|Date: Wednesday, February 04, 2015
Location: 4096 East Hall (4:00 PM to 6:00 PM)
Title: Cycles on cubic fourfolds
Abstract: Motivated by the rationality problem for cubic fourfolds, there has been a surge of new activity surrounding their Hodge structures, derived categories, and varieties of lines. I will report on recent advances concerning the "universal" Chow group of 0-cycles on a cubic fourfold (work of Voisin and joint work of mine with Colliot-Thélène and Parimala). The Chow group of 0-cycles is an interesting birational invariant of smooth projective varieties. While useless for rationally connected varieties (e.g., cubic hypersurfaces), it can exhibit nontrivial behavior upon extension of scalars. This leads to the notion of the "universal" Chow group of 0-cycles, which is related to decompositions of the diagonal introduced by Bloch and Srinivas, and provides a finer obstruction to rationality. In recent work, Voisin has proved new conditions on the discriminants of Noether--Lefschetz divisors in the moduli space of cubic fourfolds where this obstruction vanishes. However, the vanishing of this invariant for very general cubic fourfolds, or even cubic fourfolds containing a plane, is still open.
Speaker: Asher Auel
Institution: Yale University