2
votes
1answer
56 views

If the reduction is smooth and projective, can I conclude the same about the scheme

Let $X$ be a $R$-scheme, where $R$ is a dvr. Suppose that the reduction of $X$ (over the closed point of $\mathrm{Spec} \ R$) is smooth and projective. Does this imply that $X$ is smooth and ...
1
vote
1answer
103 views

Can a non-proper variety contain a proper curve

Let $f:X\to S$ be a finite type, separated but non-proper morphism of schemes. Can there be a projective curve $g:C\to S$ and a closed immersion $C\to X$ over $S$? Just to be clear: A projective ...