Recall that $\Curvesstack$ is a limit preserving algebraic stack, see Quot, Lemma \ref{quot-lemma-curves-limits}. By Limits of Stacks, Lemma \ref{stacks-limits-lemma-limit-preserving-diagonal} this implies that $\Delta : \Polarizedstack \to \Polarizedstack \times \Polarizedstack$ is limit preserving. Hence $\Delta$ is locally of finite presentation by Limits of Stacks, Proposition \ref{stacks-limits-proposition-characterize-locally-finite-presentation}. \medskip\noindent Let us prove that $\Delta$ is separated. To see this, it suffices to show that given a scheme $U$ and two objects $Y \to U$ and $X \to U$ of $\Curvesstack$ over $U$, the algebraic space $$ \mathit{Isom}_U(Y, X) $$ is separated. This we have seen in Moduli Stacks, Lemmas \ref{moduli-lemma-Mor-s-lfp} and \ref{moduli-lemma-Isom-in-Mor} that the target is a separated algebraic space. \medskip\noindent To finish the proof we show that $\Delta$ is quasi-compact. Since $\Delta$ is representable by algebraic spaces, it suffices to check the base change of $\Delta$ by a surjective smooth morphism $U \to \Curvesstack \times \Curvesstack$ is quasi-compact (see for example Properties of Stacks, Lemma \ref{stacks-properties-lemma-check-property-covering}). We choose $U = \coprod U_i$ to be a disjoint union of affine opens with a surjective smooth morphism $$ U \longrightarrow \textit{PolarizedCurves} \times \textit{PolarizedCurves} $$ Then $U \to \Curvesstack \times \Curvesstack$ will be surjective and smooth since $\textit{PolarizedCurves} \to \Curvesstack$ is surjective and smooth by Lemma \ref{lemma-polarized-curves-over-curves}. Since $\textit{PolarizedCurves}$ is limit preserving (by Artin's Axioms, Lemma \ref{artin-lemma-fibre-product-limit-preserving} and Quot, Lemmas \ref{quot-lemma-curves-limits}, \ref{quot-lemma-polarized-limits}, and \ref{quot-lemma-spaces-limits}), we see that $\textit{PolarizedCurves} \to \Spec(\mathbf{Z})$ is locally of finite presentation, hence $U_i \to \Spec(\mathbf{Z})$ is locally of finite presentation (Limits of Stacks, Proposition \ref{stacks-limits-proposition-characterize-locally-finite-presentation} and Morphisms of Stacks, Lemmas \ref{stacks-morphisms-lemma-composition-finite-presentation} and \ref{stacks-morphisms-lemma-smooth-locally-finite-presentation}). In particular, $U_i$ is Noetherian affine. This reduces us to the case discussed in the next paragraph. \medskip\noindent In this paragraph, given a Noetherian affine scheme $U$ and two objects $(Y, \mathcal{N})$ and $(X, \mathcal{L})$ of $\textit{PolarizedCurves}$ over $U$, we show the algebraic space $$ \mathit{Isom}_U(Y, X) $$ is quasi-compact. Since the connected components of $U$ are open and closed we may replace $U$ by these. Thus we may and do assume $U$ is connected. Let $u \in U$ be a point. Let $Q$, $P$ be the Hilbert polynomials of these families, i.e., $$ Q(n) = \chi(Y_u, \mathcal{N}_u^{\otimes n}) \quad\text{and}\quad P(n) = \chi(X_u, \mathcal{L}_u^{\otimes n}) $$ see Varieties, Lemma \ref{varieties-lemma-numerical-polynomial-from-euler}. Since $U$ is connected and since the functions $u \mapsto \chi(Y_u, \mathcal{N}_u^{\otimes n})$ and $u \mapsto \chi(X_u, \mathcal{L}_u^{\otimes n})$ are locally constant (see  Derived Categories of Schemes, Lemma \ref{perfect-lemma-chi-locally-constant-geometric}) we see that we get the same Hilbert polynomial in every point of $U$. Set $$ \mathcal{M} = \text{pr}_1^*\mathcal{N} \otimes_{\mathcal{O}_{Y \times_U X}} \text{pr}_2^*\mathcal{L} $$ on $Y \times_U X$. Given $(f, \varphi) \in \mathit{Isom}_U(Y, X)(T)$ for some scheme $T$ over $U$ then for every $t \in T$ we have \begin{align*} \chi(Y_t, (\text{id} \times f)^*\mathcal{M}^{\otimes n}) & = \chi(Y_t, \mathcal{N}_t^{\otimes n} \otimes_{\mathcal{O}_{Y_t}} f_t^*\mathcal{L}_t^{\otimes n}) \\ & = n\deg(\mathcal{N}_t) + n\deg(f_t^*\mathcal{L}_t) + \chi(Y_t, \mathcal{O}_{Y_t}) \\ & = Q(n) + n\deg(\mathcal{L}_t) \\ & = Q(n) + P(n) - P(0) \end{align*} by Riemann-Roch for proper curves, more precisely by Varieties, Definition \ref{varieties-definition-degree-invertible-sheaf} and Lemma \ref{varieties-lemma-degree-tensor-product} and the fact that $f_t$ is an isomorphism. Setting $P'(t) = Q(t) + P(t) - P(0)$ we find $$ \mathit{Isom}_U(Y, X) = \mathit{Isom}_U(Y, X) \cap \mathit{Mor}^{P', \mathcal{M}}_U(Y, X) $$ The intersection is an intersection of open subspaces of $\mathit{Mor}_U(Y, X)$, see Moduli Stacks, Lemma \ref{moduli-lemma-Isom-in-Mor} and Remark \ref{moduli-remark-Mor-numerical}. Now $\mathit{Mor}^{P', \mathcal{M}}_U(Y, X)$ is a Noetherian algebraic space as it is of finite presentation over $U$ by Moduli Stacks, Lemma \ref{moduli-lemma-Mor-qc-over-base}. Thus the intersection is a Noetherian algebraic space too and the proof is finished.