Abstract:
Let ${P_i}$ and ${Q_j}$ be two sets of Severi-Brauer surfaces (or two sets of
conics) over a field $k$. The main goal of this talk is to explain the
relationship between the classes of products $[\Pi P_i]$ and $[\Pi Q_j]$ in the
Grothendieck ring and the subgroups $\langle P_i\rangle$ and $\langle Q_j\rangle$ in the Brauer group $\mathrm{Br}(k)$.
Under some restrictions on the base field, the following conditions are
equivalent: (1) $\langle P_i\rangle = \langle Q_j\rangle$ in $\mathrm{Br}(k)$ (2) $[\Pi P_i] = [\Pi Q_j]$ in $K_0(\mathrm{Var}_k)$ (3)
$\Pi P_i$ and $\Pi Q_j$ are birational. In this talk, we will prove this statement. We
will also show some properties of the Grothendieck subring generated by
conics, in particular, it will be explicitly described by generators and
relations. The talk is based on the works of J. Kollar "Conics in the
Grothendieck ring" and A. Hogadi "Products of Brauer Severi surfaces".
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