hcju/stacksqa huggingface dataset:
Combining Varieties, Lemma \ref{varieties-lemma-ampleness-in-terms-of-degrees-components} and Lemmas \ref{lemma-rational-tail-negative} and \ref{lemma-rational-bridge-zero} we see that $X$ contains no rational tails or bridges. Then we see that $\omega_X^{\otimes 3}$ is globally generated by Lemma \ref{lemma-contracting-rational-tails}. Choose a $k$-basis $s_0, \ldots, s_n$ of $H^0(X, \omega_X^{\otimes 3})$. We get a morphism $$ \varphi_{\omega_X^{\otimes 3}, (s_0, \ldots, s_n)} : X \longrightarrow \mathbf{P}^n_k $$ See Constructions, Section \ref{constructions-section-projective-space}. The lemma asserts that this morphism is a closed immersion. To check this we may replace $k$ by its algebraic closure, see Descent, Lemma \ref{descent-lemma-descending-property-closed-immersion}. Thus we may assume $k$ is algebraically closed. \medskip\noindent Assume $k$ is algebraically closed. We will use Varieties, Lemma \ref{varieties-lemma-variant-separate-points-tangent-vectors} to prove the lemma. Let $Z \subset X$ be a closed subscheme of degree $2$ over $Z$ with ideal sheaf $\mathcal{I} \subset \mathcal{O}_X$. We have to show that $$ H^0(X, \mathcal{L}) \to H^0(Z, \mathcal{L}|_Z) $$ is surjective. Thus it suffices to show that $H^1(X, \mathcal{I}\mathcal{L}) = 0$. To do this we will use Lemma \ref{lemma-vanishing-on-gorenstein}. Thus it suffices to show that $$ 3\deg(\omega_X|_Y) > -2\chi(Y, \mathcal{O}_Y) + \deg(Z \cap Y) $$ for every reduced connected closed subscheme $Y \subset X$. Since $k$ is algebraically closed and $Y$ connected and reduced we have $H^0(Y, \mathcal{O}_Y) = k$ (Varieties, Lemma \ref{varieties-lemma-proper-geometrically-reduced-global-sections}). Hence $\chi(Y, \mathcal{O}_Y) = 1 - \dim H^1(Y, \mathcal{O}_Y)$. Thus we have to show $$ 3\deg(\omega_X|_Y) > -2 + 2\dim H^1(Y, \mathcal{O}_Y) + \deg(Z \cap Y) $$ which is true by Lemma \ref{lemma-no-rational-tail} except possibly if $Y = X$ or if $\deg(\omega_X|_Y) = 0$. Since $\omega_X$ is ample the second possibility does not occur (see first lemma cited in this proof). Finally, if $Y = X$ we can use Riemann-Roch (Lemma \ref{lemma-rr}) and the fact that $g \geq 2$ to see that the inquality holds. The same argument with $Z = \emptyset$ shows that $H^1(X, \omega_X^{\otimes 3}) = 0$.
Recall that we have a universal $k$-derivation $d : \mathcal{O}_X \to \Omega_{X/k}$ and hence $D = \theta \circ d$ for some $\mathcal{O}_X$-linear map $\theta : \Omega_{X/k} \to \mathcal{O}_X$. Recall that $\Omega_{X/k} \cong \omega_X$, see Lemma \ref{lemma-duality-dim-1}. By Riemann-Roch we have $\deg(\omega_X) = 2g - 2$ (Lemma \ref{lemma-rr}). Thus we see that $\theta$ is forced to be zero if $g > 1$ by Varieties, Lemma \ref{varieties-lemma-check-invertible-sheaf-trivial}. This proves part (1). If $g = 1$, then a nonzero $\theta$ does not vanish anywhere and if $g = 0$, then a nonzero $\theta$ vanishes in a divisor of degree $2$. Thus parts (2) and (3) follow if we show that vanishing of $\theta$ at a closed point $x \in X$ is equivalent to the statement that $D$ fixes $x$ (as defined above). Let $z \in \mathcal{O}_{X, x}$ be a uniformizer. Then $dz$ is a basis element for $\Omega_{X, x}$, see Lemma \ref{lemma-uniformizer-works}. Since $D(z) = \theta(dz)$ we conclude.
Let $x \in X$ be a node. Let $x', x'' \in X^\nu$ be the inverse images of $x$. (Every node is a split node since $k$ is algebriacally closed, see Definition \ref{definition-split-node} and Lemma \ref{lemma-split-node}.) Let $u \in \mathcal{O}_{X^\nu, x'}$ and $v \in \mathcal{O}_{X^\nu, x''}$ be uniformizers. Observe that we have an exact sequence $$ 0 \to \mathcal{O}_{X, x} \to \mathcal{O}_{X^\nu, x'} \times \mathcal{O}_{X^\nu, x''} \to k \to 0 $$ This follows from Lemma \ref{lemma-multicross}. Thus we can view $u$ and $v$ as elements of $\mathcal{O}_{X, x}$ with $uv = 0$. \medskip\noindent Let $D \in \text{Der}_k(\mathcal{O}_X, \mathcal{O}_X)$. Then $0 = D(uv) = vD(u) + uD(v)$. Since $(u)$ is annihilator of $v$ in $\mathcal{O}_{X, x}$ and vice versa, we see that $D(u) \in (u)$ and $D(v) \in (v)$. As $\mathcal{O}_{X^\nu, x'} = k + (u)$ we conclude that we can extend $D$ to $\mathcal{O}_{X^\nu, x'}$ and moreover the extension fixes $x'$. This produces a $D'$ in the right hand side of the equality. Conversely, given a $D'$ fixing $x'$ and $x''$ we find that $D'$ preserves the subring $\mathcal{O}_{X, x} \subset \mathcal{O}_{X^\nu, x'} \times \mathcal{O}_{X^\nu, x''}$ and this is how we go from right to left in the equality.
Let $D \in \text{Der}_k(\mathcal{O}_X, \mathcal{O}_X)$. Let $X^\nu$ be the normalization of $X$. Let $D' \in \text{Der}_k(\mathcal{O}_{X^\nu}, \mathcal{O}_{X^\nu})$ be the element corresponding to $D$ via Lemma \ref{lemma-nodal-vector-fields}. Let $C \subset X^\nu$ be an irreducible component. If the genus of $C$ is $> 1$, then $D'|_{\mathcal{O}_C} = 0$ by Lemma \ref{lemma-smooth-vector-fields} part (1). If the genus of $C$ is $1$, then there is at least one closed point $c$ of $C$ which maps to a node on $X$ (since otherwise $X \cong C$ would have genus $1$). By the correspondence this means that $D'|_{\mathcal{O}_C}$ fixes $c$ hence is zero by Lemma \ref{lemma-smooth-vector-fields} part (2). Finally, if the genus of $C$ is zero, then there are at least $3$ pairwise distinct closed points $c_1, c_2, c_3 \in C$ mapping to nodes in $X$, since otherwise either $X$ is $C$ with two points glued (two points of $C$ mapping to the same node), or $C$ is a rational bridge (two points mapping to different nodes of $X$), or $C$ is a rational tail (one point mapping to a node of $X$). These three possibilities are not permitted since $C$ has genus $\geq 2$ and has no rational bridges, or rational tails. Whence $D'|_{\mathcal{O}_C}$ fixes $c_1, c_2, c_3$ hence is zero by Lemma \ref{lemma-smooth-vector-fields} part (3).
We will prove that each condition (2) -- (8) implies (1) and we omit the verification that (1) implies (2) -- (8). \medskip\noindent Assume (2). A smooth scheme over $k$ is geometrically reduced (Varieties, Lemma \ref{varieties-lemma-smooth-geometrically-normal}) and regular (Varieties, Lemma \ref{varieties-lemma-smooth-regular}). Hence $X$ is Gorenstein (Duality for Schemes, Lemma \ref{duality-lemma-regular-gorenstein}). Thus we reduce to (3). \medskip\noindent Assume (3). Since $X$ is geometrically integral over $k$ we have $H^0(X, \mathcal{O}_X) = k$ by Varieties, Lemma \ref{varieties-lemma-regular-functions-proper-variety}. and we reduce to (4). \medskip\noindent Assume (4). Since $X$ is Gorenstein the dualizing module $\omega_X$ as in Lemma \ref{lemma-duality-dim-1} has degree $\deg(\omega_X) = -2$ by Lemma \ref{lemma-genus-gorenstein}. Combined with the assumed existence of an odd degree invertible module, we conclude there exists an invertible module of degree $1$. In this way we reduce to (6). \medskip\noindent Assume (5). Since $X$ is geometrically integral over $k$ we have $H^0(X, \mathcal{O}_X) = k$ by Varieties, Lemma \ref{varieties-lemma-regular-functions-proper-variety}. and we reduce to (6). \medskip\noindent Assume (6). Then $X \cong \mathbf{P}^1_k$ by Lemma \ref{lemma-genus-zero-positive-degree}. \medskip\noindent Assume (7). Observe that $\kappa = H^0(X, \mathcal{O}_X)$ is a field finite over $k$ by Varieties, Lemma \ref{varieties-lemma-regular-functions-proper-variety}. If $d = [\kappa : k] > 1$, then every invertible sheaf has degree divisible by $d$ and there cannot be an invertible sheaf of degree $1$. Hence $d = 1$ and we reduce to case (6). \medskip\noindent Assume (8). Observe that $\kappa = H^0(X, \mathcal{O}_X)$ is a field finite over $k$ by Varieties, Lemma \ref{varieties-lemma-regular-functions-proper-variety}. Since $\kappa \subset \kappa(x_i)$ we see that $k = \kappa$ by the assumption on the gcd of the degrees. The same condition allows us to find integers $a_i$ such that $1 = \sum a_i[\kappa(x_i) : k]$. Because $x_i$ defines an effective Cartier divisor on $X$ by Varieties, Lemma \ref{varieties-lemma-regular-point-on-curve} we can consider the invertible module $\mathcal{O}_X(\sum a_i x_i)$. By our choice of $a_i$ the degree of $\mathcal{L}$ is $1$. Thus $X \cong \mathbf{P}^1_k$ by Lemma \ref{lemma-genus-zero-positive-degree}.
By Fields, Lemma \ref{fields-lemma-separable-first} there is a subextension $k(X)/E/k(Y)$ such that $k(X)/E$ is purely inseparable and $E/k(Y)$ is separable. By Theorem \ref{theorem-curves-rational-maps} this corresponds to a factorization $X \to Z \to Y$ of $f$ with $Z$ a nonsingular proper curve. Apply Lemma \ref{lemma-purely-inseparable-smooth} to the morphism $X \to Z$ to conclude.
Let $\nu : X^\nu \to X$ be the normalization (Varieties, Lemma \ref{varieties-lemma-prepare-delta-invariant}). Choose a factorization $$ X^\nu = X_n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = X $$ as in Lemma \ref{lemma-factor-almost-isomorphism}. Let us denote $h^0_i = \dim_k H^0(X_i, \mathcal{O}_{X_i})$ and $h^1_i = \dim_k H^1(X_i, \mathcal{O}_{X_i})$. By Lemmas \ref{lemma-glue-points} and \ref{lemma-squish-tangent-vector} for each $n > i \geq 0$ we have one of the following there possibilities \begin{enumerate} \item $X_i$ is obtained by glueing $a, b \in X_{i + 1}$ which are on different connected components: in this case $\Pic(X_i) = \Pic(X_{i + 1})$, $h^0_{i + 1} = h^0_i + 1$, $h^1_{i + 1} = h^1_i$, \item $X_i$ is obtained by glueing $a, b \in X_{i + 1}$ which are on the same connected component: in this case there is a short exact sequence $$ 0 \to k^* \to \Pic(X_i) \to \Pic(X_{i + 1}) \to 0, $$ and $h^0_{i + 1} = h^0_i$, $h^1_{i + 1} = h^1_i - 1$, \item $X_i$ is obtained by squishing a tangent vector in $X_{i + 1}$: in this case there is a short exact sequence $$ 0 \to (k, +) \to \Pic(X_i) \to \Pic(X_{i + 1}) \to 0, $$ and $h^0_{i + 1} = h^0_i$, $h^1_{i + 1} = h^1_i - 1$. \end{enumerate} To prove the statements on dimensions of cohomology groups of the structure sheaf, use the exact sequences in Examples \ref{example-glue-points} and \ref{example-squish-tangent-vector}. Since $k$ is algebraically closed of characteristic prime to $\ell$ we see that $(k, +)$ and $k^*$ are $\ell$-divisible and with $\ell$-torsion $(k, +)[\ell] = 0$ and $k^*[\ell] \cong \mathbf{F}_\ell$. Hence $$ \dim_{\mathbf{F}_\ell} \Pic(X_{i + 1})[\ell] - \dim_{\mathbf{F}_\ell}\Pic(X_i)[\ell] $$ is zero, except in case (2) where it is equal to $-1$. At the end of this process we get the normalization $X^\nu = X_n$ which is a disjoint union of smooth projective curves over $k$. Hence we have \begin{enumerate} \item $h^1_n = g_{geom}$ and \item $\dim_{\mathbf{F}_\ell} \Pic(X_n)[\ell] = 2g_{geom}$. \end{enumerate} The last equality by Lemma \ref{lemma-torsion-picard-smooth-projective}. Since $g = h^1_0$ we see that the number of steps of type (2) and (3) is at most $h^1_0 - h^1_n = g - g_{geom}$. By our comptation of the differences in ranks we conclude that $$ \dim_{\mathbf{F}_\ell} \Pic(X)[\ell] \leq g - g_{geom} + 2g_{geom} = g + g_{geom} $$ and equality holds if and only if no steps of type (3) occur. This indeed means that all singularities of $X$ are multicross by Lemma \ref{lemma-multicross}. Conversely, if all the singularities are multicross, then Lemma \ref{lemma-multicross} guarantees that we can find a sequence $X^\nu = X_n \to \ldots \to X_0 = X$ as above such that no steps of type (3) occur in the sequence and we find equality holds in the lemma (just glue the local sequences for each point to find one that works for all singular points of $x$; some details omitted).
After some formal arguments this is exactly Theorem \ref{theorem-descent-quasi-coherent}. Details omitted. In Descent, Proposition \ref{descent-proposition-equivalence-quasi-coherent} we prove a more precise version of the theorem for the big Zariski, fppf, \'etale, smooth, and syntomic sites of $S$, as well as the small Zariski and \'etale sites of $S$.
By the Lemma \ref{lemma-hom-injective}, the functors $\check H^p(\mathcal{U}, -)$ are universal $\delta$-functors since they are effaceable. So are the right derived functors of $\check H^0(\mathcal{U}, -)$. Since they agree in degree $0$, they agree by the universal property of universal $\delta$-functors. For more details see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cech-cohomology-derived-presheaves}.
Choose an injective resolution $\mathcal{F}\to \mathcal{I}^\bullet$ in $\textit{Ab}(\mathcal{C})$, and consider the double complex $\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}^\bullet)$ and the maps $$ \xymatrix{ \Gamma(U, I^\bullet) \ar[r] & \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}^\bullet) \\ & \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \ar[u] } $$ Here the horizontal map is the natural map $\Gamma(U, I^\bullet) \to \check{\mathcal{C}}^0(\mathcal{U}, \mathcal{I}^\bullet)$ to the left column, and the vertical map is induced by $\mathcal{F}\to \mathcal{I}^0$ and lands in the bottom row. By assumption, $\mathcal{I}^\bullet$ is a complex of injectives in $\textit{Ab}(\mathcal{C})$, hence by Lemma \ref{lemma-forget-injectives}, it is a complex of injectives in $\textit{PAb}(\mathcal{C})$. Thus, the rows of the double complex are exact in positive degrees (Lemma \ref{lemma-hom-injective}), and the kernel of $\check{\mathcal{C}}^0(\mathcal{U}, \mathcal{I}^\bullet) \to \check{\mathcal{C}}^1(\mathcal{U}, \mathcal{I}^\bullet)$ is equal to $\Gamma(U, \mathcal{I}^\bullet)$, since $\mathcal{I}^\bullet$ is a complex of sheaves. In particular, the cohomology of the total complex is the standard cohomology of the global sections functor $H^0(U, \mathcal{F})$. \medskip\noindent For the vertical direction, the $q$th cohomology group of the $p$th column is $$ \prod_{i_0, \ldots, i_p} H^q(U_{i_0} \times_U \ldots \times_U U_{i_p}, \mathcal{F}) = \prod_{i_0, \ldots, i_p} \underline{H}^q(\mathcal{F})(U_{i_0} \times_U \ldots \times_U U_{i_p}) $$ in the entry $E_1^{p, q}$. So this is a standard double complex spectral sequence, and the $E_2$-page is as prescribed. For more details see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cech-spectral-sequence}.
Let $\tau$ be one of the topologies considered in Section \ref{section-big-small}. By Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-h1-invertible} we see that $H^1_\tau(X, \mathbf{G}_m) = H^1_\tau(X, \mathcal{O}_\tau^*) = \Pic(\mathcal{O}_\tau)$ where $\mathcal{O}_\tau$ is the structure sheaf of the site $(\Sch/X)_\tau$. Now an invertible $\mathcal{O}_\tau$-module is a quasi-coherent $\mathcal{O}_\tau$-module. By Theorem \ref{theorem-quasi-coherent} or the more precise Descent, Proposition \ref{descent-proposition-equivalence-quasi-coherent} we see that $\Pic(\mathcal{O}_\tau) = \Pic(X)$. The last equality is proved in the same way.
The necessity of exactness on stalks follows from Lemma \ref{lemma-stalk-exact}. For the converse, it suffices to show that a map of sheaves is surjective (respectively injective) if and only if it is surjective (respectively injective) on all stalks. We prove this in the case of surjectivity, and omit the proof in the case of injectivity. \medskip\noindent Let $\alpha : \mathcal{F} \to \mathcal{G}$ be a map of sheaves such that $\mathcal{F}_{\overline{s}} \to \mathcal{G}_{\overline{s}}$ is surjective for all geometric points. Fix $U \in \Ob(S_\etale)$ and $s \in \mathcal{G}(U)$. For every $u \in U$ choose some $\overline{u} \to U$ lying over $u$ and an \'etale neighborhood $(V_u , \overline{v}_u) \to (U, \overline{u})$ such that $s|_{V_u} = \alpha(s_{V_u})$ for some $s_{V_u} \in \mathcal{F}(V_u)$. This is possible since $\alpha$ is surjective on stalks. Then $\{V_u \to U\}_{u \in U}$ is an \'etale covering on which the restrictions of $s$ are in the image of the map $\alpha$. Thus, $\alpha$ is surjective, see Sites, Section \ref{sites-section-sheaves-injective}.
After base changing via $Y \to S$, we may assume that $Y = S$. Thus we may and do assume both $X$ and $Y$ are \'etale over $S$. In other words, the theorem states that the base change functor is a fully faithful functor from the category of schemes \'etale over $S$ to the category of schemes \'etale over $S'$. \medskip\noindent Consider the forgetful functor \begin{equation} \label{equation-descent-etale-forget} \begin{matrix} \text{descent data }(X', \varphi')\text{ relative to }S'/S \\ \text{ with }X'\text{ \'etale over }S' \end{matrix} \longrightarrow \text{schemes }X'\text{ \'etale over }S' \end{equation} We claim this functor is an equivalence. On the other hand, the functor \begin{equation} \label{equation-descent-etale} \text{schemes }X\text{ \'etale over }S \longrightarrow \begin{matrix} \text{descent data }(X', \varphi')\text{ relative to }S'/S \\ \text{ with }X'\text{ \'etale over }S' \end{matrix} \end{equation} is fully faithful by \'Etale Morphisms, Lemma \ref{etale-lemma-fully-faithful-cases}. Thus the claim implies the theorem. \medskip\noindent Proof of the claim. Recall that a universal homeomorphism is the same thing as an integral, universally injective, surjective morphism, see Morphisms, Lemma \ref{morphisms-lemma-universal-homeomorphism}. In particular, the diagonal $\Delta : S' \to S' \times_S S'$ is a thickening by Morphisms, Lemma \ref{morphisms-lemma-universally-injective}. Thus by \'Etale Morphisms, Theorem \ref{etale-theorem-etale-topological} we see that given $X' \to S'$ \'etale there is a unique isomorphism $$ \varphi' : X' \times_S S' \to S' \times_S X' $$ of schemes \'etale over $S' \times_S S'$ which pulls back under $\Delta$ to $\text{id} : X' \to X'$ over $S'$. Since $S' \to S' \times_S S' \times_S S'$ is a thickening as well (it is bijective and a closed immersion) we conclude that $(X', \varphi')$ is a descent datum relative to $S'/S$. The canonical nature of the construction of $\varphi'$ shows that it is compatible with morphisms between schemes \'etale over $S'$. In other words, we obtain a quasi-inverse $X' \mapsto (X', \varphi')$ of the functor (\ref{equation-descent-etale-forget}). This proves the claim and finishes the proof of the theorem.
[First proof] By Theorem \ref{theorem-etale-topological} we see that the functor is fully faithful. It remains to show that the functor is essentially surjective. Let $U \to X$ be an \'etale morphism of schemes. \medskip\noindent Suppose that the result holds if $U$ and $Y$ are affine. In that case, we choose an affine open covering $U = \bigcup U_i$ such that each $U_i$ maps into an affine open of $Y$. By assumption (affine case) we can find \'etale morphisms $V_i \to Y$ such that $X \times_Y V_i \cong U_i$ as schemes over $X$. Let $V_{i, i'} \subset V_i$ be the open subscheme whose underlying topological space corresponds to $U_i \cap U_{i'}$. Because we have isomorphisms $$ X \times_Y V_{i, i'} \cong U_i \cap U_{i'} \cong X \times_Y V_{i', i} $$ as schemes over $X$ we see by fully faithfulness that we obtain isomorphisms $\theta_{i, i'} : V_{i, i'} \to V_{i', i}$ of schemes over $Y$. We omit the verification that these isomorphisms satisfy the cocycle condition of Schemes, Section \ref{schemes-section-glueing-schemes}. Applying Schemes, Lemma \ref{schemes-lemma-glue-schemes} we obtain a scheme $V \to Y$ by glueing the schemes $V_i$ along the identifications $\theta_{i, i'}$. It is clear that $V \to Y$ is \'etale and $X \times_Y V \cong U$ by construction. \medskip\noindent Thus it suffices to show the lemma in case $U$ and $Y$ are affine. Recall that in the proof of Theorem \ref{theorem-etale-topological} we showed that $U$ comes with a unique descent datum $(U, \varphi)$ relative to $X/Y$. By \'Etale Morphisms, Proposition \ref{etale-proposition-effective} (which applies because $U \to X$ is quasi-compact and separated as well as \'etale by our reduction to the affine case) there exists an \'etale morphism $V \to Y$ such that $X \times_Y V \cong U$ and the proof is complete.
Let $\mathcal{I}$ be the category of \'etale neighborhoods of $\overline{s}$ on $S$. By Lemma \ref{lemma-higher-direct-images} we have $$ (R^nf_*\mathcal{F})_{\overline{s}} = \colim_{(V, \overline{v}) \in \mathcal{I}^{opp}} H_\etale^n(X \times_S V, \mathcal{F}|_{X \times_S V}). $$ We may replace $\mathcal{I}$ by the initial subcategory consisting of affine \'etale neighbourhoods of $\overline{s}$. Observe that $$ \Spec(\mathcal{O}_{S, s}^{sh}) = \lim_{(V, \overline{v}) \in \mathcal{I}} V $$ by Lemma \ref{lemma-describe-etale-local-ring} and Limits, Lemma \ref{limits-lemma-directed-inverse-system-affine-schemes-has-limit}. Since fibre products commute with limits we also obtain $$ X \times_S \Spec(\mathcal{O}_{S, s}^{sh}) = \lim_{(V, \overline{v}) \in \mathcal{I}} X \times_S V $$ We conclude by Lemma \ref{lemma-directed-colimit-cohomology}.
Let us construct the inverse to this functor. In Fundamental Groups, Lemma \ref{pione-lemma-sheaves-point} we have seen that given a $G$-set $M$ there exists an \'etale morphism $X \to \Spec(K)$ such that $\Mor_K(\Spec(K^{sep}), X)$ is isomorphic to $M$ as a $G$-set. Consider the sheaf $\mathcal{F}$ on $\Spec(K)_\etale$ defined by the rule $U \mapsto \Mor_K(U, X)$. This is a sheaf as the \'etale topology is subcanonical. Then we see that $\mathcal{F}_{\overline{s}} = \Mor_K(\Spec(K^{sep}), X) = M$ as $G$-sets (details omitted). This gives the inverse of the functor and we win.
For projective space one can show directly that the field $k(x_1, \ldots, x_r)$ is $C_r$ (exercise). \medskip\noindent General case. Without loss of generality, we may assume $X$ to be projective. Let $f \in k(X)[T_1, \ldots, T_n]_d$ with $0 < d^r < n$. Say the coefficients of $f$ are in $\Gamma(X, \mathcal{O}_X(H))$ for some ample $H \subset X$. Let $\mathbf{\alpha} = (\alpha_1, \ldots, \alpha_n)$ with $\alpha_i \in \Gamma(X, \mathcal{O}_X(eH))$. Then $f(\mathbf{\alpha}) \in \Gamma(X, \mathcal{O}_X((de + 1)H))$. Consider the system of equations $f(\mathbf{\alpha}) =0$. Then by asymptotic Riemann-Roch (Varieties, Proposition \ref{varieties-proposition-asymptotic-riemann-roch}) there exists a $c > 0$ such that \begin{itemize} \item the number of variables is $n\dim_k \Gamma(X, \mathcal{O}_X(eH)) \sim n e^r c$, and \item the number of equations is $\dim_k \Gamma(X, \mathcal{O}_X((de + 1)H)) \sim (de + 1)^r c$. \end{itemize} Since $n > d^r$, there are more variables than equations. The equations are homogeneous hence there is a solution by Lemma \ref{lemma-algebraically-closed-find-solutions}.
Let $\varphi : U \to X$ be an \'etale morphism. Then by properties of \'etale morphisms (Proposition \ref{proposition-etale-morphisms}), $U = \coprod_i U_i$ where each $U_i$ is a smooth curve mapping to $X$. The above sequence for $U$ is a product of the corresponding sequences for each $U_i$, so it suffices to treat the case where $U$ is connected, hence irreducible. In this case, there is a well known exact sequence $$ 1 \longrightarrow \Gamma(U, \mathcal{O}_U^*) \longrightarrow k(U)^* \longrightarrow \bigoplus\nolimits_{y \in U_0} \mathbf{Z}_y. $$ This amounts to a sequence $$ 0 \longrightarrow \Gamma(U, \mathcal{O}_U^*) \longrightarrow \Gamma(\eta \times_X U, \mathcal{O}_{\eta \times_X U}^*) \longrightarrow \bigoplus\nolimits_{x \in X_0} \Gamma(x \times_X U, \underline{\mathbf{Z}}) $$ which, unfolding definitions, is nothing but a sequence $$ 0 \longrightarrow \mathbf{G}_m(U) \longrightarrow j_* \mathbf{G}_{m, \eta}(U) \longrightarrow \left(\bigoplus\nolimits_{x \in X_0} {i_x}_* \underline{\mathbf{Z}}\right)(U). $$ This defines the maps in the Fundamental Exact Sequence and shows it is exact except possibly at the last step. To see surjectivity, let us recall that if $U$ is a nonsingular curve and $D$ is a divisor on $U$, then there exists a Zariski open covering $\{U_j \to U\}$ of $U$ such that $D|_{U_j} = \text{div}(f_j)$ for some $f_j \in k(U)^*$.