AGHD 2007  Abstracts
V. Alexeev  Torelli map and infinite grassmannians
I will give an explanation for the curious fact that the Torelli map
M_g > A_g near the boundary looks so much like the Plucker embedding
of an infinite grassmannian G(2,infty).
L. Bădescu  A BarthLefschetz theorem for submanifolds
of a product of projective spaces.
Let X be a complex submanifold of dimension d of P^{m} ×P^{n} (m > n > 2) and denote
by α: Pic(P^{m}×P^{n})> Pic(X) the restriction map of Picard groups.
Set t := max{dim π_{1}(X), dim π_{2}(X)},
where π_{1} and π_{2} are the two projections
of P^{m} ×P^{n}. We prove a BarthLefschetz type result as follows:
Theorem. If d ≥ (m+n+t+1)/2
then X is algebraically simply connected, the map α is injective and
Coker(α) is torsionfree. Moreover α
is an isomorphism if d ≥ (m+n+t+2)/2, or if d = (m+n+t+1)/2 and the normal bundle of X
in P^{m} × P^{n} is decomposable.
Explicit examples show that these bounds are optimal. The main technical ingredients in the proof are: the
KodairaLe Potier vanishing theorem in the generalized form of Sommese, the join construction and
an algebraisation result of Faltings concerning small codimensional subvarieties in P^{N}. This is a
joint work with Flavia Repetto.
I. Bauer  Action of the absolute Galois group on moduli spaces of surfaces.
(joint work with F. Catanese, F. Grunewald)
Hilbert schemes are defined over the integers, hence it follows
that the moduli spaces of surfaces of general type
are also defined over the integers, and there is an action of the
absolute Galois group Gal: = Gal (Q / Q)
on the sets of its irreducible components , respectively of its
connected components.
Considering varieties isogenous to a product (quotients of a product of
curves by the free action of a finite group G) we show that the absolute
Galois group acts faithfully on the
set of connected components. In fact,
given an element s in Gal different from the identity and from
complex conjugation, there is a surface of general type
X such that X and the conjugate variety X^{s} have non
isomorphic fundamental groups. These results
vastly generalize a phenomenon discovered by J.P. Serre in the 60's ,
that there exist Galois conjugate varieties which have
non isomorphic fundamental groups, in particular are not homeomorphic.
F. Campana  Geometry of orbifolds and classification theory.
Geometric orbifolds are pairs (X/D) with X complex projective (say)
and D a ramification divisor. They interpolate between projective manifolds (D empty) and logarithmic manifolds (D reduced integral).
They possess the usual
geometric invariants of manifolds; in particular morphisms and birational maps can
be naturally defined for them. Fibrations in this birational category enjoy additivity
properties not satisfied by the birational category of manifolds without orbifold
structure, which permit to express some invariants of the total space in terms of those
of the generic fibre and of the base, and so to inductively lift properties in towers of
fibrations. For this reason in particular, this category appears to be the natural frame
of classification theory of complex projective manifolds (without orbifold structure).
We shall extend some properties of manifolds to this larger class, and raise some
questions about rational orbifold curves and orbifold universal covers.
L. Caporaso  Degeneration of BrillNoether varieties.
BrillNoether varieties of smooth curves
are well known subvarieties of the Picard scheme
parametrizing line bundles with certain properties.
When smooth curves degenerate to singular ones
Brill Noether varieties should degenerate to geometrically meaningful
objects.
An approach to this problem, using Neron models,
will be described together with some recent results.
F. Catanese  On some classes of real and complex 3folds covered by rational curves.
L. Chiantini  On the interpolation problem on varieties.
We discuss the linear interpolation problem
for multiple points and linear systems, on general
projective varieties. Mainly we focus on the case
of multiplicity 2 and 3.
B. Fantechi  Moduli spaces with symmetric obstruction theories.
An obstruction theory is called symmetric if it admits a
symmetric isomorphism with its dual. Moduli of sheaves on a Calabi Yau
threefold and the intersection of two Lagrangians have a symmetric
obstruction theory: as a result, they have expected dimension zero,
and their virtual degree can be calculated by integrating a
constructible function  in particular, it is defined even in the
nonproper case. We discuss some properties of these spaces and their
dg structures.
S. Kebekus  On the structure of surfaces mapping to the moduli stack of
canonically polarized varieties.
This is joint work with Sándor Kovács.
Shafarevich's wellknown hyperbolicity conjecture asserts that a
family of curves over a quasiprojective 1dimensional base is isotrivial
unless the logarithmic Kodaira dimension of the base is positive. More
generally, it has been conjectured by Viehweg that the base of a smooth
family of canonically polarized varieties is of log general type if the
family is of maximal variation. Using extension properties of logarithmic
pluriforms, we relate the minimal model program of the base to the moduli
map, and give bounds for the variation of a family in terms of the
logarithmic Kodaira dimension of the base. This gives an affirmative answer
to Viehweg's conjecture for families parametrized by surfaces.
J. Kollár  Moduli spaces for varieties of general type: What remains to be done?
S. Kovács  Characterizations of projective spaces and hyperquadrics.
This is a report on joint work with Carolina Araujo and Stéphane Druel.
We confirm the following conjecture of Beauville: Let L be an ample
line bundle on the smooth projective variety X and suppose that the
pth exterior power of the vector bundle T_X contains the pth power
of L. Then either X is a projective space and L=O(1) or p=dim X and X
is a quadric hypersurface and (again) L=O(1).
A. Lopez  A (new) EnriquesFano threefold with non Qsmoothable canonical
singularities.
An EnriquesFano threefold X is a projective threefold having
as hyperplane section a smooth Enriques surface. If we assume that X
has cyclic quotient terminal singularities, there is a classification
of such threefolds due to Bayle and Sano.
We proved that any EnriquesFano threefold has genus at most 17 (proved
also indipendently by Prokhorov).
In the talk we will construct a new EnriquesFano threefold X in P^{9 }
with two properties:
1) it does not belong to BayleSano's list nor is limit of them;
2) its normalization is a QFano threefold with canonical singularities
that does not admit a Qsmoothing (that is a flat limit of threefolds
with cyclic quotient terminal singularities).
By some results in deformation theory it is known that (Namikawa) Fano
threefolds with Gorenstein terminal singularities are smoothable, while
(MinagawaSano) QFano threefolds of index at least 1 with terminal
singularities are Qsmoothable. Therefore X shows that the latter
cannot be extended to the case of canonical singularities.
This is joint work with Roberto Munoz and Andreas Knutsen.
J. McKernan  The Sarkisov Program
Conjectural the output of the minimal model program is
either a minimal model or a Mori fibre space. However in neither case
is the output unique. Recently Kawamata has show that any two minimal
models are connected by a sequence of flops.
The Sarkisov program aims to factorise any birational map between two
Mori fibre spaces as a sequence of elementary links. In the case of
surfaces these links are elementary transformations and the Sarkisov
program provides a nice framework to prove that the birational
automorphism of P^{2} is generated by a Cremona transformation and
PGL(3).
In this talk I will describe recent work with Christopher Hacon where
we extend Sarkisov's program to all dimensions.
M. Mella  Base loci of linear system and the Waring problem.
The Waring problem for forms is the quest for an additive decomposition
of homogeneus polynomials into powers of linear ones. The subject has
been widely considered in old times, (Sylvester, Hilbert, Richmond and
Palatini)
with special regards to the existence of a unique decomposition of
this type. This would give a "canonical" decomposition.
I do expect that the one described at the beginning of the XXth century are
the only possible cases in which the decomposition is unique.
The aim of this talk is to give evidence to this expectation.
Y. Miyaoka  Explicit bound of the canonical degree of a curve on a surface of
general type with K^{2} > c_{2}.
Let X be a minimal projective surface of general type defined over the
complex numbers and let C in X be an irreducible curve of geometric
genus g. Assume that K^{2}_{X} is greater than the topological Euler
number c_{2}(X). Then we prove that the “canonical degree” CK_{X} of
C is uniformly bounded in terms of the given invariants g,K_{X}
X and
c_{2}(X), thus giving an effective version of a theorem of Bogomolov on
the boundedness of the curves of fixed genus in X.
S. Mukai  Hilbert's 14th problem for the Kronecker quiver.
I was led to the following question by a systematic study of Nagata's
counterexamples to Hilbert's 14th problem:
Is the ring of invariant polynomials finitely generated for a linear
action of the 2dimensional additive group?
I will explain the background and discuss the case where the action
comes from the quiver o=>o consisting of two vertices and two arrows of
the same direction.
G. Ottaviani  Higher secant varieties to Segre and Grassmann varieties.
The dimension of the higher secant varieties to the Veronese varieties
are the expected ones with a short list of exceptions, thanks to the
AlexanderHirschowitz theorem. Their equations are still unknown. We
study the same kind of problems for Segre and Grassmann varieties. It
corresponds to find the decomposition of a general (resp.
skewsymmetric) tensor. This is joint work with C. Peterson and H. Abo.
T. Peternell  Coverings of projective and Kaehler manifolds:
selfmaps, deformations and classification.
In my talk I want to discuss the following three topics:
1. A selfmap of a projective manifold X of a surjective covering
f: X > X . Assume that f has degree at least 2. What can be said
on the structure of X and f? Particularly interesting is the case
when X is a Fano manifold with Picard number 1.
2. The deformation theory of coverings of compact Kaehler manifolds
in connection with "generic semipositivity".
3. Given a special variety X and a covering f: X > Y. What can be
said on Y? Here I want in particular discuss the case when X is a
torus.
G.P. Pirola  On the Iitaka Severi set of a projective surface.
We study dominant rational maps from a general surface in of degree d of the projiective space
to surfaces of general type. We prove special properties of these rational maps.
We show that for small degree the general surface has only the identity map.
M. Reid  Ice cream and Dedekind sums.
The contributions of isolated orbifold points to RiemannRoch can be
organised as Hilbert series called ice cream functions. The eponymous
case is
sum [3i/7]*t^i = (t^3 + t^5 + t^7)/(1t)*(1t^7),
asserting that if your income is 3/7, you get ice cream on Wednesdays,
Fridays and Sundays.
Minor modifications of this idea contain the theory
of Dedekind sums and the orbifold contributions to RiemannRoch as in
Chapter III of my [YPG].
For a preview, see the MSRI video
F. Russo  Conicconnected manifolds and rationality via special families of rational curves.
During the talk I would like to discuss the following topics:
1) Classification of conicconnected manifolds, i.e. projective manifolds X such that two general points
may be joined by an irreducible conic contained in X;
2) A criterion of rationality asserting that
the local ring of a smooth point x in X is isomorphic to the local ring of a point in P^{n} if and only
if X admits a covering family of rational 1cycles, all passing through x,
all smooth at x and such that the general cycle of the family is uniquely
determined by its tangent line at x. The last hypothesis
can be omitted when the locus of reducible 1cycles has codimension at least 2
and ``more global" versions hold in the smooth case.
3) Rationality, strong constraints for the existence and finer classification of conicconnected
projective manifolds with b_{2}=1 and for which the locus
of conics through two general points has maximal dimension equal to the secant
defect of X.
This is joint work with Paltin Ionescu.
E. Sernesi  Moduli of rational fibrations.
We obtain some relations between the numerical invariants of a surface fibered over
the projective line with general curves as fibres. We apply such relations to
give upper bounds on the genus of a general curve varying in a nontrivial linear system on a nonruled surface.
V. Shokurov  Complements on surfaces.
For algebraic surfaces, bounded complements exist for any
boundary multiplicities, that is, in the whole segment {0,1]. Moreover, we can find
complimentary indecies rather divisible.
It is expected a higher dimensional generalization of this result and methods in future.
A. Sommese  Recent Progress in Numerical Algebraic Geometry.
Following a short overview of Numerical Algebraic
Geometry, some recent ongoing work will be presented:
a) Some of the numerical issues dealt in the design of software for
Numerical Algebraic Geometry and some details about Bertini, the
software package recently released by D. Bates, J. Hauenstein, C.
Wampler, and myself.
b) work with C. Wampler on numerical computation of exceptional
sets of algebraic maps (with regards to fiber dimension) by means
of iterated fiber products.
c) work with D. Bates, C. Peterson, and C. Wampler on numerical
computation of the geometric genus of curve components of algebraic
sets.
J. Starr  Rational simple connectedness.
This is joint work with A. J. de Jong. Rational simple connectedness is
to simple connectedness as rational connectedness is to path
connectedness. As time permits, I hope to discuss two results involving
this notion.
(1) Hassett's result that every rationally simply connected fibration
over a curve satisfies "weak approximation", i.e., power series sections
are approximated to arbitrary order by polynomial sections.
(2) A theorem that a rationally simply connected fibration over a surface
has a rational section, assuming the vanishing of a Brauer obstruction
and some other hypotheses (which I will state precisely). In particular,
joint work with A. J. de Jong and Xuhua He proves that if a general fiber
is a minimal, projective homogeneous variety G/P, the Brauer obstruction
is the only obstruction. As pointed out by Philippe Gille, this implies
the last unknown case (i.e., the E8 case) of Serre's Conjecture II in
Galois cohomology for the function field of a surface.
B. van Geemen  Real multiplication on K3 surfaces and Kuga Satake varieties.
The endomorphism algebra of a K3 type Hodge structure is a totally real field
or a CM field. We give a low brow introduction to the case of a totally real
field and show existence results for the Hodge structures, for their
polarizations and for certain K3 surfaces. We discuss some examples of Kuga
Satake varieties of these Hodge structures. Finally we indicate various open
problems related to the Hodge conjecture.
A. Verra  On the Prym moduli spaces in low genus.
A survey on the Kodaira dimension of Prym moduli spaces is
given and new results of unirationality are proved for Prym moduli
spaces of curves of low genus and for some related moduli spaces.
G. Vezzosi  Derived algebraic geometry.
I will describe briefly how to do some geometry on commutative
differential graded algebras, on simplicial algebras and on commutative
ring spectra. Here the geometry has to take the intrinsic homotopical
flavor of such objects into account. Then I will describe some
applications of this 'homotopical geometry' to problems raised outside
its framework such as the problem of constructing a universal elliptic
cohomology theory and the socalled geometric Langlands correspondence.
At the end I will sketchily discuss some open problems and current
directions in this field.
J. Wiśniewski  Towards a moduli of phylogenetic trees.
I will explain recent results concerning algebraic
models of binary symmetric phylogenetic trees and make some questions
regarding their parameter space.
