2
$\begingroup$

Let $X$ be a smooth projective variety over a field $k$ and $\mathcal{E}$ a locally free sheaf of finite rank. Consider the total space $\mathrm{Tot}(\mathcal{E})=\mathcal{S}pec(\mathrm{Sym}(\mathcal{E}))$ (relative spectrum of the symmetric algebra). Is $\mathrm{Tot}(\mathcal{E})$ considered as a scheme over $k$ quasi-projective?

$\endgroup$
3
$\begingroup$

This is usually written $V(\mathcal{E})$. The projection $V(\mathcal{E}) \to X$ is affine, hence quasi-projective, and $X \to \mathrm{Spec}(k)$ is (quasi-)projective. Hence the composition is also quasi-projective.

Most of the assumptions here (variety, field, locally free etc.) are superfluous.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.