Conics in the Grothendieck ring
نویسنده
چکیده
The Grothendieck ring of k-varieties is still very poorly understood. In characteristic zero, the quotient of K0[Vark] by the ideal generated by [A ] is naturally isomorphic to the ring Z[SBk], where Z[SBk] is the free abelian group generated by the stable birational equivalence classes of smooth, projective, irreducible k-varieties and multiplication is given by the product of varieties [Lar-Lun]. (The cited paper proves this over algebraically closed fields only, but the proof works over any field of characteristic zero using the birational factorization theorem as given in [AKMW, Remark 2 after Theorem 0.3.1]. Note also that the product of two irreducible k-varieties is not necessarily irreducible, so Z[SBk] is not a monoid ring if k is not algebraically closed.) Zero divisors in the Grothendieck ring of C-varieties were found by [Poonen]. Here we give further examples of nontrivial behaviour of these rings by studying products of conics. This gives interesting examples only when the field k is not algebraically closed.
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