Program - December 20, 2018 (Aula C, second floor)
You can download the poster and the program.
9:30-10:20
Luca Cesarano
On the canonical map of smooth, ample divisors in abelian varieties
Abstract:
In this talk, we present some results and open questions concerning the behavior of the canonical map of a general smooth ample divisor in the polarization of a general non-principally polarized complex abelian variety.
The case of abelian threefold is the first geometrically meaningful case, in which new examples of canonically embedded irregular surfaces in low dimensional
complex projective spaces arise.
10:30-11:20
Pawel Boròwka
Hyperelliptic curves on (1,4) polarised abelian surfaces
Abstract:
The talk will start by recalling the fact that a smooth hyperelliptic curve can be embedded into an abelian surface if the genus of the curve is at most 5.
Using similar methods to these used to define (1,3) theta divisors, we will prove that there exists a unique hyperelliptic curve (up to translation) on a general (1,4) polarised abelian surface.
Then aim of the talk will be to study the geometry of such curves. Contrary to (1,3) case, such curves are invariant under the subgroup of translations isomorphic to the Klein group.
The main result is the fact that being a Klein covering of a genus 2 curve non-isotropic with respect to the Weil pairing is not only necessary but also sufficient condition.
The construction is explicit, i.e the Jacobian of a genus 2 curve is a principally polarised surface that is given as a quotient of a (1,4) polarised surface and the quotient map restricted to a copy of genus 5 curve is the Klein covering. The talk is based on a joint work with Angela Ortega.
12:00-12:50
Mirko Mauri
The essential skeletons of pairs and the geometric P=W conjecture
Abstract:
The geometric P=W conjecture is a conjectural description of the asymptotic behavior of the celebrated non-abelian Hodge correspondence. In particular, it is expected that the dual boundary complex of the compactification of character varieties has the homotopy type of a sphere. In a joint work with Enrica Mazzon and Matthew Stevenson, we manage to compute the first non-trivial examples of dual complexes in the compact case. This requires to develop a new theory of essential skeletons over a trivially-valued field. As a byproduct, inspired by these constructions, we show that certain character varieties appear in degenerations of compact hyper-K\"ahler manifolds. In this talk we will explain how these new non-archimedean techniques can shed new light into classical algebraic geometry problems.