Conference: Singularities in Algebraic Geometry

Lukas Braun: Some results on (non-)finitely generated quasiaffine algebras.
In this talk, I will report about work in progress on questions
related to (non-)finite generation of quasiaffine algebras. This in
particular relates to certain Cox rings, reductive invariant rings of
quasiaffine algebras, and singularities of such rings. Results
include abstract statements as well as computational methods.
Ana Belen de Belén de Felipe: Resolution of reduced curve singularities via one toric morphism.
The existence of resolution of singularities in positive
characteristic remains among the major unsolved problems in algebraic
geometry. In this talk, I will focus on reduced plane curve germs,
presenting some examples to illustrate the following conjecture stated
by Teissier (proved for plane branches by Goldin and Teissier, and for
reduced curve singularities by de Felipe, González-Pérez and
Mourtada): Let k be an algebraically closed field and let
Aⁿ(k) be the n-dimensional affine space. Given an algebraic
variety X in Aⁿ(k), there exist m and an embedding of
Aⁿ(k) in A^m(k) such that the singularities of X⊂A^m(k)
can be resolved by a toric modification of A^m(k).
Pierrette Cassou-Noguès: Sur le nombre de Milnor d'une singularité de courbe plane en caractéristique arbitraire.
Christopher Chiu: On singularities of the arc space of an algebraic variety.
The arc space X_∞ of an algebraic variety X is typically a
non-Noetherian scheme of infinite dimensions. Yet its local geometry
still carries deep information on the singularities of X. We will
first review the well-established connection between invariants of
divisorial valuations on X, and invariants of associated maximal
divisorial sets of X_∞ and their generic points. The main result of
this talk says that one of the latter invariants, the embedding
codimension, is generically constant over maximal divisorial sets. The
proof relies on an extension of the theorem of Drinfeld, Grinberg and
Kazhdan to non-rational arcs, which might be of independent interest.
Georges Comte: Bézout's bounds for rational and lacunary complex algebraic plane curves
I will explain which Bézout's bounds one can obtain in the complex
case for rational plane curves and lacunary algébraic curves. More
precisely, I will give lower and upper fewnomial bounds on the number
of intersection points in a ball of the complex plane, between a
rational curve P(C) and a lacunary algebraic curve Q=0. These
bounds depend only on the initial terms of P and on the support of Q.
This is a joint work with Sébatien Tavenas.
Eva Elduque: On the algebro-geometric realization of group homomorphisms.
Given a smooth complex quasi-projective curve with an extra orbifold
structure (which we interpret as a finite set of points with
multiplicities), we can define its orbifold fundamental group and
refer to the groups defined in this way as "curve orbifold
groups". The class of curve orbifold groups includes interesting
families of groups, like finitely generated free products of cyclic
groups and triangle groups. In this talk, we will address the
following problem: given a smooth quasi-projective variety U such that
its fundamental group surjects onto an infinite curve orbifold group,
when is this epimorphism induced by an algebraic morphism from U to a
curve C at the level of (orbifold) fundamental groups? We will also
see some geometric consequences of this result, as well as discuss
interesting examples in the case when U is the complement of a plane
curve in CP². Joint work with José Ignacio Cogolludo-Agustín.
Anne Frühbis-Krüger:TBA
Ursula Ludwig: A Morse theoretical complex computing intersection homology
In this talk we present for a compact pseudomanifold and a given
perversity in the sense of Goresky and MacPherson a Morse theoretical
cochain complex, which computes the intersection cohomology of the
space. The complex generalises the famous Morse-Thom-Smale complex
associated to a smooth Morse function on a smooth compact manifold to
pseudomanifolds equipped with socalled radial Morse functions.
Laurentiu Maxim: Higher singularities of local complete intersections
Higher analogues of rational and Du Bois singularities were introduced
recently through Hodge theoretic methods, and applied in the context
of deformation theory, birational geometry, etc. In this talk, I will
give a brief overview of these singularities, and explain how they can
be studied through the lens of characteristic classes in the case of
local complete intersections. (Joint work with Bradley Dirks and
Sebastian Olano).
Irma Pallarès Torres: Rational homology manifolds via cubical hyperresolutions.
In this talk, we will introduce a characterization of rational
homology manifolds in terms of cubical hyperresolutions and discuss
some applications in the theory of characteristic classes of singular
spaces.
Aftab Patel: The Geometry of Locally Bounded Rational Functions.
This talk concerns locally bounded rational functions defined on
non-singular real algebraic varieties — that is, rational functions
that are bounded in a neighborhood of every point in their
domain. These functions have already been studied in the past from an
algebraic point of view. However, in the context of this talk they
will be examined from geometric perspective, commencing with their
characterisation. There are two equivalent approaches to
characterising a locally bounded rational function. The first is via
the curve selection lemma. The second uses the resolution of
singularities. The most interesting observation here is that this
class of functions coincides exactly with the class of functions that
can be transformed into regular functions with values in the ground
field (we consider a real closed field here). Since these functions
are, in general, multi-valued, a careful analysis is required to
construct a suitable notion of their zero-sets. This talk will
conclude with an exposition of some other desirable properties of
these functions including a version of the Łojasiewicz inequality,
which will lead naturally to the reformulation of the zero sets of
these functions as subsets of arc-spaces. This, in turn, allows one to
obtain, in the case of two dimensions, the usual correspondence
between algebra and geometry that exists for other classes of rational
functions over real closed fields such as regulous functions. This
talk is based on joint work with Victor Delage and Goulwen Fichou.
Tomasz Pełka: Symplectic geometry "at radius zero" and applications.
Consider a degeneration of Kähler manifolds over a punctured disk
whose central fiber is snc. The A'Campo space (constructed
jointly with J. F. de Bobadilla) extends it to a symplectic fibration
over an annulus, whose restriction to the inner ("radius zero") circle
has a simple combinatorial description. In fact, it is decomposed to
pieces which are isotropic torus bundles over the strata of the
central fiber, and the monodromy acts as translation in these tori
(this description agrees with the topological construction of
N. A'Campo from the 70s, hence the name). In my talk I will survey
this construction and highlight its two applications. First, the
combinatorial description of the monodromy allows to compute its Floer
homology using McLean's spectral sequence, and thus infer a positive
answer to the Zariski multiplicity conjecture for μ-constant families
of isolated hypersurface singularities. Second, over the 0-dimensional
strata we get Lagrangian tori which have very similar properties as
those expected by the Kontsevich-Soibelman conjecture for maximal
Calabi-Yau degenerations.
Enrico Savi: The Nash-Tognoli theorem over Q & the Q-algebraicity problem for isolated singularities
Abstract and references