Chow Classes of Varieties of Secant and Tangent Lines

Abstract: (special Student Algebraic Geometry Seminar; note unusual time and location)

Given a nondegenerate smooth variety $X\subset\mathbb{P}^n$, let $\mathcal{S}(X)$ (resp. $\mathcal{T}(X)$) be the subvariety of the Grassmannian $\mathbb{G}(1, n)=\mathrm{Gr}(2, n+1)$ of lines in $\mathbb{P}^n$ consisting of secant (resp. tangent) lines to X. I will give closed-form formulae for the classes of $\mathcal{S}(X)$ and $\mathcal{T}(X)$ in the Chow ring of $\mathbb{G}(1, n)$ in terms of the “higher degrees” of the embedding, by a simple application of the Excess Intersection Formula on a flag variety. Using these formulae, one can recover classical results about the degree of the subvariety $\mathrm{Sec}(X)$ (resp. $\mathrm{Tan}(X)$) of $\mathbb{P}^n$ swept out by the lines in $\mathcal{S}(X)$ (resp. $\mathcal{T}(X)$), when it has the expected dimension. Finally, I will suggest potential extensions of these techniques to varieties of trisecant or bitangent lines.

algebraic geometry

Audience: researchers in the topic

Stanford algebraic geometry seminar

Series comments: The seminar was online for a significant period of time, but for now is solely in person. More seminar information (including slides and videos, when available): agstanford.com