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

Bilgin E.

INTERNATIONAL MATHEMATICS RESEARCH NOTICES, cilt.2014, sa.16, ss.4534-4546, 2014 (SCI-Expanded, Scopus) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 2014 Sayı: 16
  • Basım Tarihi: 2014
  • Doi Numarası: 10.1093/imrn/rnt089
  • Dergi Adı: INTERNATIONAL MATHEMATICS RESEARCH NOTICES
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Sayfa Sayıları: ss.4534-4546
  • Orta Doğu Teknik Üniversitesi Kuzey Kıbrıs Kampüsü Adresli: Evet

Özet

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.