtain this expression we consider a reformulation of {\L}ojasiewicz 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 {\L}ojasiewicz of its gradient map $\nabla f$ attains the maximum possible value.
Eprint Arxiv Math 0206244, Jun 1, 2002
This paper contains a short and simplified proof of desingularization over fields of characterist... more This paper contains a short and simplified proof of desingularization over fields of characteristic zero, together with various applications to other problems in algebraic geometry (among others, the study of the behavior of desingularization of families of embedded schemes, and a formulation of desingularization which is stronger than Hironaka's). Our proof avoids the use of the Hilbert-Samuel function and Hironaka's notion of normal flatness: First we define a procedure for principalization of ideals (i.e. a procedure to make an ideal invertible), and then we show that desingularization of a closed subscheme $X$ is achieved by using the procedure of principalization for the ideal ${\mathcal I}(X)$ associated to the embedded scheme (X). The paper intends to be an introduction to the subject, focused on the motivation of ideas used in this new approach, and particularly on applications, some of which do not follow from Hironaka's proof.
Asian Journal of Mathematics, Jun 1, 2011
These expository notes, addressed to non-experts, are intended to present some of Hironaka's idea... more These expository notes, addressed to non-experts, are intended to present some of Hironaka's ideas on resolution of singularities. We focus particularly on those ideas which have lead to the constructive proof of this Theorem.
Embedded principalization of ideals in smooth schemes, also known as Log-resolutions of ideals, p... more 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 W{\l}odarczyk. 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.
eries. (b) Conclude that the order of J at the regular ring OW;x is b, ( x (J) = b), iff x 2 V (D... more eries. (b) Conclude that the order of J at the regular ring OW;x is b, ( x (J) = b), iff x 2 V (Delta bGamma1 (J)) n V (Delta b (J)). (c) Shows that x (J) = b iff x (Delta bGamma1 (J)) = 1. (d) For J ae OW , J 6= 0, set J : W Gamma! Z the function that assigns to each closed point x, J (x) = x (J) 2 Z. Show that J is an upper-semi-continuous function along the closed points. 2. Let<F9.2
We present a procedure for computing the log-canonical threshold of an arbitrary ideal generated ... more 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.
this paper we focus on Hermite interpolation. For this case we need a carefully local analysis at... more this paper we focus on Hermite interpolation. For this case we need a carefully local analysis at the nodes with multiplicity bigger than one. Such analysis is done by using theory and some results on complete ideals (2.4). In such a way that we may Partially supported by Digicyt PB97-0471
Resolution of Singularities, 2000
... 158 Santiago Encinas and Orlando Villamayor Example 4.2. Assume k algebraically closed and ta... more ... 158 Santiago Encinas and Orlando Villamayor Example 4.2. Assume k algebraically closed and take£^ C two points in Wo AJ, set E0={Hu...,# r}, y0={£, C} and (Wo, E0) Y0 (WuEt) Here Hr+ i has two irreducible components: H^\P^ mapping to£ and//,;+ i= P^ mapping to C-Set J ...
Asian Journal of Mathematics, 2011
Page 1. arXiv:1008.4460v1 [math.AG] 26 Aug 2010 COEFFICIENT AND ELIMINATION ALGEBRAS IN RESOLUTIO... more Page 1. arXiv:1008.4460v1 [math.AG] 26 Aug 2010 COEFFICIENT AND ELIMINATION ALGEBRAS IN RESOLUTION OF SINGULARITIES. ROCÍO BLANCO AND SANTIAGO ENCINAS Dedicated to H. Hironaka in his 80th birthday. Introduction. ...
Asian Journal of Mathematics, 2011
These expository notes, addressed to non-experts, are intended to present some of Hironaka's idea... more These expository notes, addressed to non-experts, are intended to present some of Hironaka's ideas on resolution of singularities. We focus particularly on those ideas which have lead to the constructive proof of this Theorem.
Revista Matemática Iberoamericana, 2000
This paper contains a short and simplified proof of desingularization over fields of characterist... more This paper contains a short and simplified proof of desingularization over fields of characteristic zero, together with various applications to other problems in algebraic geometry (among others, the study of the behavior of desingularization of families of embedded schemes, and a formulation of desingularization which is stronger than Hironaka's). Our proof avoids the use of the Hilbert-Samuel function and Hironaka's notion of normal flatness: First we define a procedure for principalization of ideals (i. e. a procedure to make an ideal invertible), and then we show that desingularization of a closed subscheme X is achieved by using the procedure of principalization for the ideal I(X) associated to the embedded scheme X. The paper intends to be an introduction to the subject, focused on the motivation of ideas used in this new approach, and particularly on applications, some of which do not follow from Hironaka's proof. Part 3. Applications. 21 7. Weak and strict transforms of ideals: Strong Factorizing Desingularization. 21 8. On a class of regular schemes and on real and complex analytic spaces. 31 9. Non-embedded desingularization. 34 10. Equiresolution. Families of schemes. 37 11. Bodnár-Schicho's computer implementation. 43
Revista Matemática Complutense, 2013
We give an expression for the Lojasiewicz exponent of a wide class of n-tuples of ideals (I 1 , .... more We give an expression for the Lojasiewicz exponent of a wide class of n-tuples of ideals (I 1 , . . . , I n ) in O n using the information given by a fixed Newton filtration. In order to obtain this expression we consider a reformulation of Lojasiewicz exponents in terms of Rees mixed multiplicities. As a consequence, we obtain a wide class of semi-weighted homogeneous functions (C n , 0) → (C, 0) for which the Lojasiewicz of its gradient map ∇f attains the maximum possible value.
Proceedings of the London Mathematical Society, 2003
ABSTRACT Given an algorithm for resolution of singularities that satisfies certain conditions (‘a... more ABSTRACT Given an algorithm for resolution of singularities that satisfies certain conditions (‘a good algorithm’), natural notions of simultaneous algorithmic resolution, and of equi-resolution, for families of embedded schemes (parametrized by a reduced scheme $T$) are defined. It is proved that these notions are equivalent. Something similar is done for families of sheaves of ideals, where the goal is algorithmic simultaneous principalization. A consequence is that given a family of embedded schemes over a reduced $T$, this parameter scheme can be naturally expressed as a disjoint union of locally closed sets $T_j$, such that the induced family on each part $T_j$ is equi-resolvable. In particular, this can be applied to the Hilbert scheme of a smooth projective variety; in fact, our result shows that, in characteristic zero, the underlying topological space of any Hilbert scheme parametrizing embedded schemes can be naturally stratified in equi-resolvable families.
Journal of Symbolic Computation, 2011
We present a new method to achieve an embedded desingularization of a toric variety. Let W be a r... more 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 ⊂ W be a toric embedding. We construct a finite sequence of combinatorial blowing-ups such that the final strict transforms X ′ ⊂ W ′ are regular and X ′ has normal crossing with the exceptional divisor. *
Journal of Pure and Applied Algebra, 2011
We give an expression for the Lojasiewicz exponent of a set of ideals which are pieces of a weigh... more We give an expression for the Lojasiewicz 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 Lojasiewicz exponent of the gradient of a semi-weighted homogeneous function (C n , 0) → (C, 0) with an isolated singularity at the origin.
Commentarii Mathematici Helvetici, 2002
Hironaka's spectacular proof of resolution of singularities is built on a multiple and intricate ... more Hironaka's spectacular proof of resolution of singularities is built on a multiple and intricate induction argument. It is so involved that only few people could really understand it. The constructive proofs given later by Villamayor, Bierstone-Milman and Encinas-Villamayor presented important steps towards a better understanding of the reasoning. They describe an algorithmic procedure for resolution, using a local invariant to show that the situation improves under blowup. The centers of blowup are given as the locus where the invariant takes its maximal value.
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Papers by Santiago Encinas