Cubic hypersurfaces provide some fundamental examples in algebraic geometry. In this course we will start from cubic surfaces together with the geometry of 27 lines. We introduce Fano scheme of lines and rationality. Then we focus on the Hodge theory of cubic 3-folds and cubic 4-folds, including the following topics: irrationality of cubic 3-folds via intermediate Jacobians, hyperKahler geometry related to cubic 4-folds, global Torelli theorems, integrable systems from cubics. You will see that Hodge theory is a powerful tool in algebraic geometry.

Some basic knowledge of algebraic geometry and Hodge theory.

 D. Huybrechts "The geometry of cubic hypersurfaces"

 

I. V. Dolgachev "Classical Algebraic Geometry: A Modern View"

 

C. H. Clemens and P. A. Griffiths "The Intermediate Jacobian of the Cubic Threefold"

 

A. Beauville "Les singularites du diviseur Θ de la jacobienne intermediaire de l’hypersurface cubique dans P^4"

 

A. Beauville and R. Donagi "La variété des droites d'une hypersurface cubique de dimension 4"

 

C. Voisin "Théorème de Torelli pour les cubiques de P^5”

 

B. Hassett "Special Cubic Fourfolds"

 

R. Laza, “The moduli space of cubic fourfolds”

 

R. Laza, “The moduli space of cubic fourfolds via the period map”

 

E. Looijenga "The period map for cubic fourfolds”