# Pragmatic 2003

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.