E. Arrondo (Madrid, Spain)
The Picard group of small codimension
subvarieties
Abstract
Barth-Larsen theorem implies that
any smooth subvariety X of dimension r in P^N
has its Picard group isomorphic
to the Picard group of P^N when N<2r-1.
In this talk we will present a
method (introduced in the PhD thesis of Jorge Caravantes)
to decide when the same result
is true if we replace P^N with another ambient space Y of dimension N.
This method produces positive answers
when Y is a Grassmannian of (projective) lines
or a product of a projective space
by itself
(improving previous results by
Barth and van de Ven and Sommese).
We will discuss the cases in which
our method does not give such a positive answer,
namely when Y is a Grassmannian
of linear spaces of dimension at least two
(studied by Nicolas Perrin) or
a smooth quadric.
M. Beltrametti (Genova, Italy)
Hilbert curves of polarized
varieties
Abstract
Let X be a normal Gorenstein complex
projective variety.
We introduce the Hilbert variety
VX associated to the Hilbert polynomial
p(x1L1
+ ... + xrLr),
where L1,..,Lr
is
a basis of Pic(X), r being the Picard number of X and x1,...,xr
are complex variables.
After studying general properties
of VX we specialize to the Hilbert curve
of a polarized variety (X,L),
namely the plane curve of degree
dim(X) associated to p(xKX + yL).
Special emphasis is given to the
case of polarized 3-folds.
G. M. Besana (Chicago, U.S.A.)
Many pictures are worth many
thousand words: applications of algebraic geometry to vision problems
Abstract
A classical problem in computer
vision is reconstruction.
Given a series of images (projections)
of a three-dimensional scene,
with a number of corresponding
image points identified on the images,
one tries to reconstruct (up to
projective equivalence)
the position of the chosen group
of points and of the cameras (centers of projections).
When the chosen points are not
static but are allowed to move,
one can, under certain assumptions,
model the scenes as projections
from higher dimensional projective spaces.
A configuration of points is said
to be critical for a set of cameras
if there exists a non-projectively
equivalent configuration of cameras and points
giving rise to projectively equivalent
images, and thus preventing reconstruction.
Loci of critical configurations
can be naturally described
by a class of reducible determinantal
varieties.
This talk will describe the general
set-up for the problem,
introduce the class of determinantal
varieties involved,
give a few general properties of
this class,
give a description of a particular
case in P4,
and pose a number of questions
worthy of further investigation.
S. Brivio (Pavia, Italy)
On the generic finiteness of
the theta map
Abstract
Let SU(r) be the moduli space of
semistable vector bundles of
rank r and trivial determinant
on a smooth irreducible complex projective
curve of genus g >1. The
“theta map” of SU(r) is the rational map
defined by the positive generator
of the Picard group of SU(r). In this
talk we study the generic
finiteness of this map for r >2.
L. Fania (L'Aquila, Italy)
Skew-symmetric matrices and
Palatini scrolls
Abstract
Let X be the degeneracy locus of
a general morphism between vector bundles
whose structure is that of a Palatini
scroll in P(V).
In this talk I will report on a
joint work with Daniele Faenzi.
We prove that, for m > 3
and k > m-2, the Grassmannian G,
parametrizing m-spaces of
skew-symmetric forms over a vector space V of dimension 2k,
is birational to the Hilbert scheme
Hm(V)
of Palatini scrolls in P(V).
For m = 3 and k
> 3, the variety G is birational to the subscheme of Hm(V)
consisting of pairs (E,Y),
where Y is a smooth plane curve of degree k
and E is a stable rank-2
bundle on Y with det(E) = OY(k-1).
T. de Fernex (Salt Lake City, U.S.A.)
On manifolds containing
special ample subvarieties
Abstract
It is a general fact that the special
geometry of an ample divisor, or
more generally of a regular section
of an ample vector bundle, should
reflect deeply on the geometry
of the ambient variety. I will overview
part of the history of this problem,
and discuss some recent results
in this area. This talk is based
upon joint work with Mauro
Beltrametti and Antonio Lanteri.
S. Di Rocco (Stockholm, Sweden)
Toric fibrations and Cayley
polytopes
Abstract
A fibration between toric varieties,
embedded in projective
space, can be described by certain
fibered polytopes. When the fibration
has a projective space as generic
fiber, embedded linearly, the polytope
is called a strict Cayley polytope.
It turns out that this class of
polytopes encodes exceptional geometrical
properties of the corresponding
toric embeddings. Batyrev
and Nill have recently conjectured a relation
between the degree of a convex
polytope and the property of having a
Cayley-structure. A proof of the
conjecture for smooth polytopes will be
presented. The result is obtained
by translating the problem into toric
geometry. This is a joint work
with A. Dickenstein and R. Piene.
Roberto Muñoz (Madrid, Spain)
An extension of Fujita's non
extendability theorem for Grassmannians
Abstract
Quoting De Fernex's abstract:
"the special geometry of an ample
divisor, or more generally of a regular section
of an ample vector bundle, should
reflect deeply on the geometry of the ambient variety".
As an example of this fact Grassmannians
cannot appear as an ample divisor
in any smooth variety (Fujita 1979)
except for the obvious cases.
In this talk we present an extension
of this result showing that Grassmannians of lines
cannot appear as the zero locus
of a rank two ample vector bundle E on a smooth variety.
In fact we prove that, under some
mild positivity assumption on E,
the Grassmannian of lines G(1,n+1)
is the only variety
containing G(1,n) as the zero locus
of a section of E.
In the course of the proof we classify
rank two uniform vector bundles on Grassmannians.
G. Occhetta (Trento, Italy)
Rational curves and bounds on
the Picard number of Fano manifolds
Abstract
In my talk I will discuss recent
results concerning a conjecture of Mukai
which relates the dimension, the
Picard number and the (pseudo)index of a Fano manifold
(joint work with Carla Novelli).
F. Russo (Catania, Italy)
On dual defective manifolds
Abstract
Around 1985 Ein discovered notable
properties of manifolds whose dual variety is small
and proved some spectacular classification
results.
In our talk, based on a joint work
with Paltin Ionescu,
we would like to illustrate significant
improvements of those results,
inspired by various reformulations
and interpretations of Hartshorne Conjecture
on Complete Intersections for manifolds
uniruled by lines.
A. J. Sommese (Notre Dame, U.S.A.)
Zebra Fish, Cancer, and Algebraic
Geometry
Abstract
Problems of central importance
in engineering and science often
lead to systems of partial differential
equations, for which
the only hope of solution is to
compute numerical solutions.
Often the systems are intrinsically
nonlinear with several
solutions corresponding to the
same set of physical conditions.
Discretizations of such systems
of differential equations often
lead to large systems of polynomial
equations whose solutions
correspond to potential solutions
of the system of differential
equations. These naturally arising
polynomial systems are well
beyond the pale of systems previously
investigated in numerical
algebraic geometry.
This talk will describe some of
the recent work of Wenrui Hao,
Jonathan Hauenstein, Bei Hu, Yuan
Liu, Yong-Tao Zhang, and
myself in successfully solving
such systems.
First I will discuss our work on
finding steady state solutions
of a reaction-diffusion model on
zebrafish dorsal-ventral
patterning. Here, using a
new approach we found
seven solutions with the boundary
conditions, of which
three are stable: only stable solutions
have meaning in the
physical world.
Second I will discuss our successful
work on the solution of a
model for tumors. Here the boundary
of the tumor is the most
important part of the solution:
the goal is to solve this free
boundary problem as mu, a parameter
of the model called the
"tumor aggressiveness factor,"
varies. There is a family of
easily computed radially symmetric
solutions, which, for
certain discrete values of mu,
meets branches of solutions that
are not radially symmetric away
the point where the branches
meet. The problem is to compute
the solutions on the
nonradial branch. There are no
standard ways of solving this
sort of free boundary problem for
any but very small distances
from the radial solutions. Our
successful solution of this
problem required a new approach
that came to grips with some of
the numerical algebraic geometry
underlying systems of several
thousand polynomials in a like
number of variables.
Finally, I will also discuss the direction this research is going.