Cubic Threefolds and Vanishing Cycles on its Hyperplane sections

Abstract: For a general cubic threefold, a vanishing cycle on a smooth hyperplane section is an integral 2-class perpendicular to the hyperplane class with self-intersection equal to -2. The question is what is a vanishing cycle on a singular hyperplane section? We will show that there is a certain moduli space parameterizing "vanishing cycles" on all hyperplane sections and the boundary divisor answers the question. As a vanishing cycle on a smooth cubic surface is represented by the difference of two skew lines, such moduli space arises as a quotient of the Hilbert scheme of skew lines on the cubic threefold. Based on the Abel-Jacobi map on cubic threefolds studied by Clemens and Griffiths, we'll show that the moduli space is isomorphic to the blowup of the theta divisor of the at an isolated singularity.

algebraic geometry

Audience: researchers in the discipline

American Graduate Student Algebraic Geometry Seminar

Series comments: The American Graduate Student Algebraic Geometry Seminar (AGSAGS) is a virtual seminar by and for algebraic geometry graduate students.

The goal of this seminar is for graduate students to share their research through online talks and to provide an algebraic geometry graduate networking system. Grad students, postdocs, and professors are welcome to attend.

Seminars will be held on Mondays at 4 p.m. Eastern on Zoom. We hope this time is convenient for graduate students in the Americas, hence the name AGSAGS. Prior registration is required and interested participants should register here: sites.google.com/view/agsags/registration. In addition to graduate talks, there will be occasional social events.