Classes of some hypersurfaces in the Grothendieck ring of varieties

Details Classes of some hypersurfaces in the Grothendieck ring of varieties Classes of some hypersurfaces in the Grothendieck ring of varieties

2011-12-09

arxiv.org/abs/1112.2131v1

Mathematics

Algebraic Geometry

21 pages

Scientific paper

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.

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