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We study the maximal multiplicity locus of a variety $X$ over a field of characteristic $p>0$ that is provided with a finite surjective radical morphism $\delta:X\rightarrow V$, where $V$ is regular, for example, when $X\subset\mathbb{A}^{n+1}$ is a hypersurface defined by an equation of the form $T^{q}-f(x_{1},\ldots,x_{n})=0$ and $\delta$ is the projection onto $V:=\operatorname{Spec}(k[x_{1},\ldots,x_{n}])$. The multiplicity along points of $X$ is bounded by the degree, say $d$, of the field extension $K(V)\subset K(X)$. We denote by $F_{d}(X)\subset X$ the set of points of multiplicity $d$. Our guiding line is the search for invariants of singularities $x\in F_{d}(X)$ with a good behavior property under blowups $X'\rightarrow X$ along regular centers included in $F_{d}(X)$, which we call \emph{invariants with the pointwise inequality property}. A finite radicial morphism $\delta:X\to V$ as above will be expressed in terms of an $\mathcal{O}_{V}^{q}$-submodule $\mathscr{M}\subseteq\mathcal{O}_{V}$. A blowup $X'\to X$ along a regular equimultiple center included in $F_{d}(X)$ induces a blowup $V'\to V$ along a regular center and a finite morphism $\delta':X'\to V'$. A notion of transform of the $\mathcal{O}_{V}^{q}$-module $\mathscr{M}\subset\mathcal{O}_{V}$ to an $\mathcal{O}_{V'}^{q}$-module $\mathscr{M}'\subset\mathcal{O}_{V'}$ will be defined in such a way that $\delta':X'\to V'$ is the radicial morphism defined by $\mathscr{M}'$. Our search for invariants relies on techniques involving differential operators on regular varieties and also on logarithmic differential operators. Indeed, the different invariants we introduce and the stratification they define will be expressed in terms of ideals obtained by evaluating differential operators of $V$ on $\mathcal{O}_{V}^{q}$-submodules $\mathscr{M}\subset\mathcal{O}_{V}$., Comment: To be published in the Michigan Mathematical Journal

When $X$ is a $d$-dimensional variety defined over a field $k$ of characteristic zero, a constructive resolution of singularities can be achieved by successively lowering the maximum multiplicity via blow ups at smooth equimultiple centers. This is done by stratifying the maximum multiplicity locus of $X$ by means of the so called {\em resolution functions}. The most important of these functions is what we know as {\em Hironaka's order function in dimension $d$}. Actually, this function can be defined for varieties when the base field is perfect; however if the characteristic of $k$ is positive, the function is, in general, too coarse and does not provide enough information so as to define a resolution. It is very natural to ask what the meaning of this function is in this case, and to try to find refinements that could lead, ultimately, to a resolution. In this paper we show that Hironaka's order function in dimension $d$ can be read in terms of the {\em Nash multiplicity sequences} introduced by Lejeune-Jalabert. Therefore, the function is intrinsic to the variety and has a geometrical meaning in terms of its space of arcs., Comment: Minor changes, corrected errata, and additional details given in Proposition 5.9. Accepted for publication in Indiana University Mathematics Journal

We discuss Hironaka's theorem on resolution of singularities in charactetistic 0 as well as more recent progress, both on simplifying and improving Hironaka's method of proof and on new results and directions on families of varieties, leading to joint work on toroidal orbifolds with Michael Temkin and Jaros{\l}aw W{\l}odarczyk., Comment: 20 pages, submitted to proceedings of the ICM, Rio 2018

Let $X$ be a singular algebraic variety defined over a field $k$, with quotient field $K(X)$. Let $s \geq 2$ be the highest multiplicity of $X$ and $F_s(X)$ the set of points of multiplicity $s$. If $Y\subset F_s(X)$ is a regular center and $X\leftarrow X_1$ is the blow up at $Y$, then the highest multiplicity of $X_1$ is less than or equal to $s$. A sequence of blow ups at regular centers $Y_i \subset F_s(X_i)$, say $X \leftarrow X_1 \leftarrow \dotsb \leftarrow X_n$, is said to be a {\em simplification} of the multiplicity if the maximum multiplicity of $X_n$ is strictly lower than that of $X$, that is, if $F_s(X_n) $ is empty. In characteristic zero there is an algorithm which assigns to each $X$ a unique simplification of the multiplicity. However, the problem remains open when the characteristic is positive. In this paper we will study finite dominant morphisms between singular varieties $\beta: X'\to X$ of generic rank $r \geq 1$ (i.e., $[K(X'):K(X)]=r$). We will see that, when imposing suitable conditions on $\beta$, there is a strong link between the strata of maximum multiplicity of $X$ and $X'$, say $F_{s}(X)$ and $F_{rs}(X')$ respectively. In such case, we will say that the morphism is strongly transversal. When $\beta: X'\to X$ is strongly transversal one can obtain information about the simplification of the multiplicity of $X$ from that of $X'$ and vice versa. Finally, we will see that given a singular variety $X$ and a finite field extension $L$ of $K(X)$ of rank $r \geq 1$, one can construct (at least locally, in \'etale topology) a strongly transversal morphism $\beta: X'\to X$, where $X'$ has quotient field $L$., Comment: 34 pages

Given a variety $X$ over a perfect field, we study the partition defined on $X$ by the multiplicity (into equimultiple points), and the effect of blowing up at smooth equimultiple centers. Over fields of characteristic zero we prove resolution of singularities by using the multiplicity as an invariant, instead of the Hilbert Samuel function.

The purpose of this paper is to show how Rees algebras can be applied in the study of singularities embedded in smooth schemes over perfect fields. In particular, we will study situations in which the multiplicity of a hypersurface is a multiple of the characteristic. As another application, here we indicate how the use of these algebras has trivialized local-global questions in resolution of singularities over fields of characteristic zero., Comment: Revised version. To appear in Indiana Math. J

Let $k$ be \emph{any} algebraically closed field in any characteristic, let $R$ be any regular local ring such that $R$ contains $k$ as a subring, the residue field of $R$ is isomorphic to $k$ as $k$-algebras and $\dim R\geq 1$, let $P$ be any parameter system of $R$ and let $z\in P$. We consider any $\phi\in R$ with $\phi\neq 0$. In our main theorem we assume several conditions depending on $P$, $z$ and Newton polyhedrons. By our assumptions the normal fan $\Sigma$ of the Newton polyhedron $\Gamma_+(P,\phi)$ of $\phi$ over $P$ has simple structure and we can make a special regular subdivision $\Sigma^*$ of $\Sigma$ called an upward subdivision, starting from a regular cone with dimension equal to $\dim R$ and repeating star subdivisions with center in a regular cone of dimension two. Let $X$ and $\sigma:X\rightarrow\Spec(R)$ denote the toric variety over $\Spec(R)$ and the toric morphism associated with $\Sigma^*$. We consider any closed point $a\in X$ such that $\sigma(a)$ is the unique closed point of $\Spec(R)$ and the morphism $\sigma^*: R \rightarrow\mathcal{O}_{X,a}$ of local $k$-algebras induced by $\sigma$. We show that our numerical invariant of $\sigma^*(\phi)\in \mathcal{O}_{X,a}$ measuring the badness of the singularity is strictly less than the same invariant of $\phi\in R$ and the singularity $\phi$ is strictly improved by $\sigma$. We notice that this result opens a way toward the theory of resolution of singularities in arbitrary characteristic. We add several submain theorems to make bridges toward it and to show that our assumptions of the main theorem are not strong. By these results we can show that in a mathematical game with two players A and B related to the resolution of singularities of $\phi$, the player A can always win the game after finite steps. It follows "the local uniformization theorem in arbitrary characteristic and in arbitrary dimension"., Comment: This is the third version of arXiv:1004.5446, and has a new abstract, a new introduction and a lot of new descriptions. In the third version I would like to delete the strange "s" at the end of the title of previous versions. However, I have found that the arXiv server computer does not accept the submission as the third version, if I delete only one character "s" at the end of the title

In this paper we construct a combinatorial algorithm of resolution of singularities for binomial ideals, over a field of arbitrary characteristic. This algorithm is applied to any binomial ideal. This means ideals generated by binomial equations without any restriction, including monomials and $p$-th powers, where $p$ is the characteristic of the base field. In particular, this algorithm works for toric ideals. However, toric geometry tools are not needed, the algorithm is constructed following the same point of view as Villamayor algorithm of resolution of singularities in characteristic zero.

In this paper we present some results showing the good behavior of the $\tau$-invariant of a Rees algebra with integral closure and elimination (of variables)., Comment: 17 pages

We show that determinantal varieties defined by maximal minors of a generic matrix have a non-commutative desingularization, in that we construct a maximal Cohen-Macaulay module over such a variety whose endomorphism ring is Cohen-Macaulay and has finite global dimension. In the case of the determinant of a square matrix, this gives a non-commutative crepant resolution., Comment: 52 pages, 3 figures, all comments welcome

The paper consist of two parts. In the first part we introduce flags of lattices and associated injective systems of (non-normal) cones and projective systems of (non-normal) affine toric varieties. We study the associated field of multidimensional Laurent power series. In the second part we use the resolution of singularities techniques to study the geometry near a complete flag of subvarieties and the Parshin's residues. The first part plays the role of a standard coordinate neighborhood for Parshin's points., Comment: 24 pages; no pictures

The monodromy conjecture states that every pole of the topological (or related) zeta function induces an eigenvalue of monodromy. This conjecture has already been studied a lot; however, in full generality it is proven only for zeta functions associated to a polynomial in two variables. In this article we consider zeta functions associated to an ideal. First we work in arbitrary dimension and obtain a formula (like the one of A'Campo) to compute the 'Verdier monodromy' eigenvalues associated to an ideal. Afterwards we prove a generalized monodromy conjecture for arbitrary ideals in two variables., Comment: 16 pages, to appear in Ann. Inst. Fourier

This is the second of a series of papers dealing with the asymptotic behavior of certain integrals occuring in the description of the spectrum of an invariant elliptic operator on a compact Riemannian manifold carrying the action of a compact, connected Lie group of isometries, and in the study of its equivariant cohomology via the moment map.

We study blowups of affine n-space with center an arbitrary monomial ideal and call monomial ideals that render smooth blowups tame ideals. We give a combinatorial criterion to decide whether the blowup is smooth and apply this criterion to discuss a smoothing procedure proposed by Rosenberg, monomial building sets and permutohedra., Comment: 22 pages, 4 figures

We describe a class of isolated nondegenerate hypersurface singularities that give a polynomial contribution to Batyrev's stringy E-function. These singularities are obtained by imposing a natural condition on the facets of the Newton polyhedron, and they are strictly canonical. We prove that Batyrev's conjecture concerning the nonnegativity of stringy Hodge numbers is true for complete varieties with such singularities, under some additional hypotheses on the defining polynomials (e.g. convenient or weighted homogeneous). The proof uses combinatorics on lattice polytopes. The results form a strong generalisation of previously obtained results for Brieskorn singularities., Comment: 23 pages

A differential operator $D$ commuting with an $S^1$-action is said to be rigid if the non-constant Fourier coefficients of $\ker D$ and $\coker D$ are the same. Somewhat surprisingly, the study of rigid differential operators turns out to be closely related to the problem of defining Chern numbers on singular varieties. This relationship comes into play when we make use of the rigidity properties of the complex elliptic genus--essentially an infinite-dimensional analogue of a Dirac operator. This paper is a survey of rigidity theorems related to the elliptic genus, and their applications to the construction "singular" Chern numbers., Comment: 18 pages, article for MSRI Topology of Stratified Spaces Proceedings

This paper is devoted to give all the technical constructions and definitions that will lead to the construction of an algorithm of resolution of singularities for binomial ideals. We construct a resolution function that will provide a resolution of singularities for binomial ideals, over a field of arbitrary characteristic. For us, a binomial ideal means an ideal generated by binomial equations without any restriction, including monomials and $p$-th powers, where $p$ is the characteristic of the base field. This resolution function is based in a modified order function, called $E$-order. The $E$-order of a binomial ideal is the order of the ideal along a normal crossing divisor $E$. The resolution function allows us to construct an algorithm of $E$-\emph{resolution of binomial basic objects}, that will be a subroutine of the main resolution algorithm., Comment: the previous version has been divided in two parts

Two main algorithmic approaches are known for making Hironaka's proof of resolution of singularities in characteristic zero constructive. Their main difference is the use of different notions of transforms during the resolution process and the use of a stratification by the Hilbert-Samuel function in the one using the strict transform. In this article, a (hybrid-type) algorithmic approach is proposed which allows the use of the strict transform without the full impact of the complexity of the stratification by the Hilbert-Samuel function at each step of the desingularization process. This new approach is not intended to always be superior to the implemented one which uses the weak transform, instead it has its strengths precisely at the weak point of the other one and is thus a candidate to be joined with it by an appropriate heuristic., Comment: 15 Pages, revision clarifying details in sections 1-3

A sequence of monoidal transformations is defined, in terms of invariants, for a singular hypersurface embedded in a smooth scheme of positive characteristic. Some examples are added to illustrate the improvement of singularities by this procedure., Comment: This paper has been withdrawn by the authors due to most results are included in a current version which appears in arXiv:1004.1803 with a minor change of the title

We define the singular orbifold elliptic genus and $E$-function for all normal surfaces without strictly log-canonical singularities, and prove the analogue of the McKay correspondence in this setting. Our invariants generalize the stringy invariants defined by Willem Veys for this class of singularities. We show that the ability to define these invariants is closely linked to rigidity phenomena associated to the elliptic genus., Comment: 22 pages

We introduce an upper semi-continuous function that stratifies the highest multiplicity locus of a hypersurface in arbitrary characteristic (over a perfect field). The blow-up along the maximum stratum defined by this function leads to a form of simplification of the singularities, also known as a reduction to the monomial case., Comment: Several typos corrected. Minor improvements on the presentation of the published paper

We compute the symplectic reductions for the action of Sp_2n on several copies of C^2n and for all coregular representations of Sl_2. If it exists we give at least one symplectic resolution for each example. In the case Sl_2 acting on sl_2+C^2 we obtain an explicit description of Fu's and Namikawa's example of two non-equivalent symplectic resolutions connected by a Mukai flop., Comment: Some minor corrections, to be published in Mathematische Zeitschrift, 19 pages

Given a noetherian local domain $R$ and a valuation $\nu$ of its field of fractions which is non negative on $R$, we derive some very general bounds on the growth of the number of distinct valuation ideals of $R$ corresponding to values lying in certain parts of the value group $\Gamma$ of $\nu$. We show that this growth condition imposes restrictions on the semigroups $\nu(R\setminus \{0\})$ for noetherian $R$ which are stronger that those resulting from the previous paper \cite{C2} of the first author. Given an ordered embedding $\Gamma\subset ({\mathbf R}^h)_{\hbox{\rm lex}}$, where $h$ is the rank of $\nu$, we also study the shape in ${\mathbf R}^h$ of the parts of $\Gamma$ which appear naturally in this study. We give examples which show that this shape can be quite wild in a way which does not depend on the embedding and suggest that it is a good indicator of the complexity of the semigroup $\nu(R\setminus \{0\})$.

This paper has been withdrawn by the author, Comment: This paper has been withdrawn by the author

For projective varieties with a certain class of 'mild' isolated singularities and for projective threefolds with arbitrary Gorenstein canonical singularities, we show that the stringy Hodge numbers satisfy the Hard Lefschetz property. This result fits nicely with a 6-dimensional counterexample of Mustata and Payne for the Hard Lefschetz property for stringy Hodge numbers in general. We also give such an example, ours is a hypersurface singularity., Comment: 15 pages

We introduce the notion of non-lc ideal sheaves. It is an analogue of the notion of multiplier ideal sheaves. We establish the restriction theorem, which seems to be the most important property of non-lc ideal sheaves., Comment: 21 pages; v2: mistakes were corrected, v3: the proof of the main theorem was revised, v4: minor modifications, v5: minor revision following referee's comments

We present an application of elimination theory to the study of singularities over arbitrary fields, particularly to the open problem of resolution. A partial extension of a function, defining resolution of singularities over fields of characteristic zero, is discussed here in positive characteristic., Comment: 29 pages

Let $G$ be a finite abelian subgroup of $PGL(r-1,K)=\mathrm{Aut}(\P^{r-1}_K)$. In this paper, we prove that the normalization of the $G$-orbit Hilbert scheme $\Hilb^G(\P^{r-1})$ is described as a toric variety, which corresponds to the Gr\"obner fan for some homogeneous ideal $I$ of $K[x_1, ..., x_r]$., Comment: 13 pages, 3 figures

We study monomial ideals, always locally given by a monomial, like a reasonable first step to estimate in general the number of monoidal transformations of Villamayor's algorithm of resolution of singularities. The resolution of a monomial ideal $$ is interesting due to its identification with the particular toric problem $$. In the special case, when all the exponents $a_i$ are greater than or equal to the critical value $c$, we construct the largest branch of the resolution tree which provides an upper bound involving partial sums of Catalan numbers. This case will be called ``minimal codimensional case''. Partial sums of Catalan numbers (starting $1,2,5,...$) are $1,3,8,22,...$ These partial sums are well known in Combinatorics and count the number of paths starting from the root in all ordered trees with $n+1$ edges. Catalan numbers appear in many combinatorial problems, counting the number of ways to insert $n$ pairs of parenthesis in a word of $n+1$ letters, plane trees with $n+1$ vertices, $... $, etc. The non minimal case, when there exists some exponent $a_{i_0}$ smaller than $c$, will be called ``case of higher codimension''. In this case, still unresolved, we give an example to state the foremost troubles. Computation of examples has been helpful in both cases to study the behaviour of the resolution invariant. Computations have been made in Singular (see \cite{sing}) using the \emph{desing} package by G. Bodn\'ar and J. Schicho, see \cite{lib}., Comment: 17 pages, to appear in International Journal of Mathematics. Corrected format