Penn Arts & Sciences Logo

Penn Mathematics Colloquium

Wednesday, October 24, 2018 - 3:30pm

Radu Laza

SUNY Stonybrook

Location

University of Pennsylvania

DRL A4

Tea will be served at 3:00 in 4E17

Cubic hypersurfaces are among the simplest algebraic varieties, yet they have always played an outsized role in algebraic geometry and related fields. For instance, cubic curves are closely related to the rich theory of elliptic curves. Cubic surfaces with their 27 lines are the cornerstone of the Enriques-Castelnuovo theory of algebraic surfaces, and they have deep connections with singularities and exceptional Lie groups (esp. E_6). Finally, a seminal result of Clemens and Griffiths says that cubic threefolds are unirational, but not rational, thus answering in negative the well known Luroth question.

In this talk, I will focus on the equally interesting, but more challenging, four dimensional case. Namely, cubic fourfolds are objects of intense study in connection with two fundamental open problems in algebraic geometry: the rationality question and the classification of compact Hyperkahler manifolds. After reviewing some standard material and some newer results, I will then ask: ``Are there are other cubic-fourfold-like objects?’’ I will give a possible answer to this question, and then discuss the surprising relevance of this answer to the two questions mentioned above.