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

2014-2-3 10:30 - 2014-2-3 15:30
Faculty of Science Building #3 Room 413
Olivier Debarre( Ecole normale supérieure,France)
Talk 1 : On the geometry of cubic hypersurfaces in $P^5$  [10:30-12:00]

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 $\P^4$  [14:00-15:30]

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.