@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.",

}