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Geometry of Jacobians

Aula Beltrami: Wednesday, April 26, 2017

"Geometry of Jacobians"

14.15-15.15: Juan Carlos Naranjo (Universitat de Barcelona):

On the ramified Prym map

Abstract: In this talk we will report on a work in progress in collaboration with Angela Ortega. Our aim is to consider the two remaining open cases on the behaviour of the ramified Prym map. Namely, let

P_{r,g}: R_{r,g} --->A^{\Delta}_{g-1+r/2}

be the map sending an irreducible double covering : D − C of a curve C of genus g with r > 0 ramification points to the abelian variety P(D,C) := Ker : JD −>JC, called the Prym variety of the (ramified) covering. The generic Torelli theorem states that Pr,g is generically injective when the dimension of the space of coverings is less or equal to the dimension of the space of polarized abelian varieties. In a fundamental paper Marcucci and Pirola proved this theorem except for the bielliptic case (solved later by Marcucci and Naranjo) and two isolated cases: P5,2 and P2,6. We will present the prove of the generic Torelli theorem in these two situations. In the first we use the base locus of the linear system attached to the theta divisor in combination with some techniques of vector bundles on curves. Instead, in the second case we relate our map with the ́etale Prym map and we study the fibre along the locus of the intermediate Jacobians of cubic threefolds.

15.30-16.30: Sara Torelli (Pavia):

Massey products and Fujita decomposition

Abstract: The second Fujita decomposition states that on the Hodge bundle F of a fibration f : S--> B between a smooth projective surface S to a smooth projective curve B there is a splitting of vector bundles F = U ⊕ A given by a unitary flat bundle U, with A ample. Moreover, when f is semistable then U is completely determined by the geometric variation of the Hodge structure induced by f. We study the vanishing of a second order cohomological object in suitable subspaces of flat sections of U, to which correspond suitable flat subbundles of U. We prove that such a condition forces the monodromy of these subbundles to be finite and in the semistable case to be completely described in terms of morphisms of curves. In particular, the result provides a semiampleness criteria for F, when the property applies to U. This is a joint work with Gian Pietro Pirola.

17.00-18.00: Paola Porru (Pavia)

Totally geodesic submanifolds in the Jacobian locus

Abstract: In this talk we will focus on the problem of the existence of totally geodesic
submanifolds of Ag contained in the Jacobian locus. After recalling some known results we will prove that the bielliptic locus is not totally geodesic for genus at least four and we will give the construction of two new examples of totally geodesic submanifolds obtained as Galois covers
of elliptic curves.

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