1/9, IMPA. Auditorio 1
10:3011:30 
Hamid Hassanzadeh (UFRJ),
Effective criteria for graded and bigraded birational maps
Resumo. In this talk, we consider rational maps whose source is a product of two subvarieties, each one being embedded in a projective space. Our main objective is to investigate birationality criteria for such maps. First, a general criterion is given in terms of the rank of a couple of matrices that became to be known as Jacobian dual matrices. This criterion generalizes a previous criterion for projective varieties. By mention the applications and motivation, then, we focus on rational maps from
P1 x P1 to P2 in very low bidegrees and provide new matrixbased birationality criteria by analyzing the syzygies of the defining equations of the map.
This talk is based on a Joint work with L. Buse, N. Botbol, M.Chardin, T.Hoa and A. Simis
 12:0013:00 
Ugo Bruzzo (SISSA, Trieste),
The NoetherLefschetz problem for normal threefolds.
Resumo. The classical NoetherLefschetz problem is about the Picard number of surfaces in P3.
For d greater or equal 4, if S(d) is the locus of smooth surfaces in the linear system O(d) on P3, the generic surface in S(d) has Picard number one. It is possible to estimate the codimensions of the components of S(d), and show that their codimension is related to geometric properties of the surfaces they describe.
Over the last few years, in collaboration with A. Grassi, and more recently with A.F. Lopez, I have investigated generalizations to surfaces in normal, Qfactorial 3folds. My talk will be devoted to describe some of the results we obtained.

