hcju/stacksqa huggingface dataset:
We will use without further mention that the base change of a V covering is a V covering (Lemma \ref{lemma-V}). In particular it suffices to show that the morphism is submersive. Being submersive is clearly Zariski local on the base. Thus we may assume $X$ is affine. Then $\{X_i \to X\}$ can be refined by a standard V covering $\{Y_j \to X\}$. If we can show that $\coprod Y_j \to X$ is submersive, then since there is a factorization $\coprod Y_j \to \coprod X_i \to X$ we conclude that $\coprod X_i \to X$ is submersive. Set $Y = \coprod Y_j$ and consider the morphism of affines $f : Y \to X$. By Lemma \ref{lemma-refine-qcqs-V} we know that we can lift any specialization $x' \leadsto x$ in $X$ to some specialization $y' \leadsto y$ in $Y$. Thus if $T \subset X$ is a subset such that $f^{-1}(T)$ is closed in $Y$, then $T \subset X$ is closed under specialization. Since $f^{-1}(T) \subset Y$ with the reduced induced closed subscheme structure is an affine scheme, we conclude that $T \subset X$ is closed by Algebra, Lemma \ref{algebra-lemma-image-stable-specialization-closed}. Hence $f$ is submersive.
Assumptions (b), (c), and (e) of Sites, Lemma \ref{sites-lemma-bigger-site} are immediate for the functors $\Sch_\tau \to \Sch'_\tau$ and $(\Sch/S)_\tau \to (\Sch'/S)_\tau$. Property (a) holds by Lemma \ref{lemma-zariski-induced}, \ref{lemma-etale-induced}, \ref{lemma-smooth-induced}, \ref{lemma-syntomic-induced}, or \ref{lemma-fppf-induced}. Property (d) holds because fibre products in the categories $\Sch_\tau$, $\Sch'_\tau$ exist and are compatible with fibre products in the category of schemes.
Let us prove this for $\tau = fppf$. By Limits, Lemma \ref{limits-lemma-descend-finite-presentation} the category of schemes of finite presentation over $T$ is the colimit over $I$ of the categories of finite presentation over $T_i$. By Limits, Lemmas \ref{limits-lemma-descend-affine-finite-presentation} and \ref{limits-lemma-descend-flat-finite-presentation} the same is true for category of schemes which are affine, flat and of finite presentation over $T$. To finish the proof of the lemma it suffices to show that if $\{V_{j, i} \to T_i\}_{j = 1, \ldots, m}$ is a finite family of flat finitely presented morphisms with $V_{j, i}$ affine, and the base change $\coprod_j T \times_{T_i} V_{j, i} \to T$ is surjective, then for some $i' \geq i$ the morphism $\coprod T_{i'} \times_{T_i} V_{j, i} \to T_{i'}$ is surjective. Denote $W_{i'} \subset T_{i'}$, resp.\ $W \subset T$ the image. Of course $W = T$ by assumption. Since the morphisms are flat and of finite presentation we see that $W_i$ is a quasi-compact open of $T_i$, see Morphisms, Lemma \ref{morphisms-lemma-fppf-open}. Moreover, $W = T \times_{T_i} W_i$ (formation of image commutes with base change). Hence by Limits, Lemma \ref{limits-lemma-descend-opens} we conclude that $W_{i'} = T_{i'}$ for some large enough $i'$ and we win. \medskip\noindent For $\tau \in \{Zariski, \etale, smooth, syntomic\}$ a standard $\tau$-covering is a standard fppf covering. Hence the fully faithfulness of the functor holds. The only issue is to show that given a standard fppf covering $\mathcal{V}_i$ for some $i$ such that $\mathcal{V}_i \times_{T_i} T$ is a standard $\tau$-covering, then $\mathcal{V}_i \times_{T_i} T_{i'}$ is a standard $\tau$-covering for all $i' \gg i$. This follows immediately from Limits, Lemmas \ref{limits-lemma-descend-open-immersion}, \ref{limits-lemma-descend-etale}, \ref{limits-lemma-descend-smooth}, and \ref{limits-lemma-descend-syntomic}.
This follows from Lemma \ref{lemma-extend-sheaf-general} applied with $\mathcal{C} = (\Sch/S)_\tau$. Conditions (1), (2), (a), and (b) of Lemma \ref{lemma-extend} hold; we omit the details. Thus we get our unique extension $F'$ to the category of all schemes over $S$. Finally, observe that any standard $\tau$-covering is tautologically equivalent to a covering in $(\Sch/S)_\tau$, see Sets, Lemma \ref{sets-lemma-what-is-in-it} as well as Lemmas \ref{lemma-zariski-induced}, \ref{lemma-etale-induced}, \ref{lemma-smooth-induced}, \ref{lemma-syntomic-induced}, and \ref{lemma-fppf-induced}. By Sites, Lemma \ref{sites-lemma-tautological-same-sheaf} the sheaf property passes through tautological equivalence of coverings. Hence the fact that $F$ is a $\tau$-sheaf implies that property (c) of Lemma \ref{lemma-extend-sheaf-general} holds and we conclude.
Let $\{T_i \to T\}_{i \in I}$ be an fpqc covering of schemes over $S$. Set $X_i = X_{T_i} = X \times_S T_i$. Note that $\{X_i \to X_T\}_{i \in I}$ is an fpqc covering of $X_T$ (Topologies, Lemma \ref{topologies-lemma-fpqc}) and that $X_{T_i \times_T T_{i'}} = X_i \times_{X_T} X_{i'}$. Suppose that $Z_i \in \Hilbfunctor^d_{X/S}(T_i)$ is a collection of elements such that $Z_i$ and $Z_{i'}$ map to the same element of $\Hilbfunctor^d_{X/S}(T_i \times_T T_{i'})$. By effective descent for closed immersions (Descent, Lemma \ref{descent-lemma-closed-immersion}) there is a closed immersion $Z \to X_T$ whose base change by $X_i \to X_T$ is equal to $Z_i \to X_i$. The morphism $Z \to T$ then has the property that its base change to $T_i$ is the morphism $Z_i \to T_i$. Hence $Z \to T$ is finite locally free of degree $d$ by Descent, Lemma \ref{descent-lemma-descending-property-finite-locally-free}.
Let $T = \lim T_i$ be a limit of affine schemes over $S$. We have to show that $\Hilbfunctor^d_{X/S}(T) = \colim \Hilbfunctor^d_{X/S}(T_i)$. Observe that if $Z \to X_T$ is an element of $\Hilbfunctor^d_{X/S}(T)$, then $Z \to T$ is of finite presentation. Hence by Limits, Lemma \ref{limits-lemma-descend-finite-presentation} there exists an $i$, a scheme $Z_i$ of finite presentation over $T_i$, and a morphism $Z_i \to X_{T_i}$ over $T_i$ whose base change to $T$ gives $Z \to X_T$. We apply Limits, Lemma \ref{limits-lemma-descend-closed-immersion-finite-presentation} to see that we may assume $Z_i \to X_{T_i}$ is a closed immersion after increasing $i$. We apply Limits, Lemma \ref{limits-lemma-descend-finite-locally-free} to see that $Z_i \to T_i$ is finite locally free of degree $d$ after possibly increasing $i$. Then $Z_i \in \Hilbfunctor^d_{X/S}(T_i)$ as desired.
Let $T$ be a scheme over $S$ and let $Z \in \Hilbfunctor^d_{Y/S}(T)$. Claim: there is a closed subscheme $T_X \subset T$ such that a morphism of schemes $T' \to T$ factors through $T_X$ if and only if $Z_{T'} \to Y_{T'}$ factors through $X_{T'}$. Applying this to a scheme $T_{univ}$ representing $\Hilbfunctor^d_{Y/S}$ and the universal object\footnote{See Categories, Section \ref{categories-section-opposite}} $Z_{univ} \in \Hilbfunctor^d_{Y/S}(T_{univ})$ we get a closed subscheme $T_{univ, X} \subset T_{univ}$ such that $Z_{univ, X} = Z_{univ} \times_{T_{univ}} T_{univ, X}$ is a closed subscheme of $X \times_S T_{univ, X}$ and hence defines an element of $\Hilbfunctor^d_{X/S}(T_{univ, X})$. A formal argument then shows that $T_{univ, X}$ is a scheme representing $\Hilbfunctor^d_{X/S}$ with universal object $Z_{univ, X}$. \medskip\noindent Proof of the claim. Consider $Z' = X_T \times_{Y_T} Z$. Given $T' \to T$ we see that $Z_{T'} \to Y_{T'}$ factors through $X_{T'}$ if and only if $Z'_{T'} \to Z_{T'}$ is an isomorphism. Thus the claim follows from the very general More on Flatness, Lemma \ref{flat-lemma-Weil-restriction-closed-subschemes}. However, in this special case one can prove the statement directly as follows: first reduce to the case $T = \Spec(A)$ and $Z = \Spec(B)$. After shrinking $T$ further we may assume there is an isomorphism $\varphi : B \to A^{\oplus d}$ as $A$-modules. Then $Z' = \Spec(B/J)$ for some ideal $J \subset B$. Let $g_\beta \in J$ be a collection of generators and write $\varphi(g_\beta) = (g_\beta^1, \ldots, g_\beta^d)$. Then it is clear that $T_X$ is given by $\Spec(A/(g_\beta^j))$.
In this proof all unadorned products are over $S$. Let $H = \underline{\Hilbfunctor}^d_{X/S}$ and let $Z \in \Hilbfunctor^d_{X/S}(H)$ be the universal object. Consider the two objects $Z_1, Z_2 \in \Hilbfunctor^d_{X/S}(H \times H)$ we get by pulling back $Z$ by the two projections $H \times H \to H$. Then $Z_1 = Z \times H \subset X_{H \times H}$ and $Z_2 = H \times Z \subset X_{H \times H}$. Since $H$ represents the functor $\Hilbfunctor^d_{X/S}$, the diagonal morphism $\Delta : H \to H \times H$ has the following universal property: A morphism of schemes $T \to H \times H$ factors through $\Delta$ if and only if $Z_{1, T} = Z_{2, T}$ as elements of $\Hilbfunctor^d_{X/S}(T)$. Set $Z = Z_1 \times_{X_{H \times H}} Z_2$. Then we see that $T \to H \times H$ factors through $\Delta$ if and only if the morphisms $Z_T \to Z_{1, T}$ and $Z_T \to Z_{2, T}$ are isomorphisms. It follows from the very general More on Flatness, Lemma \ref{flat-lemma-Weil-restriction-closed-subschemes} that $\Delta$ is a closed immersion. In the proof of Lemma \ref{lemma-hilb-d-of-closed} the reader finds an alternative easier proof of the needed result in our special case.
Equivalence of (2) and (3). This follows from Divisors, Lemma \ref{divisors-lemma-fibre-Cartier} if we can show the equivalence of (2) and (3) when $S$ is the spectrum of a field $k$. Let $x \in X$ be a closed point. As $X$ is smooth of relative dimension $1$ over $k$ and we see that $\mathcal{O}_{X, x}$ is a regular local ring of dimension $1$ (see Varieties, Lemma \ref{varieties-lemma-smooth-regular}). Thus $\mathcal{O}_{X, x}$ is a discrete valuation ring (Algebra, Lemma \ref{algebra-lemma-characterize-dvr}) and hence a PID. It follows that every sheaf of ideals $\mathcal{I} \subset \mathcal{O}_X$ which is nonvanishing at all the generic points of $X$ is invertible (Divisors, Lemma \ref{divisors-lemma-effective-Cartier-in-points}). In other words, every closed subscheme of $X$ which does not contain a generic point is an effective Cartier divisor. It follows that (2) and (3) are equivalent. \medskip\noindent If $S$ is Noetherian, then any locally quasi-finite morphism $D \to S$ is locally of finite presentation (Morphisms, Lemma \ref{morphisms-lemma-noetherian-finite-type-finite-presentation}), whence (3) is equivalent to (4). \medskip\noindent If $X \to S$ is proper (and $S$ is arbitrary), then $D \to S$ is proper as well. Since a proper locally quasi-finite morphism is finite (More on Morphisms, Lemma \ref{more-morphisms-lemma-characterize-finite}) and a finite, flat, and finitely presented morphism is finite locally free (Morphisms, Lemma \ref{morphisms-lemma-finite-flat}), we see that (1) is equivalent to (2).
By Lemma \ref{lemma-divisors-on-curves} we see that $D_1$ and $D_2$ are relative effective Cartier divisors on $X/S$. Thus $D = D_1 + D_2$ is a relative effective Cartier divisor on $X/S$ by Divisors, Lemma \ref{divisors-lemma-sum-relative-effective-Cartier-divisor}. Hence $D \to S$ is locally quasi-finite, flat, and locally of finite presentation by Lemma \ref{lemma-divisors-on-curves}. Applying Morphisms, Lemma \ref{morphisms-lemma-image-universally-closed-separated} the surjective integral morphism $D_1 \amalg D_2 \to D$ we find that $D \to S$ is separated. Then Morphisms, Lemma \ref{morphisms-lemma-image-proper-is-proper} implies that $D \to S$ is proper. This implies that $D \to S$ is finite (More on Morphisms, Lemma \ref{more-morphisms-lemma-characterize-finite}) and in turn we see that $D \to S$ is finite locally free (Morphisms, Lemma \ref{morphisms-lemma-finite-flat}). Thus it suffice to show that the degree of $D \to S$ is $d_1 + d_2$. To do this we may base change to a fibre of $X \to S$, hence we may assume that $S = \Spec(k)$ for some field $k$. In this case, there exists a finite set of closed points $x_1, \ldots, x_n \in X$ such that $D_1$ and $D_2$ are supported on $\{x_1, \ldots, x_n\}$. In fact, there are nonzerodivisors $f_{i, j} \in \mathcal{O}_{X, x_i}$ such that $$ D_1 = \coprod \Spec(\mathcal{O}_{X, x_i}/(f_{i, 1})) \quad\text{and}\quad D_2 = \coprod \Spec(\mathcal{O}_{X, x_i}/(f_{i, 2})) $$ Then we see that $$ D = \coprod \Spec(\mathcal{O}_{X, x_i}/(f_{i, 1}f_{i, 2})) $$ From this one sees easily that $D$ has degree $d_1 + d_2$ over $k$ (if need be, use Algebra, Lemma \ref{algebra-lemma-ord-additive}).
This proof is almost exactly the same as the proof of Lemma \ref{lemma-sum-divisors-on-curves}. By Lemma \ref{lemma-divisors-on-curves} we see that $D_1$ and $D_2$ are relative effective Cartier divisors on $X/S$. By Divisors, Lemma \ref{divisors-lemma-difference-relative-effective-Cartier-divisor} there is a relative effective Cartier divisor $D \subset X$ such that $D_2 = D_1 + D$. Hence $D \to S$ is locally quasi-finite, flat, and locally of finite presentation by Lemma \ref{lemma-divisors-on-curves}. Since $D$ is a closed subscheme of $D_2$, we see that $D \to S$ is finite. It follows that $D \to S$ is finite locally free (Morphisms, Lemma \ref{morphisms-lemma-finite-flat}). Thus it suffice to show that the degree of $D \to S$ is $d_2 - d_1$. This follows from Lemma \ref{lemma-sum-divisors-on-curves}.
Let $D_{univ} \subset X \times_S \underline{\Hilbfunctor}^{d + 1}_{X/S}$ be the universal object. There is a commutative diagram $$ \xymatrix{ \underline{\Hilbfunctor}^d_{X/S} \times_S X \ar[rr] \ar[rd] & & D_{univ} \ar[ld] \ar@{^{(}->}[r] & \underline{\Hilbfunctor}^{d + 1}_{X/S} \times_S X \\ & \underline{\Hilbfunctor}^{d + 1}_{X/S} } $$ where the top horizontal arrow maps $(D', x)$ to $(D' + x, x)$. We claim this morphism is an isomorphism which certainly proves the lemma. Namely, given a scheme $T$ over $S$, a $T$-valued point $\xi$ of $D_{univ}$ is given by a pair $\xi = (D, x)$ where $D \subset X_T$ is a closed subscheme finite locally free of degree $d + 1$ over $T$ and $x : T \to X$ is a morphism whose graph $x : T \to X_T$ factors through $D$. Then by Lemma \ref{lemma-difference-divisors-on-curves} we can write $D = D' + x$ for some $D' \subset X_T$ finite locally free of degree $d$ over $T$. Sending $\xi = (D, x)$ to the pair $(D', x)$ is the desired inverse.
We have $\underline{\Hilbfunctor}^0_{X/S} = S$ and $\underline{\Hilbfunctor}^1_{X/S} = X$ thus the result is true for $d = 0, 1$. Assuming the result for $d$, we see that $\underline{\Hilbfunctor}^d_{X/S} \times_S X$ is smooth over $S$ (Morphisms, Lemma \ref{morphisms-lemma-base-change-smooth} and \ref{morphisms-lemma-composition-smooth}). Since $\underline{\Hilbfunctor}^d_{X/S} \times_S X \to \underline{\Hilbfunctor}^{d + 1}_{X/S}$ is finite locally free of degree $d + 1$ by Lemma \ref{lemma-universal-object} the result follows from Descent, Lemma \ref{descent-lemma-smooth-permanence}. We omit the verification that the relative dimension is as claimed (you can do this by looking at fibres, or by keeping track of the dimensions in the argument above).
We may replace $S$ by $T$ and $X$ by $X_T$ and assume that $S = T$ to simplify the notation. Let $\mathcal{N}$ be an invertible $\mathcal{O}_S$-module. If $f^*\mathcal{N} \cong \mathcal{O}_X$, then we see that $f_*f^*\mathcal{N} \cong f_*\mathcal{O}_X \cong \mathcal{O}_S$ by assumption. Since $\mathcal{N}$ is locally trivial, we see that the canonical map $\mathcal{N} \to f_*f^*\mathcal{N}$ is locally an isomorphism (because $\mathcal{O}_S \to f_*f^*\mathcal{O}_S$ is an isomorphism by assumption). Hence we conclude that $\mathcal{N} \to f_*f^*\mathcal{N} \to \mathcal{O}_S$ is an isomorphism and we see that $\mathcal{N}$ is trivial. This proves the first arrow is injective. \medskip\noindent Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module which is in the kernel of $\Pic(X) \to \Picardfunctor_{X/S}(S)$. Then there exists an fppf covering $\{S_i \to S\}$ such that $\mathcal{L}$ pulls back to the trivial invertible sheaf on $X_{S_i}$. Choose a trivializing section $s_i$. Then $\text{pr}_0^*s_i$ and $\text{pr}_1^*s_j$ are both trivialising sections of $\mathcal{L}$ over $X_{S_i \times_S S_j}$ and hence differ by a multiplicative unit $$ f_{ij} \in \Gamma(X_{S_i \times_S S_j}, \mathcal{O}_{X_{S_i \times_S S_j}}^*) = \Gamma(S_i \times_S S_j, \mathcal{O}_{S_i \times_S S_j}^*) $$ (equality by our assumption on pushforward of structure sheaves). Of course these elements satisfy the cocycle condition on $S_i \times_S S_j \times_S S_k$, hence they define a descent datum on invertible sheaves for the fppf covering $\{S_i \to S\}$. By Descent, Proposition \ref{descent-proposition-fpqc-descent-quasi-coherent} there is an invertible $\mathcal{O}_S$-module $\mathcal{N}$ with trivializations over $S_i$ whose associated descent datum is $\{f_{ij}\}$. Then $f^*\mathcal{N} \cong \mathcal{L}$ as the functor from descent data to modules is fully faithful (see proposition cited above).
Denote $K(T) = \Ker(\sigma_T^* : \Pic(X_T) \to \Pic(T))$. Since $\sigma$ is a section of $f$ we see that $\Pic(X_T)$ is the direct sum of $\Pic(T)$ and $K(T)$. Thus by Lemma \ref{lemma-flat-geometrically-connected-fibres} we see that $K(T) \subset \Picardfunctor_{X/S}(T)$ for all $T$. Moreover, it is clear from the construction that $\Picardfunctor_{X/S}$ is the sheafification of the presheaf $K$. To finish the proof it suffices to show that $K$ satisfies the sheaf condition for fppf coverings which we do in the next paragraph. \medskip\noindent Let $\{T_i \to T\}$ be an fppf covering. Let $\mathcal{L}_i$ be elements of $K(T_i)$ which map to the same elements of $K(T_i \times_T T_j)$ for all $i$ and $j$. Choose an isomorphism $\alpha_i : \mathcal{O}_{T_i} \to \sigma_{T_i}^*\mathcal{L}_i$ for all $i$. Choose an isomorphism $$ \varphi_{ij} : \mathcal{L}_i|_{X_{T_i \times_T T_j}} \longrightarrow \mathcal{L}_j|_{X_{T_i \times_T T_j}} $$ If the map $$ \alpha_j|_{T_i \times_T T_j} \circ \sigma_{T_i \times_T T_j}^*\varphi_{ij} \circ \alpha_i|_{T_i \times_T T_j} : \mathcal{O}_{T_i \times_T T_j} \to \mathcal{O}_{T_i \times_T T_j} $$ is not equal to multiplication by $1$ but some $u_{ij}$, then we can scale $\varphi_{ij}$ by $u_{ij}^{-1}$ to correct this. Having done this, consider the self map $$ \varphi_{ki}|_{X_{T_i \times_T T_j \times_T T_k}} \circ \varphi_{jk}|_{X_{T_i \times_T T_j \times_T T_k}} \circ \varphi_{ij}|_{X_{T_i \times_T T_j \times_T T_k}} \quad\text{on}\quad \mathcal{L}_i|_{X_{T_i \times_T T_j \times_T T_k}} $$ which is given by multiplication by some regular function $f_{ijk}$ on the scheme $X_{T_i \times_T T_j \times_T T_k}$. By our choice of $\varphi_{ij}$ we see that the pullback of this map by $\sigma$ is equal to multiplication by $1$. By our assumption on functions on $X$, we see that $f_{ijk} = 1$. Thus we obtain a descent datum for the fppf covering $\{X_{T_i} \to X\}$. By Descent, Proposition \ref{descent-proposition-fpqc-descent-quasi-coherent} there is an invertible $\mathcal{O}_{X_T}$-module $\mathcal{L}$ and an isomorphism $\alpha : \mathcal{O}_T \to \sigma_T^*\mathcal{L}$ whose pullback to $X_{T_i}$ recovers $(\mathcal{L}_i, \alpha_i)$ (small detail omitted). Thus $\mathcal{L}$ defines an object of $K(T)$ as desired.
This follows from Schemes, Lemma \ref{schemes-lemma-glue-functors}. Namely, take $I = G(k)$ and for $i = g' \in I$ take $F_i \subset G$ the subfunctor which associates to $T$ over $k$ the set of elements $g \in G(T)$ with $g'g \in F(T)$. Then $F_i \cong F$ by multiplication by $g'$. The map $F_i \to G$ is isomorphic to the map $F \to G$ by multiplication by $g'$, hence is representable by open immersions. Finally, the collection $(F_i)_{i \in I}$ covers $G$ by assumption (2)(c). Thus the lemma mentioned above applies and the proof is complete.
The meaning of the phrase ``has a $k$-rational point'' is exactly that the structure morphism $f : X \to \Spec(k)$ has a section, which verifies the first condition. By Varieties, Lemma \ref{varieties-lemma-regular-functions-proper-variety} we see that $k' = H^0(X, \mathcal{O}_X)$ is a field extension of $k$. Since $X$ has a $k$-rational point there is a $k$-algebra homomorphism $k' \to k$ and we conclude $k' = k$. Since $k$ is a field, any morphism $T \to \Spec(k)$ is flat. Hence we see by cohomology and base change (Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology}) that $\mathcal{O}_T \to f_{T, *}\mathcal{O}_{X_T}$ is an isomorphism. This finishes the proof.
This proof is a variant of the proof of Varieties, Lemma \ref{varieties-lemma-general-degree-g-line-bundle}. We encourage the reader to read that proof first. \medskip\noindent First we pick an ample invertible sheaf $\mathcal{L}_0$ and we replace $\mathcal{L}$ by $\mathcal{L} \otimes_{\mathcal{O}_{X_K}} \mathcal{L}_0^{\otimes n}|_{X_K}$ for some $n \gg 0$. The result will be that we may assume that $H^0(X_K, \mathcal{L}) \not = 0$ and $H^1(X_K, \mathcal{L}) = 0$. Namely, we will get the vanishing by Cohomology of Schemes, Lemma \ref{coherent-lemma-vanshing-gives-ample} and the nonvanishing because the degree of the tensor product is $\gg 0$. We will finish the proof by descending induction on $t = \dim_K H^0(X_K, \mathcal{L})$. The base case $t = 1$ is trivial. Assume $t > 1$. \medskip\noindent Observe that for a $k$-rational point $x$ of $X$, the inverse image $x_K$ is a $K$-rational point of $X_K$. Moreover, there are infinitely many $k$-rational points by Varieties, Lemma \ref{varieties-lemma-smooth-separable-closed-points-dense}. Therefore the points $x_K$ form a Zariski dense collection of points of $X_K$. \medskip\noindent Let $s \in H^0(X_K, \mathcal{L})$ be nonzero. From the previous paragraph we deduce there exists a $k$-rational point $x$ such that $s$ does not vanish in $x_K$. Let $\mathcal{I}$ be the ideal sheaf of $i : x_K \to X_K$ as in Varieties, Lemma \ref{varieties-lemma-regular-point-on-curve}. Look at the short exact sequence $$ 0 \to \mathcal{I} \otimes_{\mathcal{O}_{X_K}} \mathcal{L} \to \mathcal{L} \to i_*i^*\mathcal{L} \to 0 $$ Observe that $H^0(X_K, i_*i^*\mathcal{L}) = H^0(x_K, i^*\mathcal{L})$ has dimension $1$ over $K$. Since $s$ does not vanish at $x$ we conclude that $$ H^0(X_K, \mathcal{L}) \longrightarrow H^0(X, i_*i^*\mathcal{L}) $$ is surjective. Hence $\dim_K H^0(X_K, \mathcal{I} \otimes_{\mathcal{O}_{X_K}} \mathcal{L}) = t - 1$. Finally, the long exact sequence of cohomology also shows that $H^1(X_K, \mathcal{I} \otimes_{\mathcal{O}_{X_K}} \mathcal{L}) = 0$ thereby finishing the proof of the induction step.