Advanced Topics in Algebraic Geometry: Cubic hypersurfaces (V5A3)

    Fall 2017/18, Mo 10:15 c.t. - 12:00, Friday 14:15 c.t. - 16:00 Uhr, Endenicher Allee 60, SR 1.007

    Notes

    Comments very welcome!

    The course deals with some of the many aspects of the geometry of (smooth) cubic hypersurfaces. This includes classical aspects (like the existence of 27 lines on each smooth cubic surface) every algebraic geometer should know but also modern aspects (derived categories, Torelli theorems, moduli spaces, etc.) Cubic hypersurfaces can be studied from various angles and the class will provide he opportunity to introduce and then discuss many techniques in the special case of cubic hypersurfaces.

    Although elliptic curves can be viewed as cubic hypersurfaces, and indeed provide the intuition for many constructions in higher dimensions, they will barely mentioned in this class.

    The course should be accessible for students with a background in algebraic geometry (say at the level of Hartshorne's book) and complex geometry (I'll be freely using the basic facts from Hodge theory). The seminar by Soldatenkov will cover intermediate Jacobian which will also come up here.

    Here is a rough plan:

  1. Invariants of smooth cubic hypersurfaces:

    Numerical invariants (Betti and Hodge numbers, Chern numbers, deformation parameter) Cohomology (Lefschetz hyperplane), Zeta function Discriminant divisor Monodromy of the family of smooth cubics

  2. Fano varieties of lines:

    Existence Deformation theory

  3. Cubic surfaces:

    27 lines Alternative description (blow up of P^2, 2:1 covers of P^2, Sylvester, determinantal) Moduli spaces of cubics surfaces

  4. Cubic threefolds:

    Intermediate Jacobian Fano surface of lines Smooth cubic threefolds are never rational

  5. Cubic fourfolds:

    Moduli space Hodge structure Relation to K3 surface