Which is canonical $|D|\cong \mathbb{P}(\Gamma(X,\mathscr{O}_X(D))^{\vee}$) or $|D|\cong \mathbb{P}(\Gamma(X,\mathscr{O}_X(D))$)?

Question

Sorry for my bad English.

Let $X$ be a smooth projective curve over an algebraically closed field $k$, and $D$ is divisor on $X$.

By The stacks project, $|D|\cong \mathbb{P}(\Gamma(X,\mathscr{O}_X(D))^{\vee})$ .

On the other hand, Hartshorne say $|D|\cong \mathbb{P}(\Gamma(X,\mathscr{O}_X(D)))$.

Which is canonical?

Please tell me, thanks.

Answer 1

Your confusion here is actually just about the definition of $\mathbb{P}(V)$ for a vector space $V$. In elementary contexts, it is common to define $\mathbb{P}(V)$ as the quotient of $V\setminus\{0\}$ by the action of nonzero scalars. However, in algebraic geometry, it is instead defined as the quotient of $V^\vee\setminus\{0\}$ by the action of nonzero scalars. Or more precisely, as a scheme, $\mathbb{P}(V)$ is defined as $\operatorname{Proj}(\operatorname{Sym}(V))$. Note that this means that closed points of $\mathbb{P}(V)$ come from maximal homogeneous proper subideals of the irrelevant ideal of $\operatorname{Sym}(V)$. Such ideals are generated by codimension $1$ linear subspaces $W$ of $\operatorname{Sym}^1(V)=V$, which (by identifying the quotient $V/W$ with $k$) correspond to nonzero homomorphisms $V\to k$ up to scalar multiples. That is, they correspond to elements of $V^\vee\setminus\{0\}$ up to scalar multiples. So, with this definition of $\mathbb{P}(V)$, the correct statement is $|D|\cong \mathbb{P}(\Gamma(X,\mathscr{O}_X(D))^{\vee})$