Hyperplane sections and Moduli

Abstract: One way to produce new varieties from a fixed subvariety of projective space is to intersect with linear subspaces. When we consider a cubic threefold $X$ in $\mathbb{P}^4$, we can consider hyperplane sections: to every hyperplane (considered as a point in the dual projective space) we can associate a cubic surface namely the intersection $X \cap H$. One natural question is to ask, given a cubic surface $Y$, how many times does it appear as a hyperplane section of $X$ (up to projective equivalence)? More rigorously, we can define a rational map which takes a hyperplane $H$ to the class of the intersection, considered as a point in the moduli space of cubic surfaces (GIT). One can check that this is a generically finite surjective map, and thus answering our question is equivalent to calculating the degree of this map. Although the question is enumerative, the techniques involved are particularly interesting: the wonderful blow-up technique of De Concini-Procesi, plus the dual perspective of the moduli of cubic surfaces. This is a work in progress, and we will actually consider a slight modification resulting in easier computations.

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.