Theorem 99.14.15 (Algebraicity of the stack of polarized schemes). The stack $\mathcal{P}\! \mathit{olarized}$ (Situation 99.14.1) is algebraic. In fact, for any algebraic space $B$ the stack $B\textit{-Polarized}$ (Remark 99.14.5) is algebraic.
Proof. The absolute case follows from Artin's Axioms, Lemma 98.17.1 and Lemmas 99.14.7, 99.14.9, 99.14.8, 99.14.12, and 99.14.13. The case over $B$ follows from this, the description of $B\textit{-Polarized}$ as a $2$-fibre product in Remark 99.14.5, and the fact that algebraic stacks have $2$-fibre products, see Algebraic Stacks, Lemma 94.14.3. $\square$
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