Abstracts

COURSES

Marco Golla: Milnor fibres from a symplectic perspective.

Symplectic topology is a flexible version of algebraic geometry over C, and many algebro-geometric constructions have an analogue in symplectic topology. We will look at smoothings and deformations of surface singularities from this perspective: I will talk about contact structures and symplectic fillings via open books and Lefschetz fibrations, with an emphasis on canonical contact structures on link of singularities.

David Rydh: Stacks and resolution of singularities.

The goal of this course is to explain how using stacks makes resolution of singularities and weak factorization simpler and more transparent. In particular, I will give a gentle introduction to stacks and stack-theoretic modifications such as root stacks and weighted blow-ups and explain some relations to logarithmic geometry. I will also present the weighted embedded resolution algorithm of Abramovich--Temkin--Włodarczyk as well as related algorithms such as weak factorization and destackification. If there is time, I will also say something about stack-theoretic modifications in positive characteristic and how these could shed light on resolution of singularities in positive characteristic.

Mattia Talpo: Log geometry and degenerations.

Logarithmic (log) geometry is a version of algebraic geometry where one has an extra structure sheaf (of monoids) on a variety, that keeps track of some information of interest, typically a boundary locus or some infinitesimal information about a deformation. The theory was originally born for arithmetic applications, but it later found use in several other contexts. I will give an introduction to log geometry, with a view towards its interest for the study of degenerations, and its connections to toric and tropical geometry.

TALKS

Maria Alberich Carramiñana: The minimal Tjurina number of irreducible germs of plane curve singularities.

The minimal Tjurina number in the equisingularity class of any plane branch can be computed from its semigroup of values since 1997 by using Peraire's algorithm. We give a closed formula for the minimal Tjurina number of an equisingularity class in terms of the sequence of multiplicities of the strict transform along a resolution. The proof is based on a formula by Genzmer in 2016 for the dimension of the generic component of the moduli space of plane branches.

This is a joint work with P. Almirón, G. Blanco and A. Melle-Hernández.

Angelica Benito: Small irreducible components of arc spaces in positive characteristic.
Joint work with O. Piltant and A. Reguera

In 1968, J. Nash initiated the study of the space of arcs X_∞ of a (singular) algebraic variety X over a field of characteristic zero, with the purpose of understanding the structure of the various resolutions of singularities of X. His work was done shortly after Hironaka’s proof of Resolution of Singularities in characteristic zero. Nash proved, using Resolution of Singularities that the space of arcs X^{Sing}_{∞} centered in the singular locus of X has a finite number of irreducible components.

This ideas extends, with some important differences, to the case of positive characteristic p > 0. The first difference is that resolution of singularities is still an open problem if char k = p > 0 and dim X ≥ 4. Another difference is that, in contrast with characteristic zero, (Sing X)_∞ may contain some of the irreducible components of X^{Sing}_{∞}. Understanding these “small” components is the main purpose of the talk.

In this talk, we will propose some questions which would have an affirmative answer if a resolution of singularities existed:

Q1: Has X^{Sing}_{∞} a finite number of irreducible components?

Q2: Given a variety X, does there exist a proper and birational morphism Y ⟶ X such that Y_∞ is irreducible?

We will give partial answers and explain the status of these problems.

David Bourqui: Degenerations of families of arcs and torus actions.

Arcs on an algebraic variety are naturally organized into families according to the valuation they define through their contact order along regular functions. To understand how these families degenerate in general is a natural yet difficult question, which has strong connections with the famous Nash problem. I will report on recent progress obtained in the case of varieties equipped with a torus action of complexity one .

(joint work with K. Langlois and H. Mourtada)

Pedro Daniel González Pérez: Resolving singularities of curves with one toric morphism.

We give an explicit positive answer, in the case of reduced curve singularities, to a question of B. Teissier about the existence of a toric embedded resolution after reembedding. In the case of a curve singularity (C, O) contained in a non singular surface S such a reembedding may be defined in terms of a sequence of maximal contact curves of the minimal embedded resolution of C. We prove that there exists a toric modification, after reembedding, which provides an embedded resolution of C. We use properties of the semivaluation space of S at O to describe how the dual graph of the minimal embedded resolution of C may be seen on the local tropicalization of S associated to this reembedding.
This is a joint work with Hussein Mourtada and Ana Belén de Felipe.

Tomasz Pełka: Equimultiplicity of μ-constant families.

I will present my recent joint work with J. F. de Bobadilla, proving that a family of isolated hypersurface singularities with constant Milnor number has constant multiplicity. The key idea is to endow the A'Campo model of "radius zero" monodromy with an explicit fiberwise symplectic structure. As a consequence, using a Morse-Bott approach we can prove that the fixed point Floer homology of the Milnor monodromy is the same for each member of a μ-constant family. The result follows since, as discovered by McLean, fixed point Floer homology captures the multiplicity.

Jean-Baptiste Campesato: Motivic, logarithmic, and topological Milnor fibrations. [CANCELLED]

We compare the topological Milnor fibration and the motivic Milnor fibre of a regular complex function with only normal crossing singularities by introducing their common extension: the complete Milnor fibration for which we give two equivalent constructions. The first one extends the classical Kato-Nakayama log-space, and the second one, more geometric, is based on a the real oriented version of the deformation to the normal cone.

In particular, we recover the topological Milnor fibration by quotienting the motivic Milnor fibration with suitable powers of (0,+∞). Conversely, we also show that the stratified topological Milnor fibration determines the classical motivic Milnor fibre.

(joint work with Goulwen Fichou and Adam Parusiński)