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Titles and Abstracts
Carolina
Araujo - Explicit
log Fano structures on blowups of
projective spaces
In this talk we will discuss
blowups of projective spaces at general
points. We will determine which of these
blowups $X$ are log Fano, i.e., when there is
a $\mathbb{Q}$-divisor $\Delta$ on $X$ such
that $(X, \Delta)$ is klt and $-(K_X+\Delta)$
is ample. This is a joint work with Alex
Massarenti.
Gilberto
Bini - A New
Calabi-Yau Threefold
In this talk we will describe the geometry of
a new Calabi-Yau threefold Y that is obtained
as a crepant resolution of a complete
intersection with 72 ordinary double points in five
dimensional projective space. The Picard
number and the number of moduli of Y coincide and they are both
equal to 10.
This is joint work with Matteo Penegini.
Paolo
Cascini - Birational
geometry and singularities in positive
characteristic
Many of the results in the Minimal Model
Program over complex projective varieties
depend on Kodaira's vanishing theorem and its
generalizations over singular varieties.
Because of the failure of these tools over an
algebraically closed field of positive
characteristic, it is not known whether these
results generalize to this case. The main tool
available for the study of varieties over a
field of positive characteristic is the
Frobenius morphism. In particular, it is
crucial to understand singularities from this
point of view. I will survey some recent
progress in this direction.
Cinzia
Casagrande - Fano 4-folds,
flips, and smooth blow-ups of
points
Let X be a
(smooth, complex) Fano 4-fold. Suppose
that there exist a smooth projective
4-fold Y and
a point p in Y such
that X and
the blow-up of Y at p are
isomorphic in codimension 1;
then the second Betti number of X is
at most 12. We
will discuss this result and explain the
main ideas of the proof.
Filippo Favale
- A surface of
general type which is birational to
a surface of degree $10$ of $\mathbb
P^3$
I will
describe the construction of a family
of minimal surfaces of general type
with $p_g=q=0$ and $K^2=3$, whose
general member $S$ is birational to a
surfaces $\Sigma$ of degree $10$ in
$\mathbb P^3$.
The motivation for studying this type
of surfaces comes from some geometric
properties that are really
interesting. First of all the order of
$\pi_1(S)$ is the (conjectured)
maximum order for the fundamental
group of such surfaces (in this case
is $\mathbb Z_4 \ltimes\mathbb Z_4$,
and these surfaces are the first
discovered with this property).
Moreover, the birational map from $S$
to $\Sigma$ is not a morphism and it
is given by the linear system
$|2K_S+\eta|$ where $\eta$ is a
torsion element. Again, this property
has never been observed before.
Davide Frapporti -
On mixed
surfaces
Let C be a Riemann surface of genus at
least 2 and G be a finite group acting
on the product $C \times C$ exchanging
the factors and such that the index 2
subgroup $G^0$ of the elements that do
not exchange the factors acts freely.
We call the quotient surface
$X:=(C\times C)/G$ a mixed surface. In
the talk we investigate these surfaces
and explain how their geometry is
encoded in the group G. Based on this,
we present an algorithm to classify
the mixed surfaces with given
geometric genus, irregularity, and
self-intersection of the canonical
class. In particular we give the
classification of the mixed surfaces
with $K^2>0$ and $p_g=q$ and we
discuss their minimality. As a
byproduct, we obtain the first
examples of minimal surfaces of
general type with $K^2=7$ and
$p_g=q=1, 2$. This is a joint work
with N. Cancian.
Andreas Höring -
Families of
singular rational curves with
degenerations
A rational curve C on a projective
manifold X is said to be minimal if
its deformations dominate X and if for
a general point in X the deformations
of C passing through this point form a
proper family. Minimal rational curves
have been studied for a long time and
have numerous applications to the
classification of projective
manifolds.
In this talk I will introduce a class of
rational curves that admits some mild
degenerations if we fix two general
points. We prove that these curves are
immersed in the fixed points which is
the analogon of a theorem of Kebekus
for minimal rational curves. I will
then explain the role of these
rational curves for characterizations
of hyperquadrics. This is work in
progress with Thomas Dedieu.
Paolo Lella -
Symmetry and
equations of the Hilbert scheme
The Hilbert scheme is classically
realized as a subscheme of a suitable
Grassmannian. However, computing the
equations of a given Hilbert scheme in
terms of Plücker coordinates is a hard
achievement. This is mainly due to the
dimension of the projective space
given by the Plücker embedding and to
some redundancy of the Grassmannian.
In this talk, I will recall the
equations proposed by
Iarrobino-Kleiman and
Bayer-Haiman-Sturmfels, explaining why
it is in fact impossible to compute
them explicitly. Then, I will
introduce a simpler set of equations
(less equations of lower degree)
obtained taking into account the
symmetries of the Hilbert scheme and I
will show some explicit results. This
is a joint work with J. Brachat, B.
Mourrain and M. Roggero.
Massimiliano Mella
- On unirational conic
bundles
A variety is unirational if it is
dominated by a rational variety.
I aim to investigate the unirationality
problem for conic bundles over
an arbitrary field.
Roberto Muñoz
- Weak Zariski
decomposition on projective bundles
Weak Zariski decomposition on
projective bundles The Zariski
decomposition of a pseudoeffective
divisor on a smooth projective surface
is a key ingredient for the study of
linear series on surfaces. Since it
does not directly apply to higher
dimension, several attempts to
generalize this notion appear in the
literature. One of these
generalizations is the weak Zariski
decomposition (WZD): a divisor has a
WZD if, up to a birational
transformation, can be written
numerically as the sum of an effective
and anef divisor. In this talk we will
study this notion and, as a natural
application of the description of the
nef and pseff cones of certain
projective bundles, we will provide
some results on the existence of WZD's
for divisors on projective bundles
over varieties of Picard number one.
This is joint work with F. Di Sciullo
and L. E. Solá-Conde
Matteo
Penegini - Characterization
of
the 5-canonical
birationality of algebraic
threefolds with $p_g = 3$
In this talk I present a
characterization for the
birationality of the
$5$--canonical map of a
minimal algebraic threefolds
of general type with $p_g
\geq 3$. Our result is
an analogue of the
characterization of the
$4$-canonical map for
surfaces of general type due
to Bombieri. This is a joint
work with Meng Chen.
Sönke
Rollenske - Geometry and
moduli of stable surfaces
The moduli space of stable
surfaces is a modular
compactification of the
Gieseker moduli space of
surfaces of general type. I
will report joint work with
Wenfei Liu, Marco Franciosi
and Rita Pardini where we
study the geometry of stable
surfaces, especially
Gorenstein stable surfaces
with $K_X^2=1$
Edoardo
Sernesi - Syzygies
of special line bundles
on curves
I will introduce geometric
conditions called
$(\Delta_q)$ on a special
very ample line bundle $L$
on a projective curve C. I
will show that
$(\Delta_3)$ implies that
$L$ has the well known
property $(M_3)$,
generalizing a similar
result proved by Voisin
for $L=K$.
(joint
work with M.
Aprodu)
Luis
Solá Conde - Characterizing
rational homogeneous
spaces among Fano
manifolds
Many important aspects of
the geometry of rational
homogeneous spaces, a
class including
''classical varieties''
such as projective spaces,
quadrics and
Grassmannians, can be
understood in terms of the
representation theory of
their groups of
automorphisms. On the
other hand, in the context
of Mori theory, rational
homogeneous spaces appear
within the broader class
of Fano manifolds, hence
it is a natural question
to characterize them in
terms of Mori-theoretical
properties, such us the
positivity of their
tangent bundles, or the
behaviour of their
famillies of rational
curves. In this talk I
will make an account on
some recent results in
this direction, obtained
within a joint project
with R. Muñoz, G.
Occhetta, K. Watanabe and
J. Wisniewski.
Luca
Tasin - Algebraic
structures and Chern
numbers.
In 1952, F. Hirzebruch
posed the question about
the topological invariance
of Chern numbers of
complex projective
varieties. D. Kotschick in
2012 solved the problem
and asked the
following question: which
Chern numbers are
determined up to finite
ambiguity by the
underlying smooth
manifold?
We will show that in
dimension higher than 3
only few Chern numbers are
bounded by the underlying
manifold. Then we will
analyse the 3-dimensional
case, where the minimal
model program plays a
major role in our approach
to this problem.
Filippo
Viviani - Fourier-Mukai
and autoduality for
compactified Jacobians
To every reduced
(projective) curve X with
planar singularities one
can associate many fine
compactified Jacobians,
depending on the choice of
a polarization on X, which
are birational (possibly
non-isomorphic) singular
Calabi-Yau projective
varieties, each of which
yields a modular
compactification of a
disjoint union of copies
of the generalized
Jacobian of X.
We define a Poincaré sheaf
on the product of any two
(possibly equal) fine
compactified Jacobians of
X and show that the
associated integral
transform is an
equivalence of their
derived categories, hence
it defines a Fourier-Mukai
transform.
This generalizes the
classical result of S.
Mukai for Jacobians of
smooth curves and the more
recent result of D.
Arinkin for compactified
Jacobians of integral
curves with planar
singularities, and it
provides further
evidence for the
classical limit of the
geometric Langlands
conjecture (as formulated
by Donagi and Pantev).
As a corollary, we prove
that there is a canonical
isomorphism (called
autoduality) between the
generalized Jacobian of X
and the connected
component of the identity
of the Picard scheme of
any fine compactified
Jacobian of X and
that algebraic
equivalence and numerical
equivalence coincide on
any fine compactified
Jacobian.
This is a joint work with
M. Melo and A. Rapagnetta.
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