Holomorphic flows on analytic spaces

The existence of a holomorphic flow on an analytic space imposes restrictions upon the nature of the singularities of the space. We will talk about a result that states that, in Stein singular surfaces endowed with a holomorphic flow, the singularities are either quasihomogeneous or cyclic quotient ones

Normal singularities with torus actions

A T-variety is a normal variety endowed with a regular action of the algebraic torus T. In 2006, Altmann and Hausen gave a combinatorial description of affine T-varieties that generalizes the usual description of affine toric varieties via polyhedral cones. We call this the AH-description of a T-variety. In this talk, for every affine variety T-variety X, we compute a partial desingularization of X up to a toroidal variety in terms of the AH-description. We apply this partial desingularization to characterize the singularities of X. In particular, we give a criterion for X to have only rational singularities and partial criteria for X to have only Cohen-Macaulay and/or Du Bois singularities.

This talk is based on and joint work with H. Suess and current research with A. Laface and J. Moraga.

Resolution of singularities of the cotangent sheaf of a singular variety.

The subject of the talk is resolution of singularities of differential forms on an algebraic or analytic variety. We address the problem of finding a resolution of singularities $\sigma: X \to X_0$ of a singular algebraic or analytic variety $X_0$ such that the pulled back cotangent sheaf of $X_0$ (i.e., the pull-back of the differential forms defined in $X_0$) is given, locally in $X$, by monomial differential forms (with respect to a suitable coordinate system). This problem is related with monomialization of maps, the $L^2$ cohomology of singular varieties and reduction of singularities of vector-fields. In a work in collaboration with Bierstone, Grandjean and Milman, we give a positive answer to the problem when $\dim X_0 \leq 3$.

On Higher order Whitney conditions

For a germ of complex analytic singularity $(X,0)$ we use the concepts of generalized tangent spaces and the related Higher Nash blowups to define higher order Whitney conditions. We will talk about their basic properties and give a couple of examples. This is joint work with Roberto Callejas-Bedregal and Daniel Duarte.

Equisingularity of map germs from a surface to the plane.

Let $(X,0)$ be an ICIS of dimension 2 and let $f:(X,0)\to (\mathbb C^2,0)$ be a map germ with an isolated instability. We look at the invariants that appear when $X_s$ is a smoothing of $(X,0)$ and $f_s\to B_\epsilon$ is a stabilization of $f$. We find relations between these invariants and also give necessary and sufficient conditions for a $1$-parameter family to be Whitney equisingular.

Joint work with J.J. Nuño-Ballesteros and J. N. Tomazella.

Multiplier ideals of irreducible plane curves

Multiplier ideals are a recent and important tool in singularity theory and birational geometry. In particular they give information of the singularity of a divisor. However they are difficult to compute in practice because in the algebraic geometry context they are defined through resolution of singularities. We give an effective method to built them in the case of irreducible plane curves using equisingularity invariants of the curve as the Newton pairs and the approximate roots.

Optimal bounds for T-singularities in stable surfaces

Kollár and Shepherd-Barron (1988) introduced a natural compactification to the Gieseker moduli space of surfaces of general type, which is analogous to the Deligne-Mumford (1969) compactification of the moduli space of curves of genus $g>1$. This compactification is coarsely represented by a projective scheme (due to Kollár 1990) because of Alexeev's proof of boundedness (1994). Thus we have a proper KSBA moduli space of stable surfaces, which includes classical canonical surfaces of general type. In particular, after fixing the self-intersection of the canonical class, we have a finite list of singularities appearing on stable surfaces. It is hard to give that list. T-singularities $1/dn^2(1,dna-1)$ (with $gcd(n,a)=1$ and $n>1$) form a remarkable set of singularities in stable surfaces, since they are precisely the ones showing up in normal degenerations of canonical surfaces in the KSBA compactification. This talk is about optimal bounds for singularities on stable surfaces W with one T-singularity. The bound depends linearly on $K_W^2$, and it is optimal when W is not rational. I will show examples achieving the bound for each Kodaira dimension of the minimal resolution of $W$. I will mention what we know when $W$ is rational, putting together our approach with Alexeev's proof. This is a joint work with Julie Rana.

Characteristic p, Hypersurfaces Singularities and Milnor Number

The aim of this talk is to discuss some aspects of hypersurfaces singularities in characteristic p, mainly isolated singularities. We will see how some problems with the well known definition m $\mu$ as the codimension of the gradient ideal are related with counter-examples of Bertini's Theorem in characteristic p. We also use a bit of Commutative Algebra to present a consistent definition for $\mu$ in arbitrary characteristic. If time permits we can also discuss an extension of the $\mu^*$ sequence introduced by Teissier to characteristic $p$ hypersurface singularities.

Local uniformization and reduction to rank one

In this talk we will give an overview about the local uniformization problem. This problem can be seen as a local version of the resolution of singularities problem for algebraic varieties. Local uniformization is one of the two main steps in Zariski's approach for resolution of singularities, which is still open in positive characteristic. We will also present our joint work with Mark Spivakovsky, where we show that in order to prove local uniformization, it is enough to prove it for rank one valuations. We will also discuss our recent work which generalizes the reduction to the rank one case for valuations centered on an algebraic variety not necessarily reduced.

Reduction of singularities for vector fields and line foliations in dimension 3

In dimension 2 Seidenberg theorem asserts that the singularities
of a line foliation can be ``simplified'' by performing (unramified) blow-ups.
Moreover all the final models provided by Seidenberg theorem possess at least one eigenvalue
different from zero. In dimension 3 however, this analogous statement no longer holds as
shown by Sanz and Sancho. Recently this topic has been the object of two major works:
Cano, Roche, and Spivakovsky have worked out a reduction procedure using unramified
blow-ups though some of their final models have all eigenvalues equal to zero. On the other
hand, McQuillan and Panazzolo have successfully used ramified blow-ups to obtain final
models having one eigenvalue different from zero but the resulting ambient space is an orbifold,
as opposed to a smooth manifold.

In this talk, we will build on these works to obtain a reduction of singularities theorem that
is arguably sharp. We will also provide an application of this statement to the reduction of
singularities of vector fields defined on compact complex manifolds of dimension 3. This is
joint work with H. Reis.

On the complexity of polynomial perturbations of integrable systems

We present the results of a recent joint work with Pavao Mardesic, Dmitry Novikov and Jessie Pontigo-Herrera.

We will consider polynomial perturbations of polynomial integrable systems. Fixing a  loop on a regular fiber, and fixing a perturbation, there is a displacement function associated to them. The first nonzero term of the displacement function is known as the first non zero Melnikov function and it depends on the perturbation. This function is expressed by an iterated integral. We will explain how to provide a universal bound for the complexity (depth) of such integral in terms of the geometry of the unperturbed system.

Betti numbers of tropical varieties

Surprisingly enough there exist planar tropical cubic curves of genus g for any non-negative integer g. During this talk, I will present this construction and some generalizations in higher dimension and higher degree. 

Joint work with Benoît Bertrand and Erwan Brugallé.

The semiring of values associated to an algebroid curve

We introduce the semiring $\Gamma$ of values with respect to the tropical operations associated to an algebroid curve. As a set, $\Gamma$ determines and is determined by the well known semigroup of values $S$. We prove that $\Gamma$ is always finitely generated in contrast to $S$. In particular, for a plane curve, we present a straightforward way to obtain $\Gamma$ in terms of the semiring of each branch of the curve and the mutual intersection multiplicity of its branches. In the analytical case, this allows us to connect directly the results of Zariski and Waldi that characterize the topological type of the curve. The principal ingredient is the concept of Standard Basis for the local ring of the curve that give us a computational method to compute the minimal system of generators of $\Gamma$.

On Segre numbers of homogeneous map germs

Segre numbers and Segre cycles of ideals were independently introduced by Tworzewski, by Achilles and Manaresi and by Gaffney and Gassler. They are generalization of the Lê numbers and Lê cycles, introduced by Massey. In this article we give Lê-Iomdine type formulas for these cycles and numbers of arbitrary ideals. As a consequence we give a Plücker type formula for the Segre numbers of ideals generated by weighted homogeneous functions, in terms of their weights and degree. As an application of these results, we compute, in a purely combinatorial manner, the Segre numbers of the ideal which defines the critical loci of a map germ defined by a sequence of central hyperplane in $\mathbb{C}^{n+1}$.

Indefinite fibrations on differentiable 4-manifolds

A broken Lefschetz fibration (BLF, for short) is a smooth map of a closed oriented 4-manifold onto a closed surface whose singularities consist of Lefschetz critical points together with indefinite folds (or round singularities). Such a class of maps was first introduced by Auroux-Donaldson-Katzarkov (2005) in relation to near-symplectic structures. In this talk, we give a set of explicit moves for BLFs, and give an elementary and constructive proof to the fact that any map into the 2-sphere is homotopic to a BLF with embedded round image. We also show how to realize any given null-homologous 1-dimensional submanifold with prescribed local models for its components as the round locus of a BLF. These algorithms allow us to give a purely topological and constructive proof of a theorem of Auroux-Donaldson-Katzarkov on the existence of broken Lefschetz pencils with embedded round image on near-symplectic 4-manifolds. We moreover establish a correspondence between BLFs and Gay-Kirby trisections of 4-manifolds, and show the existence of simplified trisections on all 4-manifolds. This is a joint work with R. Inanc Baykur (University of Massachusetts).

Surfaces with non isolated singularities

In this talk, we speak about the topological triviality and Whitney equisingularity of families of surfaces parametrized by finitely determined map germs from $\mathbb{C}^2$ to $\mathbb{C}^3$.

Sigularities of equidistant submanifolds (support front)

In classical differential geometry the "support function" of a given closed convex curve enables to describe the equidistant curves and their singularities. We show that the graph of the support function of a plane curve contains all local and global geometric information of the initial curve, of its equidistants and of its evolute (caustic). Moreover, to any plane curve (without convexity restrictions) corresponds a curve on the unit cylinder (the graph of a "multivalued function") and vice-versa. We establish the correspondence between Euclidean differential geometry of plane curves and projective differential geometry of curves on the unit cylinder the "support map", which sends any plane curve to a curve on the unit cylinder. We give the geometric construction of the natural isomorphism between the surface (in space-time) formed by the union of equidistants of a plane curve with the dual surface of the graph of the support function (the subvariety formed by the planes of $\mathbb R^3$ which are tangent to the graph). All our constructions and results hold in Euclidean spaces of higher dimensions for submanifolds of any dimension. Theorem. For any class of singularities $X$ the set of singularities of type $X$ of the evolute of a smooth submanifold $M$ of $\mathbb R^n$ is isomorphic to the set of singularities of type $X$ in the front formed by the hyperplanes of $\mathbb R^{n+1}$ which are tangent to the image of $M$ by the support map (in the unit cylinder $C_n$) by the support map. The talk will be elementary and with many pictures.

Singularities arising in the multiplication of polynomials.

 (Joint work with Marc Chaperon and Enrique Vega).

The multiplication of two general real monic polynomials of degree n and m defines a mapping $\mathbb{R}^{n+m} \to \mathbb{R}^{n+m}$ which is a local diffeomorphism at a point $(P,Q)$ if, and only if, the polynomials are relatively prime. The singularity type at a point where the greatest common divisor of P and Q is of positive degree was studied for a certain number of cases (Chaperon-LdM). In this talk we will give a general view of the theory and introduce a new singularity type obtained in joint work with Enrique Vega, who will describe it in detail.

Algebraically closed fields containing the ring of power series

We introduce a family of fields of series with support in strongly convex rational cones and coefficients in a field of positive characteristic. All these fields are algebraically closed and contain the ring of power series in several variables. As a consequence of our main theorem we extend the McDonald theorem to positive characteristic.

TBA

TBA

DEFORMING SPACES OF M-JETS OF HYPERSURFACES SINGULARITIES

Let $K$ be an algebraically closed field of characteristic zero, and $V$ a hypersurface defined by an irreducible polynomial $f$ with coefficients in $K$. In this talk we prove that an Embedded Deformation of $V$ which admits a Simultaneous Embedded Resolution induces, under certain mild conditions, a deformation of the reduced scheme associated to the space of $m$-jets $V_m$ , $m \geq 0$.

ON THE TOPOLOGY OF SMOOTH MAP-GERMS NEAR A CRITICAL POINT

We study the topology of differentiable map-germs $(R^n,0) \buildrel {f} \over{\to} (R^k,0)$, $n\geq k\geq2$, near a critical point. The starting point is Milnor's celebrated theorem (see the text below) stating that if $f$ is a complex valued holomorphic function, then one has two locally trivial fibrations which are essentially equivalent. The first is a fibration $\mathbb S_e \setminus f^{-1}(0) \buildrel {\frac{f}{|f|}} \over{\longrightarrow} \mathbb S^1$, where $\mathbb S_e$ is a sufficiently small sphere around $ 0 \in C^m$, while the second is a local tube fibration. Milnor also discusses in his book the real analytic case restricted to the rather stringent condition that $f$ has an isolated critical point. This gave rise to interesting work by several authors. Later Pichon-Seade extended the discussion to real analytic functions with an isolated critical value, which is more general but still rather stringent. Here we study the general case of smooth functions (not necessarily analytic) and arbitrary critical set, subject to two conditions which are stringent but rather general. The first of these grants that we have a local {\it tube fibration} away from the discriminant. The second condition grants that up to a homeomorphism, this local tube fibration determines an equivalent fibration on small spheres.

A new singularity in the multiplication of polynomials.

 (Joint work with Santiago López de Medrano).

The multiplication of monic polynomials of degree $n$ and $m$ defines a mapping $\mathbb{R}^{n+m} \to\mathbb{R}^{n+m}$. The singularities of this mapping at a point $(P,Q)$ have been studied by M.Chaperon and SLdM, and depend on the $mcd(P,Q)$. In this talk we will give the general idea and describe the singularity that appears in a case not studied before.

Equivariant indices of 1-forms on varieties

For a $G$-invariant holomorphic 1-form with an isolated singular point on a germ of a complex-analytic $G$-variety with an isolated singular point ($G$ is a finite group) one has notions of the equivariant homological index and of the (reduced) equivariant radial index as elements of the ring of complex representations of the group. During my talk I will show that on a germ of a smooth complex-analytic $G$-variety these indices coincide. This permits to consider the difference between them as a version of the equivariant Milnor number of a germ of a $G$-variety with an isolated singular point. The talk is based on the joint work with Sabir M. Gusein-Zade

Modified multiple point spaces

We introduce a new approach to multiple point spaces of high corank mappings. Our new spaces satisfy many natural properties, are easily computable, and in many cases they coincide with Kleiman’s multiple point schemes. Finally, our spaces characterize A-finite determinacy for arbitrary corank maps in a wide range of dimensions, in the same manner as Mond’s multiple point spaces do for corank one germs.

Detecting bifurcation values of polynomial maps.

Let $f:\mathbb{R}^n \to \mathbb{R}^p$ be a polynomial mapping, $n \geq p$. The bifurcation values of $f$ is the smallest subset $B(f) \subset \mathbb{R} ^p$ such that $f$ is a locally trivial fibration over $\mathbb{R}^p \setminus B(f)$. We present an effective estimation of the nontrivial part of $B(f)$.

SEMISTABLE FIBRATIONS OVER THE PROJECTIVE LINE WITH FIVE SINGULAR FIBRES

We consider a non-isotrivial semistable fibration over the projective line with five singular fibres. In addition, we suppose that the fibration is expressed as the pullback of a pencil on a minimal surface. We will show that $(K_X+F)^2=0$ whenever the genus of the general fibre is sufficiently large.

Stability of relative degree

Let $S$ be an unbounded subset of $R^n$. Consider a polynomial $f$. Let $\deg_S f$ be the smallest degree of a polynomial $h$ such that $f<h$ on $S$. We call such a number the degree of $f$ relative to $S$. Analogously, one can define a multiplicity at $0$ relative to a set $S$ such that $0$ lies in its closure. Consider a real polynomial mapping $(g_1,\dots, g_k): R^n\to R^k$ and its sublevel set $S_c$, where $c\in R^k$, given by inequalities $g_1<c_1,\dots,g_k<c_k$. We show that there exists a semialgebraic set $V_g\subset R^k$ of positive codimension such that if $c, C$ are contained in the same connected component of $R^k\setminus V_g$, then the relative degrees coincide i.e. $\deg_{S_c}\equiv \deg_{S_C}$. Analogous property is true for the relative multiplicity. To prove this, we will construct an appropriate compactification of $R^n$ via resolution of singularities. We will discuss the relation of $V_g$ with bifurcation values at infinity of $g$, the moment problems and Positivstellensaetze.

This is joint work with V. Grandjean.

Kato-Matsumoto type results for the image Milnor fiber

We bound the (homological) connectivity of the image Milnor fiber of a holomorphic map germ $f : ( \mathbb{C}^{n},0 ) \to (\mathbb{C}^{n+1},0)$ by means of the dimension of the locus of instability. Our methods apply in the case of corank 1.

This is joint work with Guillermo Penafort-Sanchis.

Poincaré-Hopf Index Formula Associated With Certain Real Algebraic Surfaces

In this talk we will discuss some aspects about the geometric structure of the real algebraic surfaces $z-f(x,y)=0$, where $f\in\mathbb{R}[x,y]$.

We will present an index formula for the field of asymptotic lines involving the number of connected components of the projective Hessian curve of $f$ and the number of godrons. As an application of this, we obtain upper bounds, respectively, for the number of godrons having an interior tangency and when they have an exterior tangency. We will analyse the extension to the real projective plane of both fields of asymptotic lines and the Poincaré index at its singular points at infinity. Also, we will show certain criteria to determine when the parabolic curve is compact and when the unbounded component of its complement is hyperbolic or elliptic.

The local Euler obstruction of generic determinatal varieties

In the 1970s MacPherson proved the existence and uniqueness of Chern classes for possibly singular complex algebraic varieties. The local Euler obstruction, defined by MacPherson in that paper, was one of the main ingredients in his proof. The computation of the local Euler obstruction is not easy; various authors propose formulas which make the computation easier. In a paper that published in 2000, Brasselet, Lê and Seade give a Lefschetz type formula for the local Euler obstruction. The formula shows that the local Euler obstruction, as a constructible function, satisfies the Euler condition relative to generic linear forms. In order to understand this invariant better, some authors worked on some more specific situations. For example, in the special case of toric surfaces, an interesting formula for the Euler obstruction was proved by Gonzalez-Sprinberg, his formula was generalized by Matsui and Takeuchi for normal toric varieties. A natural class of singular varieties to investigate the local Euler obstruction and the generalizations of the characteristic classes is the class of generic determinantal varieties. Roughly speaking, generic determinantal varieties are sets of matrices with a given upper bound on their ranks. Their significance comes, for instance, from the fact that many examples in algebraic geometry are of this type, such as the Segre embedding of a product of two projective spaces. In a joint work with M. Ruas and T. Gaffney, we prove a formula that allow us to compute the local Euler obstruction of generic determinantal varieties using only Newton binomials. Using this formula we also compute the Chern-Schwartz-MacPherson classes of such varieties.

Singular Improper Affine Spheres from a Lagrangian Submanifold

Given a Lagrangian submanifold L of an affine symplectic space $\mathbb{R}^{2n}$, one can define uniquely, up to additive constants, a center-chord and a special improper affine sphere, as hypersurfaces in $\mathbb{R}^{2n+1}$. For each of these two improper affine spheres associated to a Lagrangian submanifold $L$, its set of singularities contains $L$ and, furthermore, these improper affine spheres may present other singularities arbitrarily close to $L$. We classify the simple stable Lagrangian and Legendrian singularities that may occur in this context.

Milnor fibration to discriminant of dimension one of real analytical map

Milnor, in his well known book [4], proves the existence of, what we call today, the Milnor Fibration, for polinomial function $f: (\mathbb{C}^n, \underline{0}) \longrightarrow (\mathbb{C},0)$. Later, in [3], Lê presented a new version of this result, what is known as the Milnor-Lê Fibration. The first fibration is a projection on the unitary sphere and the second is in the Milnor tube, $N(\epsilon,\delta)= \mathbb{B}_\epsilon^{n} \cap f^{-1}(\mathbb{S}_\delta^1)$. The approach used in our problem is closer to the last fibration. The isolated singularity case was wide investigated and still has interesting open questions, although much more questions remains open in the non-isolated case. This talk is dedicate tho the non-isolated critical values case. In our work, we study the discriminant of a function $f$, denoted by $\Delta_f$. In the category of subanalytic subsets. Applying results on Whitney stratifications, with $w$ -regularity ([1]) it was possible to guarantee the main result of lecture:

Theorem: Let $f:(\mathbb{R}^n,0)\longrightarrow (\mathbb{R}^k,0), \ n < k \leq 2$ be a real analytic map-germs with dim $\Delta_f = 1$. Then $f_{|_{\mathcal{A}}}: \mathcal{A} \longrightarrow \mathcal{Y}$ is a locally trivial topological fibration, where $\mathcal{A}:= f^{-1}(\Delta_f \cap \stackrel{\circ}{\mathbb{D}_\eta^*}) \cap \mathbb{B}_\epsilon^n$, $\mathcal{Y} := \Delta_f \cap \stackrel{\circ}{\mathbb{D}_\eta^*}$, with $\epsilon >0$ sufficiently small.

In our work we consider linear discriminant ([2]). Improving the number of interesting examples of Milnor fibrations for the real case. We search more applications for the results obtained.

This research was developed with Aurélio Mengon Neto and Nivaldo Grulha.

References

[1] J-L. Verdier, Stratification de Whitney et Théorème de Bertini-Sard. Invent. Math., 36, p. 295-312, 1976.

[2] J.~L. Cisneros-Molina, A. Menegon Neto, J.~Seade, and J.~Snoussi, The $d_h$-regularity for real analytic map-germs.

[3] D.~T. Lê,  Some remarks on relative monodromy.  In Real and complex singularities Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976, p. 397-403. Sijthoff and Noordhoff, Alphen aan den Rijn, 1977.

[4] J. Milnor, Singular points of complex hypersurfaces. Annals of Mathematics Studies, No. 61 Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968.

On specialization of Milnor classes for singular varieties

Chern classes had played an important role in Complex Geometry and Algebraic Geometry ever since its introduction by S-S Chern in 1946. Chern classes are characteristic classes associated to smooth complex manifolds or smooth algebraic varieties.In this talk I will address the two most important generalization which have appeared in the literature for singular varieties, the so called Fulton-Johnson and Schwartz-MacPherson classes. The difference between those two classes is the so called Milnor class, which is a class supported on the singular locus of the variety. In this talk I will be mostly concerned with the specialization of this Milnor class when the variety is deformed in a flat one-parameter family of varieties.

The Bi-Lipschitz Equisingularity on Determinantal Surfaces in $\mathbb{C}^4$

T. Gaffney started the study about Bi-Lipschitz equisingularity from a perspective of his previous works in Whitney equisingularity. In these works, the study of the equisingularity condition is developed in two directions. The first one is the study of a suitable notion of integral closure for modules, applying to the jacobian module of a singularity. The other one is going through analytic invariants which control a particular stratification condition. In his work, Gaffney got a sufficient algebraic condition so that a family of curves is Bi-Lispchitz equisingular, using the concept of integral closure of ideals. Further, some equations was obtained relating the Milnor number, the multiplicity and the second Segre number of a plane curve $X$. We conjecture that similar equations may be obtained in the case of families of determinantal surfaces in $\mathbb{C}^4$ using the Milnor number for this kind of surface defined by Pereira and Ruas. We also hope to get similar conditions so that this family is Bi-Lipschitz equisingular. This is a work in progress with M. Pereira and N. Grulha.

Motivic String Amplitudes

The talk aims to discuss the connections between p-adic string amplitudes with the theory of local zeta functions. We will present the motivic versions ( in the sense of motivic integration) of the p-adic string amplitudes and discuss some applications.