


Seminari de Geometria Algebraica 2019/2020 


Conferenciant

Títol 
Data i hora 

Women in Geometry and Topology

25 a 27 de setembre, CRM

Joaquim Roé
UAB

NewtonOkounkov polygons
Since the formalization of NewtonOkounkouv bodies by KavehKhovanski and LazarsfeldMustata
about ten years ago, we have learned a lot about their shape and how it reflects the properties
of varieties and line bundles, espcially concerning cohomology and positivity. After Jow's proof
that the set of all NewtonOkounkov bodies of a given line bundle is a complete numerical
invariant (and local variants of this result) it is natural to ask about this set, for instance,
how do bodies of a given line bundle vary by chosing different flags/valuations? And how can one
extract information about the line bundle from its collection of bodies? I will look at some
well known results on NewtonOkounkov bodies from this point of view, and report on work in
progress with MoyanoFernández, Nickel and Szemberg for the case of surfaces and NewtonOkounkov
polygons.

Divendres 11 d'octubre, 15h10, Aula T2, FMIUB

Irene Spelta
Università di Pavia, Itàlia
Contact:
jcnaranjo at ub.edu

POSPOSAT

Divendres 18 d'octubre, 15h00, Aula T2, FMIUB

Olivier Martin
U. Chicago  Collège de France
Contact:
jcnaranjo at ub.edu

The degree of irrationality of most abelian surfaces is 4
The degree of irrationality of a complex projective \(n\)dimensional variety \(X\) is the minimal
degree of a dominant rational map from \(X\) to \(n\)dimensional projective space. It is a birational
invariant that measures how far \(X\) is from being rational. Accordingly, one expects the computation
of this invariant in general to be a difficult problem. Alzati and Pirola showed in 1993 that the
degree of irrationality of any abelian \(g\)fold is at least \(g+1\) using inequalities on holomorphic
length. Tokunaga and Yoshihara later proved that this bound in sharp for abelian surfaces and
Yoshihara asked for examples of abelian surfaces with degree of irrationality at least \(4\).
Recently, Chen and Chen, Stapleton showed that the degree of irrationality of any abelian surface
is at most \(4\). In this work I provide the first examples of abelian surfaces with degree of
irrationality \(4\). In fact, I show that most abelian surfaces have degree of irrationality \(4\).
For instance, a very general \((1,d)\)polarized abelian surface has degree of irrationality \(4\) if
\(d\) does not divide \(4\). We will give the complete proof since it is very short and uses nothing
beyond Mumford's theorem on rational equivalences of zerocycles on surfaces with \(p_g>0\).

Divendres 25 d'octubre, 15h00, Aula T2, FMIUB

Simone Marchesi
UB

Terao's conjecture holds for RUA arrangements
One of the most famous open problems in the topic of hyperplane
arrangements is the so called Terao's conjecture, which states that
the freedom of an arrangement depends on its combinatorics. After giving
an introduction to the problem, we will focus our attention on the
special case of triangular arrangements in the projective
plane, i.e. with all the lines passing through three fixed points. We
prove that the conjecture holds for a central family, calles Roots of
Unity Arrangements, which also provides a counterexample to the
weak conjecture. This is a joint work with Jean Vallès.

Divendres 8 de novembre, 15h00, Aula T2, FMIUB

Irene Spelta
Università di Pavia, Itàlia
Contact:
jcnaranjo at ub.edu

Infinitely many totally geodesic subvarieties via Galois coverings of Elliptic curves
We will speak about totally geodesic subvarieties of \(\mathcal{A}_{g}\) which are generically
contained in the Torelli locus. ColemanOort conjecture says that for genus \(g\) large
enough such varieties should not exist. Nevertheless if \(g\leq7\) there are examples
obtained as families of Jacobians of Galois coverings of curves \(f:C\to C'\), where
\(C'\) is a smooth curve of genus \(g'\). All of them satisfy a sufficient condition,
which we will denote by (\(\ast\)).
We will describe how strong is the condition (\(\ast\)). First, it gives us a bound on
the genus \(g'\) which we use to say that there are only 6 families of Galois coverings
of curves of \(g'\geq1\) which yield Special subvarieties of \(\mathcal{A}_{g}\).
Then we use (\(\ast\)) again to study the Prym maps of the families described above:
we will prove that they are fibered, via their Prym map, in curves which are totally
geodesic. In this way, we get infinitely many new examples of totally geodesic subvarieties
of \(\mathcal{A}_{2},\ \mathcal{A}_{3}\) and \(\mathcal{A}_{4}\).

Divendres 15 de novembre, 15h00, Aula T2, FMIUB

Simone Melchiorre Chiarello
U. Geneva, Suïssa
Contact:
eva.miranda at upc.edu

Quasisymplectic reductions and equivariant cohomology
Symplectic reductions provide an elegant framework for the study of moduli spaces.
In particular, Atiyah and Bott in their famous paper "The YangMills equations over Riemann surfaces" proved that the moduli space of
flat unitary connections is a particular symplectic reduction, of an infinitedimensional affine space via an infinitedimensional group.
This gives good information about the cohomology of such moduli spaces, thanks to Kirwan's surjectivity theorem.
However, when we try to display the same moduli space via finitedimensional manifolds, we fall into the domain of quasisymplectic reductions.
I will talk about the issues which arise in this setting, which are closely related to higher algebraic structures on equivariant cohomology.
This exposes a work in progress with A. Alekseev.

Divendres 22 de novembre, 15h00, Aula T2, FMIUB

Francesc Planas
UPC

Noetherian rings of low global dimension and syzygetic prime ideals
In this talk we present some recent work on Noetherian rings of low global dimension Concretely, let \(R\) be a
Noetherian ring. We prove that \(R\) has global dimension at most two if, and only if, every prime ideal of \(R\) is of linear
type. Similarly, one proves that \(R\) has global dimension at most three if, and only if, every prime ideal of \(R\) is syzygetic. As a
consequence, one derives a characterization of these rings using the AndréQuillen homology. The extension of this result to the
nonNoetherian setting is an open question by now.

Divendres 29 de novembre, 15h00, Aula T2, FMIUB

Roberta di Gennaro
U. Napoli Federico II, Itàlia
Contact:
miro at ub.edu

TBA

Divendres 17 de gener, 15h00, Aula T2, FMIUB



