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Let $\mathscr{E}$ be a locally free sheaf on a scheme $X$. I always thought that $\mathbb{P}(\mathscr{E})$ meant $\mathrm{\bf Proj}(\textrm{Sym}(\mathscr{E}))$, the global Proj associated to the sheaf of symmetric algebra of $\mathscr{E}$. But, when I was reading about Hilbert schemes, I came across with the notation $\mathbb{P}((\mathrm{Sym}^{2}\mathscr{E}^{\vee}))$. It appears as the scheme parameterizing conics in a projective space; $\mathscr{E}$ denotes the universal vector bundle on the Grassmannian of projective planes $\mathbb{G}(2, n)$. I am a little bit confused. How is defined $\mathbb{P}((\mathrm{Sym}^{2}\mathscr{E}^{\vee}))$?

Thank you.

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1 Answer 1

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I think this is defined exactly as you say. $\operatorname{Sym}^2\mathscr E^\vee$ is the degree two piece of the sheaf of graded algebras $\operatorname{Sym}\mathscr E^\vee$, and is in particular an $\mathscr O_X$-module. Moreover, $\mathscr E$ is locally free implies that $\mathscr E^\vee$ and $\operatorname{Sym}^2 \mathscr E^\vee$ are locally free. Thus $\mathbb P(\operatorname{Sym}^2 \mathscr E^\vee) := \mathbf{Proj}\big(\operatorname{Sym}(\operatorname{Sym}^2 \mathscr E^\vee)\big)$.

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