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Let $i:X\to \overline{X}$ be any dense open immersion of $X$ in a projective scheme. For every morphism $\phi:X\to X$, denote by $\overline{\Gamma}_\phi\subset \overline{X}\times_{\text{Spec}(\mathbb{C})}\overline{X}$ the closure of the graph of $\phi$. For every choice of ample invertible sheaf $\mathcal{L}$ on $\overline{X}\times_{\text{\Spec}(\mathbb{C})}\overline{X}$, there is an associated Hilbert polynomial $P(t)\in \mathbb{Q}[t]$ of $\overline{\Gamma}_\phi$ with respect to $\mathcal{L}$. For every choice of $P(t)$, denote by $\text{Hilb}^{P(t)}$ the associated Hilbert scheme parameterizing closed subschemes $Z\subset \overline{X}\times_{\text{Spec}(\mathbb{C})}\overline{X}$ with Hilbert polynomial $P(t)$. There is an open subscheme $U$ that parameterizes those $Z$ such that both projections $Z \cap (X\times_{\text{Spec}(\mathbb{C})} X) \to X$ are isomorphisms. As an open subscheme of a projective scheme, $U$ has only finitely many irreducible components. Also there are only countably many possible numerical polynomials $P(t)$. Thus, the group $Q(X)$ is finite or countably infinite.

Let $i:X\to \overline{X}$ be any dense open immersion of $X$ in a projective scheme. For every morphism $\phi:X\to X$, denote by $\overline{\Gamma}_\phi\subset \overline{X}\times_{\text{Spec}(\mathbb{C})}\overline{X}$ the closure of the graph of $\phi$. For every choice of ample invertible sheaf $\mathcal{L}$ on $\overline{X}\times_{\text{Spec}(\mathbb{C})}\overline{X}$, there is an associated Hilbert polynomial $P(t)\in \mathbb{Q}[t]$ of $\overline{\Gamma}_\phi$ with respect to $\mathcal{L}$. For every choice of $P(t)$, denote by $\text{Hilb}^{P(t)}$ the associated Hilbert scheme parameterizing closed subschemes $Z\subset \overline{X}\times_{\text{Spec}(\mathbb{C})}\overline{X}$ with Hilbert polynomial $P(t)$. There is an open subscheme $U$ that parameterizes those $Z$ such that both projections $Z \cap (X\times_{\text{Spec}(\mathbb{C})} X) \to X$ are isomorphisms. As an open subscheme of a projective scheme, $U$ has only finitely many irreducible components. Also there are only countably many possible numerical polynomials $P(t)$. Thus, the group $Q(X)$ is finite or countably infinite.

Let $i:X\to \overline{X}$ be any dense open immersion of $X$ in a projective scheme. For every morphism $\phi:X\to X$, denote by $\overline{\Gamma}_\phi\subset \overline{X}\times_{\text{Spec}(\mathbb{C})}\overline{X}$ the closure of the graph of $\phi$. For every choice of ample invertible sheaf $\mathcal{L}$ on $\overline{X}\times_{\text{\Spec}(\mathbb{C})}\overline{X}$, there is an associated Hilbert polynomial $P(t)\in \mathbb{Q}[t]$ of $\overline{\Gamma}_\phi$ with respect to $\mathcal{L}$. For every choice of $P(t)$, denote by $\text{Hilb}^{P(t)}$ the associated Hilbert scheme parameterizing closed subschemes $Z\subset \overline{X}\times_{\text{Spec}(\mathbb{C})}\overline{X}$ with Hilbert polynomial $P(t)$. There is an open subscheme $U$ that parameterizes those $Z$ such that both projections $Z \cap (X\times_{\text{Spec}(\mathbb{C})} X) \to X$ are isomorphisms. As an open subscheme of a projective scheme, $U$ has only finitely many irreducible components. Also there are only countably many possible numerical polynomials $P(t)$. Thus, the group $Q(X)$ is finite or countable.

Let $i:X\to \overline{X}$ be any dense open immersion of $X$ in a projective scheme. For every morphism $\phi:X\to X$, denote by $\overline{\Gamma}_\phi\subset \overline{X}\times_{\text{Spec}(\mathbb{C})}\overline{X}$ the closure of the graph of $\phi$. For every choice of ample invertible sheaf $\mathcal{L}$ on $\overline{X}\times_{\text{\Spec}(\mathbb{C})}\overline{X}$, there is an associated Hilbert polynomial $P(t)\in \mathbb{Q}[t]$ of $\overline{\Gamma}_\phi$ with respect to $\mathcal{L}$. For every choice of $P(t)$, denote by $\text{Hilb}^{P(t)}$ the associated Hilbert scheme parameterizing closed subschemes $Z\subset \overline{X}\times_{\text{Spec}(\mathbb{C})}\overline{X}$ with Hilbert polynomial $P(t)$. There is an open subscheme $U$ that parameterizes those $Z$ such that both projections $Z \cap (X\times_{\text{Spec}(\mathbb{C})} X) \to X$ are isomorphisms. As an open subscheme of a projective scheme, $U$ has only finitely many irreducible components. Also there are only countably many possible numerical polynomials $P(t)$. Thus, the group $Q(X)$ is finite or countably infinite.

Let $i:X\to \overline{X}$ be any dense open immersion of $X$ in a projective scheme. For every morphism $\phi:X\to X$, denote by $\overline{\Gamma}_\phi\subset \overline{X}\times_{\text{Spec}(\mathbb{C})}\overline{X}$ the closure of the graph of $\phi$. For every choice of ample invertible sheaf $\mathcal{L}$ on $\overline{X}\times_{\text{\Spec}(\mathbb{C})}\overline{X}$, there is an associated Hilbert polynomial $P(t)\in \mathbb{Q}[t]$ of $\overline{\Gamma}_\phi$ with respect to $\mathcal{L}$. For every choice of $P(t)$, denote by $\text{Hilb}^{P(t)}$ the associated Hilbert scheme parameterizing closed subschemes $Z\subset \overline{X}\times_{\text{Spec}(\mathbb{C})}\overline{X}$ with Hilbert polynomial $P(t)$. There is an open subscheme $U$ that parameterizes those $Z$ such that both projections $Z\to \overline{X}$ are isomorphisms over the dense open $X$. As an open subscheme of a projective scheme, $U$ has only finitely many irreducible components. Also there are only countably many possible numerical polynomials $P(t)$. Thus, the group $Q(X)$ is finite or countable.

Let $i:X\to \overline{X}$ be any dense open immersion of $X$ in a projective scheme. For every morphism $\phi:X\to X$, denote by $\overline{\Gamma}_\phi\subset \overline{X}\times_{\text{Spec}(\mathbb{C})}\overline{X}$ the closure of the graph of $\phi$. For every choice of ample invertible sheaf $\mathcal{L}$ on $\overline{X}\times_{\text{\Spec}(\mathbb{C})}\overline{X}$, there is an associated Hilbert polynomial $P(t)\in \mathbb{Q}[t]$ of $\overline{\Gamma}_\phi$ with respect to $\mathcal{L}$. For every choice of $P(t)$, denote by $\text{Hilb}^{P(t)}$ the associated Hilbert scheme parameterizing closed subschemes $Z\subset \overline{X}\times_{\text{Spec}(\mathbb{C})}\overline{X}$ with Hilbert polynomial $P(t)$. There is an open subscheme $U$ that parameterizes those $Z$ such that both projections $Z \cap (X\times_{\text{Spec}(\mathbb{C})} X) \to X$ are isomorphisms. As an open subscheme of a projective scheme, $U$ has only finitely many irreducible components. Also there are only countably many possible numerical polynomials $P(t)$. Thus, the group $Q(X)$ is finite or countable.