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Conics in the Grothendieck ring. (English) Zbl 1082.14022
Let \(K_0(\text{Var}_k)\) denote the Grothendieck ring of algebraic \(k\)-schemes, with addition and multiplication given by disjoint sums and products, respectively. This paper computes the subring generated by the smooth conics \(C\subset\mathbb P^2\), which are the 1-dimensional Severi–Brauer varieties. There is a technical assumption on the ground field \(k\), but number fields, functions fields of complex surfaces, and more generally \(C_2\)-fields are allowed.
To describe the result, let \(G\) be a finite subgroup inside the 2-torsion of the Brauer group \(\text{Br}(k)\). Choose a basis \(C_1,\ldots,C_n\in G\) consisting of smooth conics, where \(G\) is regarded as vector space over the field with two elements, and let \(C(G)=[C_1\times\ldots\times C_n]\) be the class in the Grothendieck ring. Then the subring of the Grothendieck ring generated by smooth conics is the free abelian group generated by elements of the form \(C(G) \cdot [\mathbb P^1]^m\), for varying \(G\) and \(m\). There is also an explicit formula for multiplication.
The computation depends on another result of the paper, which characterizes when two schemes of the form \(C_1\times\ldots\times C_n\) and \(C'_1\times\ldots\times C'_{n'}\), where all factors are smooth conics, have the same class in the Grothendieck ring. It turns out that this holds if and only if the following equivalent conditions hold: The two schemes are (1) birational, (2) stably birational, or (3) we have \(n=n'\) and the subgroups of the Brauer groups generated by the factors \(C_1,\ldots,C_n\) and \(C_1',\ldots, C'_n\) are the same.
The proofs are based on work of M. Larsen and V. A. Lunts [Mosc. Math. J. 3, 85–95 (2003; Zbl 1056.14015)], and a nice geometric description of the Brauer product of two smooth conics \(C_1,C_2\) as a subscheme of the Hilbert scheme of divisors on \(C_1\times C_2\) of bidegree (1,1).

14F22 Brauer groups of schemes
14G27 Other nonalgebraically closed ground fields in algebraic geometry
Full Text: DOI
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