affine morphism of schemes

Question

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?

Answer 1

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.