Seminar on Arithmetic Algebraic Geometry : On the geometry of cubic hypersurfaces in $P^5$
 Date

201423 10:30 
201423 15:30
 Place
 Faculty of Science Building #3 Room 413

Speaker/Organizer
 Olivier Debarre( Ecole normale supérieure,France)

 Talk 1 : On the geometry of cubic hypersurfaces in $P^5$ [10:3012:00]
Let $X$ be a smooth complex cubic hypersurface in $P^5$. The variety $F(X)$ which parametrizes lines contained in $X$ is smooth of dimension 4 and is an example of a hyperk\"ahler (or holomorphic symplectic) manifold. Moreover, as discovered by Beauville and Donagi, there is an isomorphism between the primitive Hodge structures on $H^4(X)$ and $H^2(F(X))$. I will explain these facts and their consequences on the structure of the moduli space for cubic hypersurfaces and their period map.
Talk 2 : On the geometry of quadratic line complexes in $\P^4$ [14:0015:30]
A quadratic line complex is the intersection, in its Pl\"ucker embedding, of the grassmannian of lines in $\P^4$ with a quadric. In general, this is a smooth Fano variety of dimension 5. Again, one can associate with $X$ a hyperk\"ahler fourfold $Y$ (called a double EPW septic) and it is a challenge to try to understand the relationship between the Hodge structures of $X$ and $Y$, their period maps, and their moduli spaces.