Let $k$ be a field of characteristic 0 and let $X$ and $Y$ be smooth, projective and geometrically integral $k$-schemes of finite type. Assume that both $X$ and $Y$ have 0-cycles of degree 1. Does $X\times\_{k}Y$ have a 0-cycle of degree 1? 



Products of varieties of index 1
The referee for one of my papers gave the following argument:


Let $k$ be any field and let $X$ and $Y$ be smooth, proper and geometrically integral k-schemes of finite type. Let $x$ be a $0$-cycle on $X$ and $y$ a $0$-cycle on $Y$. Then $x\times Y$ and $X \times y$ are cycles on $X \times Y$ of complementary dimensions, so that $z=(x \times Y).(X \times y)$ is a
$0$-cycle on $X \times Y$. Then, by intersection theory, ${\rm deg}(z)={\rm deg}(x){\rm deg}(y)$. In particular, if both $X$ and $Y$ have $0$-cycles of degree $1$, then so does $X \times Y$.


