Classes of some hypersurfaces in the Grothendieck ring of varieties

2011/12/09 by Emel Bilgin, Bilgin, Emel
Computer Science · Mathematics · #14E08 #14G05 #14J70 #14N25 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #math.AG #msc:14E08 #msc:14G05 #msc:14J70 #msc:14N25

paper · pdf · doi:10.48550/arxiv.1112.2131

21 pages

arxiv created 2011/12/09 · openalex publication_date 2011/12/09 · arxiv updated 2011/12/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X be a projective hypersurface in P_kⁿ of degree d <= n. In this paper we study the relation between the class [X] in K₀(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 K₀(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.

Citations

• Smoothness, Semistability, and Toroidal Geometry

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