Motivic Limits for Fano Varieties of k-Planes

Abstract

Abstract We study the probability that an $(n - m)$-dimensional linear subspace in $\mathbb{P}^n$ or a collection of points spanning such a linear subspace is contained in an m-dimensional variety $Y \subset \mathbb{P}^n$. This involves a strategy used by Galkin–Shinder to connect properties of a cubic hypersurface to its Fano variety of lines via cut-and-paste relations in the Grothendieck ring of varieties. Generalizing this idea to varieties of higher codimension and degree, we can measure growth rates of weighted probabilities of k-planes contained in a sequence of varieties with varying initial parameters over a finite field. In the course of doing this, we move an identity motivated by rationality problems involving cubic hypersurfaces to a motivic statistics setting associated with cohomological stability.