Scientific activities
Igor Dolgachev: An introduction to the theory of vector bundles on
algebraic curves.
Duration of talks 2 weeks. First week 2-hour lectures, the second week one
hour lectures.
Generalities about vector bundles on curves, their moduli spaces, different constructions of vector bundles. More details about the moduli space of rank 2 bundles over a curve of
genus 3 (the Coble quartic) and an introduction to the problem of the
description of the moduli space of rank 3 vector bundles over a curve of
genus 2.
Lecture notes (33 pages) were made available to students.
Alessandro Verra: An introduction to the theory of determinantal
representations of hypersurfaces.
Duration of talks 2 weeks. First week 2-hour lectures, the second week one
hour lectures.
About an article of A. Beauville published in Fulton's volume of the
Mich. Math. J. . The main emphasis was to make the exposition less formal
and more geometric. Many new connections with projective geometry of
scrolls, cubic 4-folds, Prym varieties, the Gauss maps for theta divisors
Problems.
1. (Borislav Gajic, Milena Radnovic) Cremona equivalence of dterminantal
representations of plane nonsingular curves.
A determinantal representation of a plane curve defines an embedding of a
curve to the projective space of solutions of the system of linear
equations defined by the corresponding matrix of linear forms. All
determinantal representations are parametrized by an open subset of the
Jacobian variety of the curve. The problem is to find Cremona
trransformations between the projective spaces of solutions which
transform one embedded curve to another. The special problem is to find a
geometric explanation of some concrete formulas obtained by analysts which
relate different determinantal representations of the same curve.
2. (Elisa Oby, Alessandra Dragotto) Degenerate cubo-cubic Cremona
transformations in $P^3$. A nonsingular curve $X$ of genus 3 and degree 6 emebedded in $P^3$ by the linear system $|K_X+D|$ defines a Cremona transformation given by the
linear system of cubic surfaces through the curve. Its inverse is also
given by cubics through another model of the same curve embedded in $P^3$
by the linear system $|2K_X-D|$. The first embedding can be considered as
given by left null spaces of a determinantal representation of a plane
quartic. The second model corresponds to the right null spaces. The
problem is to find Cremona transformations corresponding to degenerate
curves of arithmetic genus 3 and degree 6. For example, to consider the
case when the curve degenerates to the union of 4 skew lines and their 2
transversals. Another special example to consider is the case when the
curve aquires a double point and one puts an embedded point with support
at thios singular point to get a curve in the same component of the
Hilbert scheme.
3.( Nguen Minh, Slawek Rams) The Coble cubic.
The Coble cubic is a unique cubic hypersurface in $P^8$ whose singular
locus is equal to the Jacobian variety of a curve $X$ of genus 2 embedded
by the linear system $|3\Theta|$, where
$\Theta$ is the theta divisor on the Jacobian. It is known that the moduli
space $SU_X(3)$ of rank 3 vector bundles on $X$ with trivial determinant is
isomorphic to the double cover of the dual projective space $P^8$
identified with $|3\Theta|$. Its branch divisor is a hypersurface of
degree 6. The main problem is verify a conjecture of Dolgachev that the
dual hypersurface of the Coble cubic coincides with the branch sextic. A
more general problem is to find a relationship between the beautiful
classical geometry of the space $|3\Theta|^*$ containing the Jacobian of
$X$ and the modern theory of vector bundles on curves.
4. (Damiano Fulghesu, Alessandra Bernardi) The genertalized Kummer 4-fold.
The generalized Kummer variety of a curve of genus $g$ is defined as the
fibre over 0 of the addition map $Jac(X)^{(n)}\to Jac(X)$ defined on the
$n$th symmetric product of the ordinary Jacobian variety of $X$. When $n
= 2$, the definition coincides with the classical definition of the Kummer
variety of $X$. It is known that these varieties admit a nonsingular
birationale model which admits a structure of a holomorphic symplectic
manifold. These manifolds are subjects of intensive current research in
mathematics.
The problem is to find an explicit description of these varieties in the
case when $X$ is of genus 2 and $n = 3$. It is related to Problem 3 since
the variety in this case is a subvariety of the branch divisor of degree 6. The problem is to find the degree of the 4-fold, describe its
singularities and find the equations defining it in $P^8$.
5. (Rogier Swierstra, Paolo Stellari) Cubic 4-folds and odd theta
characteristics on degenerate plane curves of degree 6.
It is known that the projecton from a plane contained in cubic
hypersurface in $P^5$ defines a quadric bundle over projective plane such
that the set of singular fibres is a plane curve of degree 6. The
converse construction is known in the case when the sextic curve is
nonsingular. The additional data to reconstruct the cubic is an odd theta
charactertistic on the curve. The problem is to carry out this
construction when the sextic curve is singular. In the special case the
problem asks to describe the corresponding cubics when the sextic
degenerates to the union of 6 lines or a curve with 10 nodes.
6. (Remke Kloosterman, Michela Artebani, Marco Pacini) Theta
characteristics on some families of plane quartics.
Let $(Q,t)$ be a pair where $Q$ is a smooth plane quintic and $t$ is a
theta characteristic different from $O_Q(1)$. It is known that that the Prym
variety of the pair $(Q,t(-1)$ is the intermediate Jacobian of a cubic 3-fold if
$t$ is odd and a genus 5 Jacobian if $t$ is even.
The family of plane quartics $C$ having contact intersection $CQ = 2d$,
with $d$ in the linear system defined by $t(1)$ is related to the theta divisor $T$ of
the previous Prym. More precisely the the family of pairs $(C,e)$, where $e$ is an even
theta charcateristic on $C$ should be a birational model of $T/<-1>$. The
problem consists in describing in detail the family of pairs $(C,e)$ ( a family of
spin curves according to recent language) including the singular case and comparing
the model realized by such a family with $T/<-1>$.
7. (Mesut Sahin, Gioia Failla) Higher rank vector bundles and symmetric
determinantal representations of plane curves.
It is known that a symmetric determinantal representation of a nonsingular
plane curve of degree d is defined by a choice of an even theta
characteristic which is an orthogonal vector bundle of rank 1. A
generalization is to find a symmetric determinanat of a matrix with linear
forms as entries equal to a power of the equation defining the curve. This
involves a choice of an orthogonal vector bundle of higher rank on the
curve. There is a beautiful projective geometry interpretation of this
construction in terms of linear sustems of singular quadrics. The problem
is to find such representations for curves of lower degree (say 3 and 4).
8. (Francesco Leon Trujillo) Polar maps and arrangements of hyperplanes in
a projective space.
Given an arrangement of hyperplanes $H_i$ in a projective space $P^n$ one
defines the polynomial $F$ equal to the product of linear forms defining
the hyperplanes.Its partial derivatives define a rational map from $P^n$
to $P^n$. The problem is to classify arrangements when the degree of the
map is low, say 1 or 2.
9. (Alessandro Arsie, Concettina Galati) Geometric $k$-normality for
irreducible nodal curves on surfaces.
Let $S$ in $P^n$ be a smooth, non degenerate surface and let $X$ be an
irreducible, non degenerate curve on $S$, having $d$-nodes as the only
singularities. Let $\phi : C \to X$ be the normalization of $X$. Then, if $H$ denotes the
hyperplane section of $S$, $X$ is said to be {\it geometrically linearly
normal} if $h^0(C, O_C(\phi^*(H)) =n+1$. The characterization of
geometrically linearly normal curves on a given $S$ has been studied by
several authors. The problem is to investigate the $k$-geometric
normality of nodal curves, for $k >1$. This approach has also several
fascinating relations with singularities of the so called
Severi varieties parametrizing irreducible and nodal curves on $S$. The
most interesting case is when $S$ is of general type.