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18/06/2020  17h18h (en commun avec Nice) Online conf. access code required  
Same access code. Send a mail to O. Debarre or F. Han or A. Höring to get it. The notion of ample subscheme can be traced back to the work of Hartshorne and was recently formalized by Ottem. In this talk, I will discuss an extension of the GrothendieckLefschetz theorem to ample subvarieties and some applications to abelian varieties. I will then address a conjecture of Sommese on the extension of fiber structures from an ample subvariety to its ambient variety. Using cohomological methods, I will outline a solution of the conjecture which relies on strengthening the positivity assumption in a suitable arithmetic sense; the same methods can be applied to verify the conjecture in special cases. A different approach based on deformation theory of rational curves leads to a proof of the conjecture for smooth fibrations with rationally connected fibers and a classification theorem for projective bundles and quadric fibrations. The talk is based on joint work with Chung Ching Lau.  
25/06/2020  16h17h (en commun avec Nice) Online conf. access code required  
Same access code. Send a mail to O. Debarre or F. Han or A. Höring to get it. Abstract: Given a smooth projective variety, it is natural to ask (1) How can we determine when it is rational? and (2) If it is not rational, can we measure how far it is from being rational? When the variety is a smooth hypersurface in projective space, these questions have attracted a great deal of attention both classically and recently. An interesting case is when the degree of the hypersurface is at most the dimension of the projective space (the "Fano" range). In joint work with David Stapleton, we show that smooth Fano hypersurfaces of large dimension can have arbitrarily large degrees of irrationality, i.e. they can be arbitrarily far from being rational. 