E. Arrondo (Madrid, Spain)
The Picard group of small codimension subvarieties
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
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
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
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
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
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
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
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
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
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
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
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
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.