We address the question of the classification under blow-Nash equivalence of simple Nash function germs. We state that this classification coincides with the real analytic classification. We prove moreover that a simple germ can not be blow-Nash equivalent to a nonsimple one. The method is based on the computation of relevant coefficients of the real zeta functions associated to a Nash germ via motivic integration.
We make progress towards the classification of simple Nash germs invariant under the involution changing the sign of the first coordinate, with respect to equivariant blow-Nash equivalence, which is an equivariant Nash version of blow-analytic equivalence, taking advantage of invariants for this relation, the equivariant zeta functions.
We compute the Nash blow-up of a cominuscule Schubert variety. In particular, we show that the Nash blow-up is algebraically isomorphic to another Schubert variety of the same Lie type. As a consequence, we give a new characterization of the smooth locus and, for Grassmannian Schubert varieties, determine when the Nash blow-up is a resolution of singularities. We also study the induced torus action on the Nash blow-up and give a bijection between its torus fixed points and Peterson translates on the Schubert variety.
We address the following question, raised by T. Fukui. Is the corank an invariant of the blow-analytic equivalence between real analytic function germs? We give a partial positive answer in the particular case of the blow-Nash equivalence. The proof is based on the computation of some virtual Poincare polynomials and zeta functions associated to a Nash function germ.
We propose a refinement of the notion of blow-Nash equivalence between Nash function germs, which is an analog in the Nash setting of the blow-analytic equivalence defined by T.-C. Kuo. The new definition is more natural and geometric. Moreover, this equivalence relation still does not admit moduli for a Nash family of isolated singularities. Some previous invariants are no longer invariants for this new relation, however, thanks to a Denef & Loeser formula coming from motivic integration in a Nash setting, we managed to derive new invariants for this equivalence relation.
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.
A semialgebraic map $f:X\to Y$ between two real algebraic sets is called blow-Nash if it can be made Nash (i.e. semialgebraic and real analytic) by composing with finitely many blowings-up with non-singular centers. We prove that if a blow-Nash self-homeomorphism $f:X\rightarrow X$ satisfies a lower bound of the Jacobian determinant condition then $f^{-1}$ is also blow-Nash and satisfies the same condition. The proof relies on motivic integration arguments and on the virtual Poincaré polynomial of McCrory-Parusiński and Fichou. In particular, we need to generalize Denef-Loeser change of variab…
We define invariants of the blow-Nash equivalence of real analytic function germs, in a similar way that the motivic zeta functions of Denef & Loeser. As a key ingredient, we extend the virtual Betti numbers, which were known for real algebraic sets, as a generalized Euler characteristics for projective constructible arc-symmetrics sets. Actually we prove more: the virtual Betti numbers are not only algebraic invariant, but also Nash-invariant of arc-symmetric sets. Our zeta functions enable to sketch the blow-Nash equivalence classes of Brieskorn polynomials of two variables.
To any Nash germ invariant under right composition with a linear action of a finite group, we associate its equivariant zeta functions, inspired from motivic zeta functions, using the equivariant virtual Poincaré series as a motivic measure. We show Denef-Loeser formulae for the equivariant zeta functions and prove that they are invariants for equivariant blow-Nash equivalence via equivariant blow-Nash isomorphisms. Equivariant blow-Nash equivalence between invariant Nash germs is defined as a generalization involving equivariant data of the blow-Nash equivalence.
On a Computational Approach to the Nash Blowup Problem
Centro de Investigación en Matemáticas（数学研究中心）；Centro de Ciencias Matematicas, UNAM（墨西哥国立自治大学数学科学中心）；Instituto de Matemáticas, Universidad de Talca（塔尔卡大学数学研究所）
Given an arbitrary projective birational morphism of varieties, we provide a natural and explicit way of constructing relative compactifications of the maps induced on the main components of the jet schemes. In the case the morphism is the Nash blow-up of a variety, such relative compactifications are shown to be given by the Nash blow-ups of the main components of the jet schemes.
Approximation of real analytic functions by Nash functions is a classical topic in real geometry. In this paper, we focus on the Nash approximation of an analytic desingularization of a Nash function germ obtained by a sequence of blowings-up along smooth analytic centers. We apply the result to prove that Nash function germs that are analytically equivalent after analytic desingularizations are Nash equivalent after Nash desingularizations. Results are based on a precise Euclidean description of a sequence of blowings-up combined with Néron Desingularization.
In this paper we present new proofs using real spectra of the finiteness theorem on Nash trivial simultaneous resolution and the finiteness theorem on Blow-Nash triviality for isolated real algebraic singularities. That is, we prove that a family of Nash sets in a Nash manifold indexed by a semialgebraic set always admits a Nash trivial simultaneous resolution after a partition of the parameter space into finitely many semialgebraic pieces and in the case of isolated singularities it admits a finite Blow-Nash trivialization. We also complement the finiteness results with recursive bounds.
This paper shows the affirmative answer to the local Nash problem for a toric singularity and analytically pretoric singularity. As a corollary we obtain the affirmative answer to the local Nash problem for a quasi-ordinary singularity.
In this paper we give a positive answer to a question of Nash concerning the arc space of a singularity, for the class of quasi-ordinary hypersurface singularities, extending to this case previous results and techniques of Shihoko Ishii.
Nash proved that every irreducible component of the space of arcs through a singularity corresponds to an exceptional divisor that occurs on every resolution. He asked if the converse also holds: does every such exceptional divisor correspond to an arc family? We prove that the converse holds for toric singularities but fails in general.
This paper is devoted to the approximation of differentiable semialgebraic functions by Nash functions. Approximation by Nash functions is known for semialgebraic functions defined on an affine Nash manifold M, and here we extend it to functions defined on Nash subsets X of M whose singularities are monomial. To that end we discuss first "finiteness" and "weak normality" for such sets X. Namely, we prove that (i) X is the union of finitely many open subsets, each Nash diffeomorphic to a finite union of coordinate linear varieties of an affine space and (ii) every function on X which is Nash on…
It is a long-standing question whether an arbitrary variety is desingularized by finitely many normalized Nash blow-ups. We consider this question in the case of a toric variety. We interpret the normalized Nash blow-up in polyhedral terms, show how continued fractions can be used to give an affirmative answer for a toric surface, and report on a computer investigation in which over a thousand 3- and 4-dimensional toric varieties were successfully resolved.
We consider the question whether one can construct an embedded resolution of singularities of a singular variety $X\subset \textbf{A}^n$ from the data of the irreducible components of the spaces of jets (of $X$) centered at the singular locus of $X.$ We show that the answer is no in general and that it is yes for some birational models of rational triple surface singularities.
Let $X\subset \mathbb{R}^n$ be a compact semialgebraic set and let $f:X\to \mathbb{R}$ be a nonzero Nash function. We give a Solernó and D'Acunto-Kurdyka type estimation of the exponent $\varrho\in[0,1)$ in the Łojasiewicz gradient inequality $|\nabla f(x)|\ge C|f(x)|^\varrho$ for $x\in X$, $|f(x)|<\varepsilon$ for some constants $C,\varepsilon>0$, in terms of the degree of a polynomial $P$ such that $P(x,f(x))=0$, $x\in X$. As a corollary we obtain an estimation of the degree of sufficiency of non-isolated Nash functions singularities
In this paper we introduce the notion of Blow-semialgebraic triviality consistent with a compatible filtration for an algebraic family of algebraic sets, as an equisingularity for real algebraic singularities. Given an algebraic family of 3-dimensional algebraic sets defined over a nonsingular algebraic variety, we show that there is a finite subdivision of the parameter algebraic set into connected Nash manifolds over which the family admits a Blow-semialgebraic trivialisation consistent with a compatible filtration. We show a similar result on finiteness also for a Nash family of 3-dimension…
In this paper we apply Shokurov's inductive method to study terminal and canonical singularities. As an easy consequence of the Minimal Model Program we show that for any three-dimensional log terminal singularity there exists some special, so called, plt blow-up. We discuss properties of them and construct some examples.
The Nash problem on arcs for normal surface singularities states that there are as many arc families on a germ (S,O) of a singular surface as there are essential divisors over (S,O). It is known that this problem can be reduced to the study of quasi-rational singularities. In this paper we give a positive answer to the Nash problem for a family of non-rational quasi-rational hypersurfaces. The same method is applied to answer positively to this problem in the case of E_6 and E_7 type singularities, and to provide new proof in the case of D_n, n> =4, type singularities.
Let (S,0) be a germ of complex analytic normal surface. On its minimal resolution, we consider the reduced exceptional divisor E and its irreducible components E_i. The Nash map associates to each irreducible component C_k of the space of arcs through 0 on S the unique component of E cut by the strict transform of the generic arc in C_k. Nash proved its injectivity and asked if it was bijective. As a particular case of our main theorem, we prove that this is the case if E.E_i <0 for any i.
We address Nash problem for surface singularities using wedges. We give a refinement of the characterisation of A. Reguera of the image of the Nash map in terms of wedges. Our improvement consists in a characterisation of the bijectivity of the Nash mapping using wedges defined over the base field, which are convergent if the base field is the field of complex numbers, and whose generic arc has transverse lifting to the exceptional divisor of a resolution of singularities. This improves the recent results of M. Lejeune-Jalabert and A. Reguera for the surface case. In the way to do this we find…
It is shown that, for any reduced algebraic variety in characteristic zero, one can resolve all but simple normal crossings (snc) singularities by a finite sequence of blowings-up with smooth centres which, at every step, avoids points where the transformed variety together with the exceptional divisor has only snc singularities. The proof follows the philosophy of arXiv:1107.5595 that the desingularization invariant can be used together with natural geometric information to compute local normal forms of singularities.
In this paper we explore the generalized Nash problem for arcs on a germ of smooth surface: given two prime divisors above its special point, to determine whether the arc space of one of them is included in the arc space of the other one. We prove that this problem is combinatorial and we explore its relation with several notions of adjacency of plane curve singularities.
Let X be a singular real rational surface obtained from a smooth real rational surface by performing weighted blow-ups. Denote by Aut(X) the group of algebraic automorphisms of X into itself. Let n be a natural integer and let e=[e_1,...,e_l] be a partition of n. Denote by X^e the set of l-tuples (P_1,...,P_l) of distinct nonsingular curvilinear infinitely near points of X of orders (e_1,...,e_l). We show that the group Aut(X) acts transitively on X^e. This statement generalizes earlier work where the case of the trivial partition e=[1,...,1] was treated under the supplementary condition that …
It is known that the weights of a complex weighted homogeneous polynomial $f$ with isolated singularity are analytic invariants of $(\mathbb C^d,f^{-1}(0))$. When $d=2,3$ this result holds by assuming merely the topological type instead of the analytic one. G. Fichou and T. Fukui recently proved the following real counterpart: the blow-Nash type of a real singular non-degenerate convenient weighted homogeneous polynomial in three variables determines its weights. The aim of this paper is to generalize the above-cited result with no condition on the number of variables. We work with a character…
In this paper we give a necessary combinatorial condition for a negative--definite plumbing tree to be suitable for rational blow--down, or to be the graph of a complex surface singularity which admits a rational homology disk smoothing. New examples of surface singularities with rational homology disk smoothings are also presented; these include singularities with resolution graph having valency four nodes.
The Nash multiplicity sequence was defined by M. Lejeune-Jalabert as a non-increasing sequence of integers attached to a germ of a curve inside a germ of a hypersurface. M. Hickel generalized this notion and described a sequence of blow ups which allows us to compute it and study its behavior. In this paper, we show how this sequence can be used to compute some invariants that appear in algorithmic resolution of singularities.
We extend the Cox-Hu-Keel construction of the Cox rings to any proper birational morphisms of normal noetherian schemes. It allows the representation of any proper birational morphism by a map of schemes with mild singularities with torus actions. In a particular case, the notion generalizes the combinatorial construction of Satriano and the recent construction of multiple weighted blow-ups on Artin-stacks by Abramovich-Quek. The latter can be viewed as an extension of stack theoretic blow-ups by Abramovich, Temkin, and Wlodarczyk, a similar construction of McQuillan and the author's recent co…
Let $(X,O)$ be a germ of a normal surface singularity, $π: \tilde X\longrightarrow X$ be the minimal resolution of singularities and let $A=(a_{i,j})$ be the $n\times n$ symmetrical intersection matrix of the exceptional set of $\tilde X$. In an old preprint Nash proves that the set of arcs on a surface singularity is a scheme ${\cal H}$, and defines a map ${\cal N}$ from the set of irreducible components of ${\cal H}$ to the set of exceptional components of the minimal resolution of singularities of $(X,O)$. He proved that this map is injective and ask if it is surjective. In this paper we co…
We show that the minimum weight of a weighted blow-up of $\mathbf A^d$ with $\varepsilon$-log canonical singularities is bounded by a constant depending only on $\varepsilon $ and $d$. This was conjectured by Birkar. Using the recent classification of $4$-dimensional empty simplices by Iglesias-Valiño and Santos, we work out an explicit bound for blowups of $\mathbf A^4$ with terminal singularities: the smallest weight is always at most $32$, and at most $6$ in all but finitely many cases.
It is known that ambient bilipschitz equivalence preserves tangent cones. This paper explores the behavior of the Nash cone and, in particular, exceptional rays under ambient bilipschitz equivalence for real surfaces in $R^3$ with isolated singularity.
This paper deals with the Nash problem, which claims that there are as many families of arcs on a singular germ of surface $U$ as there are essential components of the exceptional divisor in the desingularisation of this singularity. Let $\mathcal{H}=\bigcup \bar{N_α}$ be a particular decomposition of the set of arcs on $U$, described later on. We give two new conditions to insure that $\bar{N_α}\not \subset \bar{N_β}$, $α\not = β$; more precisely,for the first one, we claim that if there exists $f \in {\mathcal{O}}_{U}$ such that $ord_{E_α}(f)<ord_{E_β}(f)$, where $E_α, E_β$ are exceptional d…
The blow-up of a singularity at the module of derivations
Instituto de Matemáticas, Unidad Cuernavaca, Universidad Nacional Autónoma de México（墨西哥国立自治大学库埃纳瓦卡数学研究所）；Instituto de Ingeniería y Tecnología, UACJ（奇瓦瓦自治科技大学工程与技术研究所）
We establish an algorithm for resolution of singularities of an idealistic filtration in dimension 3 (at the local level) in positive characteristic, incorporating the method recently developed by Benito-Villamayor into our framework. Although (a global version of) our algorithm only implies embedded resolution of surfaces in the smooth ambient space of dimension 3, a classical result known before, we introduce some new invariant which effectively measures how much singularities are improved in the process of our algorithm and which strictly drops after each blow up. This is in contrast to the…
We prove that for every compact, connected, differentiable 3--manifold $M$ there is a compact complex manifold $X$ which can be obtained from projective 3--space by a sequence of smooth, real blow ups and downs such that $M$ is diffeomorphic to the set of real points of $X$. By earlier results, such an $X$ can almost never be projective.
In this paper we characterize the Blowing-up maps of ordinary singularities for which there exists a natural Gysin morphism, i.e. a bivariant class $θ\in Hom_{D(Y)}(Rπ_*\mathbb Q_X, \mathbb Q_Y)$, compatible with pullback and with restriction to the complement of the singularity.
In this paper we investigate the relation betwen the Nash modification and the Bi-Lipschtiz equivalent germs in the cases of two germs and for a family of hypersurfaces with isolated singularities.
The philosophy of the article is that the desingularization invariant together with natural geometric information can be used to compute local normal forms of singularities. The idea is used in two related problems: (1) We give a proof of resolution of singularities of a variety or a divisor, except for simple normal crossings (i.e., which avoids blowing up simple normal crossings, and ends up with a variety or a divisor having only simple normal crossings singularities). (2) For more general normal crossings (in a local analytic or formal sense), such a result does not hold. We find the small…
