Classes of some hypersurfaces in the Grothendieck ring of varieties

TL;DR

This paper explores the relationship between the class of certain projective hypersurfaces in the Grothendieck ring of varieties and the existence of rational points, using geometric methods for specific cases.

Contribution

It establishes conditions linking the class of hypersurfaces in the Grothendieck ring to the presence of rational points for particular hypersurface types.

Findings

X(k) is nonempty iff [X] is equivalent to 1 mod the affine line class for some hypersurfaces.

Identifies specific hypersurfaces where class and rational points are related.

Provides geometric proofs for cases like unions of hyperplanes, quadrics, and certain cubics.

Abstract

Let X be a projective hypersurface in P_k^n of degree d <= n. In this paper we study the relation between the class [X] in K_0(Var_k) and the existence of k-rational points. Using elementary geometric methods we show, for some particular X, that X(k) is nonempty if and only if [X] is equivalent to 1 modulo the class of the affine line in K_0(Var_k). More precisely we consider the following cases: a union of hyperplanes, a quadric, a cubic hypersurface with a singular k-rational point, and a quartic which is a union of two quadrics one of which being smooth.