Q:

Products of varieties of index 1

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? 

A:

A zero cycle of degree 1 on such an $X$ can be written as $D_1-D_2$ for two effective cycles with $\deg D_1=\deg D_2+1$. Similarly, we have $E_1-E_2$ for $Y$. Then the obvious product (somewhat loosely written) $D_1\times E_1-D_1\times E_2-D_2\times E_1+D_2\times E_2$ has degree 1.

A:

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 $X \times Y$.

