Lemma 27.13.1. Let $S = \mathbf{Z}[T_0, \ldots , T_ n]$ with $\deg (T_ i) = 1$. The scheme

represents the functor which associates to a scheme $Y$ the pairs $(\mathcal{L}, (s_0, \ldots , s_ n))$ where

$\mathcal{L}$ is an invertible $\mathcal{O}_ Y$-module, and

$s_0, \ldots , s_ n$ are global sections of $\mathcal{L}$ which generate $\mathcal{L}$

up to the following equivalence: $(\mathcal{L}, (s_0, \ldots , s_ n)) \sim (\mathcal{N}, (t_0, \ldots , t_ n))$ $\Leftrightarrow $ there exists an isomorphism $\beta : \mathcal{L} \to \mathcal{N}$ with $\beta (s_ i) = t_ i$ for $i = 0, \ldots , n$.

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