@article{aca1fe62f2ee4662b6540c8e4c74ec66,
title = "Conics in the Grothendieck ring",
abstract = "The subring of the Grothendieck ring of k-varieties generated by smooth conics is described, giving many zero divisors. The proof uses only elementary projective geometry.",
keywords = "Conics, Grothendieck ring, Severi-Brauer varieties",
author = "J{\'a}nos Koll{\'a}r",
note = "Funding Information: Proof. If P = P(C1,C2) then the embedding C1 × C2 ↪→ P gives a smooth quadric. Conversely, if Q ⊂ P is a smooth quadric which is isomorphic to the product of 2 conics C1,C2 (over the base field), then P = P(C1,C2). Thus we need to find a decomposable quadric in P. As noted in [Artin, 4.5], the existence of a quadric implies that the Grassmannian of lines G(1,P ) is embedded as a quadric in P5. If the base field is finite, then P P3 and we are done. For an infinite base field k, a general k-line in P5 intersects G(1,P ) in 2 points which correspond to skew lines L ∪ L′ ⊂ Pk¯. Thus P contains a conjugate pair of skew lines. The linear system of quadrics in P containing L ∪ L′ has dimension 3, hence there is a smooth k-quadric Q ⊂ P which contains L ∪ L′. The two families of lines on Q are now both defined over k; one family contains both L, L′ the other contains neither. □ I thank A.J. de Jong and M. Larsen for useful comments and suggestions. Partial financial support was provided by the NSF under grant number DMS02-00883.",
year = "2005",
month = dec,
day = "1",
doi = "10.1016/j.aim.2005.01.004",
language = "English (US)",
volume = "198",
pages = "27--35",
journal = "Advances in Mathematics",
issn = "0001-8708",
publisher = "Academic Press Inc.",
number = "1 SPEC. ISS.",
}