F. BASTIANELLI :
Title : Effective cycles on symmetric products of curves
Abstract : I will be concerned with the convex-geometric proerties of the cone of n-dimensional pseudoeffective
cycles in the d-fold symmetric product $C_d$ of a smooth curve $C$. In particular, I will show
that the cone generated by n_dimensional diagonal cycles is a perfect face
of the pseudoeffective cone, and I will determine its extermal rays.
Moreover, I will describe a series of cones related to the contractibility properties
of the Abel-Jacobi morphism, which form a maximal chain of perfect faces of
the pseudoeffective cone, provided that $C$ is a very general curve and $d$
is sufficiently large. This is a joint work with A. Kouvidakis, A.F. Lopez and F. Viviani.
M. BOLOGNESI :
Title:
A variation on a
theme of Faber and Fulton
Abstract :
In this talk we study
the geometry of GIT congurations of n ordered points on P1 both
from
the the birational and the biregular viewpoint. In particular, we
prove that any extremal ray of
the Mori cone of effective curves
of the quotient (P1)^n//PGL(2), taken with the symmetric
polarization,
is generated by a one dimensional boundary stratum
of the moduli space. Furthermore, we develop
some technical
machinery that we use to compute its group of automorphisms. (joint
work with A.Massarenti)
F. CAMPANA :
Title: Dense entire curves in rationally connected manifolds
Abstract : we show in particular that for any given sequence of points on a rationally
connected manifold X, there is a holomorphic map from the complex line to X
whose image contains all these points. (Joint with Joerg Winkelmann)
L. EIN
Title : Singularities and syzygies of secant varieties of curves I
A. KURONYA :
Title :
Effective
global generation on manifolds with numerically trivial canonical
bundle
Abstract : If
L is a line bundle on a projective manifold, then the
existence
of effective bounds for its tensor powers to have global
sections
or become globally generated have been a central problem in
algebaic
geometry for the last 150 years. While the case of curves
follows
from Riemann-Roch, satsifactory answers for surfaces only
arrived
about thirty years ago. Research in the area has been mostly
motivated by Fujita's conjectures predicting the global
generation and
very ampleness of certain adjoint line bundles.
In this talk we will
consider the case of effective global
generation for projective
manifolds with numerically trivial
canonical bundle. This is an account
of joint work with Yusuf Mustopa.
J. LIU :
Title: Manifolds admitting locally rigid hypersurfaces as ample divisors
Abstract: Polarized manifolds containing special subvarieties
as ample divisors are fundamental objects in adjunction theory.
In this talk I will focus on the polarized pairs $(X,A)$
where $A$ is a locally rigid smooth Fano hypersurface
of a rational homogeneous space $G/P$ of Picard number
one and show how the VMRT theory gives a classification of such pairs.
A.F. LOPEZ :
Title:
On the existence of Ulrich
vector bundles on some irregular surfaces
Abstract:
The problem of existence of
Ulrich vector bundles is open already in
the case of
surfaces. In the talk we will outline the existence of
rank
two Ulrich vector bundles on surfaces that are either of maximal
Albanese dimension or with irregularity $1$, under many
embeddings. In
particular this gives the first known
examples of Ulrich vector
bundles on irregular surfaces
of general type.
W. NAHM :
J. CH. OTTEM :
Title: Deformations of hypersurfaces
Abstract: I will discuss an old question due to Kollár and Mori: when is a smooth specialization of
a family of hypersurfaces again a hypersurface? This is joint work with Stefan Schreieder.
G. PACIENZA :
Title: On the cone conjecture for singular irreducible holomorphic symplectic varieties.
Abstract: I will report on a joint work in progress with Ch. Lehn and G. Mongardi in which we explore
the possibility of extending the Kawamata-Morrison cone conjecture to singular irreducible holomorphic symplectic varieties.
The conjecture for smooth IHS varieties was recently established by Amerik-Verbitsky, based upon previous works due to Markman, Markman-Yoshioka,
and Amerik-Verbitsky.
G. PARESCHI :
Title:
Inequalities
for coherent sheaves and divisors on abelian
varieties
Abstract:
I will discuss some old and
new lower bounds for the Euler
characteristic of coherent
sheaves on abelian varieties satisfying
certain vanishing
conditions. An application of them is the proof of a
conjecture
of O. Debarre and Ch. Hacon on singularities of divisors of
low degree on simple
abelian varieties.
J. PARK :
Title: Singularities and syzygies of secant varieties of curves II
Abstract: This talk is a continuation of Lawrence Ein's talk. We consider the k-th
secant variety of a smooth projective complex curve of any genus embedded by the
complete linear system of a very ample line bundle of high degree. I give a sketch of the
proofs of the following results: (1) The k-th secant variety has mild singularities naturally
appearing in birational geometry. (2) The k-th secant variety is arithmetically
Cohen-Macaulay, and satisfies the N_{k+2,p}-property. One of the main technical
ingredients is a vanishing theorem on Cartesian product of curves. This talk is based on
joint work with Lawrence Ein and Wenbo Niu.
J. ROE :
Title : Squeezing the juice out of Newton-Okounokov bodies
Abstract : Since the formalization of Newton-Okounkouv bodies by Kaveh-Khovanski
and Lazarsfeld-Mustata 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 Newton-Okounkov 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 od bodies ? I will look at some well known results on Newton-Okounkov
bodies from this point of view, and report on work in progress with Moyano-Fernàndez,
Nickel and Szemberg for the case surfaces and Newton-Okounkov polygons.
E. ROUSSEAU :
Title :Symmetries of transversely projective foliations
Abstract: Algebraic curves with positive canonical line bundle are known to have
finitely many automorphisms. In a joint work with F. Lo Bianco, J.V. Pereira, and F.
Touzet, we look for foliated versions of this classical fact. Such generalizations are
obtained on projective manifolds for transversely hyperbolic foliations and more generally
for transversely projective ones, whereas some counter-examples are given for non Kähler
manifolds. This study also provides interesting consequences for the distribution of entire
curves on manifolds equipped with such foliations.
T. SZEMBERG
Title: Postulation in projective spaces and unexpected hypersurfaces
Abstract:
Given a suscheme X in projective space, it is very classical problem to determine the
dimension of the vector space of all homogeneous polynomials of some fixed degree d
vanishing along X. Even in the simplest case, when X is supported on finite set of points
in the projective plane , a complete answer is not known and it is subject to open
conjectures due to Nagata (1959) and Segre-Harbourne-Gimiliano-Hirshowitz (1964-78).
Recently Cook II, Harbourne,Miliore and nagel observed that the situation becomes even
more interesting if one replaces the space of all polynomials of certain degree by its
carfully chosen subspaces, I will focus on examples exhibiting these new phenomena.
This is based on joint work with Bauer, Malara and Szpond.
J. SZPOND :
Title: Unexpected curves and Togliatti-type surface
Abstract: I will report on a recent work on a direct link between the theory of
unexpected hypersurfaces and varieties with defective osculating behavior.
For a smooth, complex projective variety X embedded in a projective
space, and a positive integer m, one defines the m-th osculating space
to X at P as the linear subspace determined at a point P by the partial
derivatives of order less then or equal to m of coordinate functions,
with respect to a system of local parameters for X at P, evaluated at P.
If the dimension of the m-th osculating space at a general (hence any)
point P of X is lower than expected (hence certain cohomology group is
not vanishing) then X is said to be hypo-osculating of order m.
Recently Cook II, Harbourne, Migliore and Nagel introduced the notion
of unexpected hypersurfaces. In the simplest case, we say that a finite
set of points Z in the projective plane admits in a general point P an
unexpected curve C of degree d and multiplicity m at P, if the existence
of C does not follow from a naive dimension count (i.e. again a certain
cohomology group is not vanishing).In my talk I will discuss how these two
phenomena are related to each other. This will be illustrated by an example
attached to the B3 root system.