Lemma 109.33.3. Let $K$ be a field. Let $C_ i$, $i = 1, \ldots , n$ be smooth, projective, geometrically irreducible curves over $K$. Let $P_ i \in C_ i(K)$ be a rational point and let $Q_ i \in C_ i$ be a point such that $[\kappa (Q_ i) : K] = 2$. Then $[P_1 \times \ldots \times P_ n]$ is nonzero in $\mathop{\mathrm{CH}}\nolimits _0(U_1 \times _ K \ldots \times _ K U_ n)$ where $U_ i = C_ i \setminus \{ Q_ i\}$.

Proof. There is a degree map $\deg : \mathop{\mathrm{CH}}\nolimits _0(C_1 \times _ K \ldots \times _ K C_ n) \to \mathbf{Z}$ Because each $Q_ i$ has degree $2$ over $K$ we see that any zero cycle supported on the “boundary”

$C_1 \times _ K \ldots \times _ K C_ n \setminus U_1 \times _ K \ldots \times _ K U_ n$

has degree divisible by $2$. $\square$

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