Classes of some hypersurfaces in the Grothendieck ring of varieties

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.

21 pages

References in corpus (1)

• Unramified cohomology, $\A^1$-connectedness, and Chevalley-Warning problem in Grothendieck ring

Cited by in corpus (2)

• A counterexample to the geometric Chevalley-Warning conjecture
• Unramified cohomology, $\A^1$-connectedness, and Chevalley-Warning problem in Grothendieck ring