The Stacks project

Lemma 109.25.2. Let $g \geq 2$. The stack $\overline{\mathcal{M}}_ g$ is quasi-compact.

Proof. We will use the notation from Section 109.4. Consider the subset

\[ T \subset |\textit{PolarizedCurves}| \]

of points $\xi $ such that there exists a field $k$ and a pair $(X, \mathcal{L})$ over $k$ representing $\xi $ with the following two properties

  1. $X$ is a stable genus $g$ curve, and

  2. $\mathcal{L} = \omega _ X^{\otimes 3}$.

Clearly, under the continuous map

\[ |\textit{PolarizedCurves}| \longrightarrow |\mathcal{C}\! \mathit{urves}| \]

the image of the set $T$ is exactly the open subset

\[ |\overline{\mathcal{M}}_ g| \subset |\mathcal{C}\! \mathit{urves}| \]

Thus it suffices to show that $T$ is quasi-compact. By Lemma 109.4.1 we see that

\[ |\textit{PolarizedCurves}| \subset |\mathcal{P}\! \mathit{olarized}| \]

is an open and closed immersion. Thus it suffices to prove quasi-compactness of $T$ as a subset of $|\mathcal{P}\! \mathit{olarized}|$. For this we use the criterion of Moduli Stacks, Lemma 108.11.3. First, we observe that for $(X, \mathcal{L})$ as above the Hilbert polynomial $P$ is the function $P(t) = (6g - 6)t + (1 - g)$ by Riemann-Roch, see Algebraic Curves, Lemma 53.5.2. Next, we observe that $H^1(X, \mathcal{L}) = 0$ and $\mathcal{L}$ is very ample by Algebraic Curves, Lemma 53.24.3. This means exactly that with $n = P(3) - 1$ there is a closed immersion

\[ i : X \longrightarrow \mathbf{P}^ n_ k \]

such that $\mathcal{L} = i^*\mathcal{O}_{\mathbf{P}^1_ k}(1)$ as desired. $\square$

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