All Questions

One can construct non-isomorphic smooth projective varieties which define the same motive by blowing up $\mathbb{P}^2$ at five points. I think I learned this here at MathOverflow. But these examples ...

Let $X$ be a (say, smooth projective) variety over a field $k$. For which $K$ it is known that the ("ordinary", that is, not higher) Chow groups of $X$ map onto that of $X_K$ bijectively? ...

Let $A$ be an abelian variety over $\mathbb{C}$ and $A^*$ the dual Abelian variety. The class of $A$ and the class of $A^*$ in $\mathcal{M}_{\mathbb{C}}=K_0(Var_\mathbb{C})[\mathcal{L}^{- 1}]$ are ...

Let $\mathcal{Q}$ be an irreducible quadric in $\mathbb{P}^n(k)$, with $n \geq 2$ and $k$ a finite field. Let $K_0(V_k)$ be the Grothendieck ring of $k$-varieties. It is well known (it appears) that ...