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arXiv · 1112.2131

Classes of some hypersurfaces in the Grothendieck ring of varieties

Emel Bilgin
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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.

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Emel Bilgin. 2011-12-09. Classes of some hypersurfaces in the Grothendieck ring of varieties. https://arxiv.org/abs/1112.2131

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