Let $k$ be a field. Let $X$, $Y$ and $C$ be varieties over $k$ such that $X\times_k C $ is isomorphic to $Y\times_k C$. Assume that $C$ is a curve. (The varieties $X$, $Y$ and $C$ are assumed to be smooth projective and geometrically connected over $k$.)
Does it follow that $X$ is isomorphic to $Y$?
My guess is that $X\to X\times C\cong Y\times C\to Y$ gives an isomorphism. (The first map is the inclusion and the second map is the projection. For the first map to be defined you might want to assume $C(k)$ is non-empty.) Anyway, the inverse map should be given by the reverse construction: $Y\to Y\times C\cong X\times C\to X$.
Is this correct? (No if $X=Y=C$. But let's assume $\dim X = \dim Y \geq 2$.)