All Questions

Let $X$ be an arbitrary smooth projective variety over a field $k$. Do there exist: a smooth complete intersection $X'$ in a projective space. a surjective morphism of $k$-varieties $X'\to X$ ?

Is there a name for a normal, projective variety such that every effective divisor is ample? Examples of such varieties are projective space, weighted projective spaces, and simple Abelian varieties ...

Let $X$ be a smooth projective variety over a finite field, with a closed immersion to some other smooth projective variety $S$, with $S$ liftable. Suppose there exists an fpqc cover $S'\to S$, such ...