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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).

##### MSC:
 14F22 Brauer groups of schemes 14G27 Other nonalgebraically closed ground fields in algebraic geometry
##### Keywords:
Grothendieck ring; conics; Severi-Brauer variety
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##### References:
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