Motivic limits for Fano varieties of $k$-planes

TL;DR

This paper investigates the probability of linear subspaces being contained in algebraic varieties, extending motivic and probabilistic methods to higher codimension and degree, linking rationality questions to motivic statistics.

Contribution

It generalizes existing strategies to higher codimension and degree, connecting rationality problems with motivic statistics and cohomological stability.

Findings

Derived growth rates of probabilities for $k$-planes in varieties

Extended the Galkin--Shinder approach to higher codimension

Linked rationality problems to motivic statistics

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.