Alessandro Verra

The unirationality of the moduli spaces of curves of genus 14 or lower

We prove the unirationality of the moduli space of complex curves of genus 14. The method essentially relies on linkage of curves. In particular it is shown that a general curve of genus 14 admits a projective model D , in a six-dimensional projective space, which is linked to a general curve C of degree 14 and genus 8 by a complete intersection of quadrics. Using this property we are able to obtain, after a reasonable amount of further work, the unirationality result in the case of genus 14. Moreover, some variations of the same method, involving the Hilbert schemes of curves of very low genus, are used to obtain the same result for the known cases of genus 11, 12, 13.

The classification of universal Jacobians over the moduli space of curves

We carry out a complete birational classification of the degree g universal Jacobian P_g over the moduli space of curves, highlighting the transition cases g=10, 11. The universal Jacobian is unirational when g 11, the variety P_g has Kodaira dimension 3g-3, that is, the maximum allowed by Iitakas easy addition formula for fibre spaces. In particular, we disprove the expectation that P_g and M_g have the same Kodaira dimension for all genera.

Reye constructions for nodal Enriques surfaces

A classical Reye congruence X is an Enriques surface of rational equivalence class (3, 7) in the grassmannian G(1, 3) of lines of P 3 . X is the locus of lines of P 3 which are included in two quadrics of W = web of quadrics. A generaIization to G(1, t) is given (1) for each t > 2 there exist Enriques surfaces X of class (t, 3t − 2) in G(1, t), (2) the determinant of the dual of the universal bundle on X is O X (2E+R+K X ), with E = isolated elliptic curve, R 2 = -2, E.R = t, (3) X parameterizes lines of P t which are included in a codimension 2 subsystem of W, W = linear system of quadrics of dimension ( 2 t ). The paper includes a description of the variety of trisecant lines to a smooth Enriques surface of degree 10 P 5

The theta divisor of

Let

SU_C(2,2d)^s

X

is very ample if

be the moduli space of semistable rank 2 vector bundles over a smooth curve C of genus

C

g \ge 2

is not hyperelliptic

and

On the Theta divisor of SU(2,1)

\theta : X \to PH^0(L)^*

Quaternionic pryms and Hodge classes

be the map associated to the generalized theta divisor L on X. We prove that for C not hyperelliptic, the map