On the Classes of Hypersurfaces of Low Degree in the Grothendieck Ring of Varieties

Bilgin E.

INTERNATIONAL MATHEMATICS RESEARCH NOTICES, vol.2014, no.16, pp.4534-4546, 2014 (SCI-Expanded, Scopus) identifier identifier

  • Publication Type: Article / Article
  • Volume: 2014 Issue: 16
  • Publication Date: 2014
  • Doi Number: 10.1093/imrn/rnt089
  • Journal Name: INTERNATIONAL MATHEMATICS RESEARCH NOTICES
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.4534-4546
  • Middle East Technical University Northern Cyprus Campus Affiliated: Yes

Abstract

Let X be a projective hypersurface in P-n(k) 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)not equal empty set if and only if [X]equivalent to 1 modulo L 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.