1. science
2. mathematics
3. geometry

GAeL XXIII schedule
9:00
Monday
Tuesday
Wednesday
Thursday
Friday
Taelman
Fulton
Charles
Fulton
Taelman
Gagliardi(10:40-11:10)
Zach
Lindner
Dachs-Cadefau
Co↵ee break
Co↵ee break
Co↵ee break
Co↵ee break
Fulton
Charles
Charles
Fulton
Lunch
Lunch
Gallet
Garcia-Raboso
Gross (15:10-15:40)
Walters
Registration
9:30
10:00
10:30
11:00
11:30
12:00
Co↵ee break
Taelman
12:30
Lunch
13:00
13:30
Lunch
Lunch
14:00
and
Free Afternoon
14:30
15:00
Charles
Taelman
15:30
16:00
Co↵ee break
Co↵ee break
Lavanda
Hui
Moschetti (16:40-17:10)
Witasjek
16:30
17:00
Co↵ee break
Heuberger (16:50-17:20)
GAeL XXIV (17:10-17:40)
17:30
18:00
18:30
19:00
19:30
20:00
Poster session
Laface (15:50-16:20)
Pokora (16:10-16:40)
Conference dinner
(17:45-...)
Goodbye Co↵ee
(16:20-...)
Senior talks
Fran¸
cois Charles
Rational curves and rational equivalence on K3 surfaces
While they are not uniruled, K3 surfaces always contain rational curves,
and they are expected to contain infinitely many. The goal of these lectures will be to give an overview of the geometry of these curves, using
methods from complex and arithmetic geometry. While known results
are sometimes incomplete, we will try to give conjectural motivations
and will discuss the relationship of higher-dimensional analogous results with conjectures of Beauville-Voisin on rational equivalence.
William Fulton
Degeneracy loci
In the mid 19th century, Cayley found formulas for loci of matrices
with less than maximal rank. Such loci are defined by more equations
than their codimension, so the earlier formula of B´ezout doesn’t apply.
Finding such formulas has stimulated the development of intersection
theory. Today we look for formulas in terms of Chern classes. The story
includes situations governed by Lie groups other than the general linear
group, and relations with equivariant cohomology.
Lenny Taelman
Around the Woods-Hole trace formula
The Woods-Hole trace formula is a coherent version of the Lefschetz
trace formula (or fixed-point formula). It relates the action of an endomorphism on the coherent cohomology of an algebraic variety to the
fixed points of that endomorphism. It was conjectured by Shimura,
and proven in a working group at a conference in Woods Hole, Massachusetts in 1964.
In these lectures we will give a crash course in derived categories and
Grothendieck-Serre duality, and use these to give a simplified version
of Illusie’s proof of the Woods-Hole trace formula in SGA 5. Finally
we will discuss several applications of this trace formula.
Junior talks
Ferran Dachs-Cadefau
Poincar´e series of multiplier ideals in two-dimensional local rings with
rational singularities
Multiplier ideals and jumping numbers are invariants that encode relevant information about the structure of the ideal to which they are
associated. In general, jumping numbers and multiplier ideals of a fixed
ideal are determined by the divisors appearing in the resolution of the
ideal. In this talk, we give a formula to compute the multiplicity of
jumping numbers of an m-primary ideal in a 2-dimensional local ring
with rational singularities. This formula leads to a simple way to detect
whether a given rational number is a jumping number. Another consequence of the formula is that it allows us to give an explicit rational
expression for the Poincar´e series of the multiplier ideals introduced
by Galindo and Monserrat in 2010. This Poincar´e series encodes in a
unified way the jumping numbers and its multiplicities. This is a joint
work with Maria Alberich Carrami˜
nana, Josep Montaner and V´ıctor
Gonz´alez Alonso.
Giuliano Gagliardi
The generalized Mukai conjecture for symmetric varieties
(joint work with Johannes Hofscheier)
The generalized Mukai conjecture due to Bonavero, Casagrande, Debarre, and Druel gives an inequality relating the dimension, the Picard
number, and the pseudo-index of smooth Fano varieties and explicitly
describes the cases of equality. We associate to any complete spherical
variety X a certain nonnegative rational number }(X), which we conjecture to satisfy the inequality }(X)  dim X rank X with equality
holding if and only if X is isomorphic to a toric variety. We show that,
for spherical varieties, our conjecture implies the generalized Mukai
conjecture. We are able to prove our conjecture for symmetric varieties.
Matteo Gallet
Liaison Hexapods
joint with G. Nawratil and Josef Schicho
This work is devoted to the construction of a family of mechanical
devices, which belong to the class of so-called hexapods - also known
as Stewart Gough platforms. The geometry of this kind of mechanical
manipulators is defined by the coordinates of the 6 platform anchor
points p1 , . . . , p6 in R3 and of the 6 base anchor points P1 , . . . , P6 in
R3 . All pairs of points (pi , Pi ) are connected by a rigid body, called
leg, so that for all possible configurations the distance di from pi to
Pi is preserved. In general, a hexapod admits only a finite number
of possible configurations; we say that a hexapod is movable if, once
we fix the position of the base points Pi , the platform points pi are
allowed to move in an (at least) one dimensional set of configurations
respecting the constraints given by the legs; in this case, each pi moves
on the sphere with center Pi and radius di .
The classification of movable hexapods is still an open problem. Therefore it seems interesting to provide new instances of this kind of objects.
We present an algorithm that takes a general 6-tuple of base points as
an input and produces a movable hexapods with the given base points.
To do this, we employ concepts and techniques from algebraic geometry
(moduli space of points, liaison theory).
Alberto Garc´ıa-Raboso
A twisted nonabelian Hodge correspondence
Let X be a smooth complex projective variety. The nonabelian Hodge
correspondence of Simpson establishes an equivalence between categories of vector bundles on X equipped with two di↵erent kinds of
operators: flat connections on one side, and Higgs fields on the other.
I will discuss categories of twisted vector bundles on X equipped with
operators generalizing flat connections and Higgs fields, and prove an
equivalence between them.
Andreas Gross
Correspondence Theorems via Tropicalizations of Moduli Spaces
Tropical geometry relates objects from algebraic geometry, in its most
basic form subvarieties of algebraic tori, to integral polyhedral complexes, that is purely combinatorial objects. The connection from algebraic to tropical geometry usually involves a degeneration process and
therefore loses information. One of the most celebrated results of tropical geometry is Mikhalkin’s correspondence theorem, which showed
(for plane curves) that enough information is maintained to reduce algebraic enu- merative problems to tropical ones. The proof used ad hoc
methods, and since then many new techniques have been developed in
tropical enumerative geometry to pave the way for more conceptual
and general approaches. In this talk I will discuss how our current
knowledge of tropical moduli spaces, their intersection theory, and the
tropicalization map can be used to obtain correspondence theorems for
rational curves in toric varieties.
Liana Heuberger
Singularities of anticanonical divisors
Let X be a Fano variety. A result by Shokurov states that in dimension
three the linear system | KX | is non empty and every general element
D in | KX | is smooth. In dimension four, one can construct Fano
varieties X so that every such D is singular, however we show it has at
most terminal singularities. A consequence of this result is that locally
the singularities are ordinary double points.
Chun Yin Hui
Type A images of Galois representations and maximality
Given a compatible system of n-dimension `-adic Galois representations
arising from ´etale cohomology of any complete, non-singular variety, we
prove that for sufficiently large prime `, type A Galois image (in the
Cartan-Killing classification) is in some sense maximal in its Zariski
closure in GL(n). This is joint work with Michael Larsen.
Roberto Laface
On singular K3 surfaces defined over Q
After the groundbreaking work of Shioda-Mitani [4] and Shioda-Inose
[3], the theory of K3 surfaces has been enriched of an arithmetic flavour
due to the study of singular K3 surfaces. In 1977, Shioda and Inose
proved that every singular K3 surface can be defined over a number
field; later, Shafarevich [2] proved that for every positive integer n there
is a finite number of singular K3 surfaces defined over a field of degree n
over the rational numbers. Arithmetically speaking, one wants to classify all K3 surfaces with equation over Q: by using class field theory,
Sch¨
utt [1] narrowed down the investigation to 101 cases (plus possibly
one exception). With this motivation in mind, I will present some of
the techniques and the results in this direction: in particular, I will
stress the interplay between singular abelian surfaces and singular K3
surfaces, and I will start picturing the classification of Q-models in the
case of singular K3 surfaces of low class number.
References:
[1] Matthias Sch¨
utt. Fields of definition of singular K3 surfaces. Commun. Number Theory Phys., 1(2):307321, 2007.
[2] I. R. Shafarevich. On the arithmetic of singular K3-surfaces. In
Algebra and analysis (Kazan, 1994), pages 103108. de Gruyter, Berlin,
1996.
[3] T. Shioda and H. Inose. On singular K3 surfaces. In Complex analysis and algebraic geometry, pages 119136. Iwanami Shoten, Tokyo,
1977.
[4] Tetsuji Shioda and Naoki Mitani. Singular abelian surfaces and
binary quadratic forms. In Classification of algebraic varieties and
compact complex manifolds, pages 259287. Lecture Notes in Math.,
Vol. 412. Springer, Berlin, 1974.
Elena Lavanda
A Specialization map between D-modules and the pro´etale fundamental
group
Let R be a complete discrete valuation ring of equal characteristic p > 0
with algebraically closed residue field k and fraction field K. Let X be
a stable curve over R with smooth generic fibre XK . The aim of the
talk is to generalize the construction of the specialization morphism
defined by Grothendieck in SGA1, and to compare di↵erent fundamental groups on the fibres of X.
Consider the category of stratified bundles on the geometric generic
fibre XK . Via Tannakian duality, we associate to this category a group
scheme ⇡ strat (XK ). On the closed fibre Xk we consider, instead, the
continuous representations of the protale fundamental group of Xk ,
defined by B. Bhatt and P. Scholze. Again by Tannakian formalism,
we define an associated group scheme ⇡ pro´et (Xk )cts .
In this talk we show how to construct a specialization morphism from
⇡ strat (XK ) to ⇡ pro´et (Xk )cts . Moreover, we prove that the profinite completion of this specialization morphism coincides with the specialization
map defined by Grothendieck.
Niels Lindner
Smoothness and factoriality of projective hypersurfaces
Over the complex numbers, a general hypersurface in projective space
is smooth by Bertini’s theorem. The situation is di↵erent in positive
characteristic: The density of singular hypersurfaces in projective space
over a finite field is positive. Nevertheless, hypersurfaces with a small
amount of singularities compared to their degree form a set of density
one. Hypersurfaces of dimension three tend to have a big singular
locus if they fail to be factorial, i. e. if they contain a Weil divisor
not linearly equivalent to any Cartier divisor. Thus it seems that a
“general” hypersurface in P4 over a finite field is maybe not smooth,
but still factorial.
Riccardo Moschetti
Fourier-Mukai functors and indecomposable complexes on dual numbers
“Are all exact ‘geometric’ functors of Fourier-Mukay type?” By now,
due to the recent work of Rizzardo-van den Bergh, this well-known
problem has become “What are the minimal hypotheses on an exact
‘geometric’ functor for being of Fourier-Mukai type?” We propose an
approach to this problem which makes use of indecomposable objects
and we show that exact fully faithful functors from the category of
perfect complexes on the spectrum of the dual numbers to the bounded
derived category of coherent sheaves of a noetherian separated scheme
are of Fourier-Mukai type. The same circle of ideas is used to describe
the space of stability conditions for the dual numbers. This is a joint
work with F. Amodeo.
Piotr Pokora
Negative curves on blow ups of P2 along arbitrary sets of points
The theory of curves on algebraic surfaces is a classical subject of algebraic geometry. This theory develops in many direction, one of them
can be connected with the notion of the positivity (via Seshadri constants), but on the other hand we still know very little about the negativity of curves. One of the most fundamental information about the
way a curve is embedded into a surface is its self-intersection. In the
middle of the 20th century the Italian School of Algebraic Geometry
has formulated the following conjecture, which is now known as the
Bounded Negativity Conjecture.
Conjecture 1 (BNC). Let X be a smooth complex projective surface.
Then there exists b(X) 2 Z such that for every reduced curve C ⇢ X
one has C 2
b(X).
This conjecture is known to be true for instance for the projective plane
and minimal surfaces with the Kodaira dimension equal to 0.However,
we don’t know whether blow ups of these surfaces also have bounded
negativity. In my talk I would like to introduce Harbourne constants
which measure e↵ect the local negativity and after that I will present
two results which show that certain (large) classes of curves on blow
ups of the complex projective plane have bounded negativity. At the
end of my talk (and if time permits) I will present a certain relation of
the BNC and Zariski decompositions.
Robin Walters
Two related Bernstein-Sato polynomial computations.
The Bernstein-Sato polynomial, or b-function, is an important invariant in singularity theory, which is difficult to compute in general. I’ll
describe two b-function computations. The first computation is for
f (M, v), which tests whether v is a cyclic vector for M . This computation is complete and is achieved by making a connection to Eric
Opdam’s 1989 computation of a related polynomial. Secondly, I will
talk about ongoing work with Asilata Bapat to compute the b-function
for the Vandermonde determinant ⇠. We also make use of Opdam’s
work, in this case to produce a lower bound for the b-function of ⇠.
In our second result, we use duality of some D-modules to show that
the roots of this b-function of ⇠ are symmetric about -1. Finally, we
use results about jumping coefficients together with Kashiwara’s proof
that the roots of a b-function are rational in order to prove an upper
bound for the b-function of ⇠ and give a conjectured formula.
Jacub Witasjek
Base point freeness of line bundles in positive characteristic
Since the geometry of a variety is reflected in its maps to other varieties, showing base point freeness of line bundles is a fundamental
problem in birational geometry. In my talk, I will give a brief overview
of fascinating techniques for proving when line bundles in positive characteristic are base point free. Further, I will present results from joint
work with Diletta Matinelli and Yusuke Nakamura about base point
free theorems, and discuss the recent progress in the positive characteristic minimal model program.
Matthias Zach
Vanishing Cycles for Isolated Cohen-Macaulay codimension 2 Singularities
Vanishing cycles can be understood as the generators of the homology
of a smooth deformed fiber of an (isolated) singularity. For isolated
hypersurface or more general complete intersection singularities, these
vanishing cycles are well known. As a first step beyond these cases,
I would like to exhibit a technique to calculate the vanishing cycles
for isolated Cohen-Macaulay codimension 2 singularities of low CohenMacaulay type.