hcju/stacksqa huggingface dataset:
Our basic tool will be Derived Categories of Schemes, Lemma \ref{perfect-lemma-chi-locally-constant-geometric} which shows that $$ 1 - g_C = \chi(C, \mathcal{O}_C) = \chi(X_k, \mathcal{O}_{X_k}) $$ Choose a sequence of effective Cartier divisors $$ X_k = D_m \supset D_{m - 1} \supset \ldots \supset D_1 \supset D_0 = \emptyset $$ such that $D_{j + 1} = D_j + C_{i_j}$ for each $j$. (It is clear that we can choose such a sequence by decreasing one nonzero multiplicity of $D_{j + 1}$ one step at a time.) Applying Lemma \ref{lemma-add-component} starting with $\chi(\mathcal{O}_{D_0}) = 0$ we get \begin{align*} 1 - g_C & = \chi(X_k, \mathcal{O}_{X_k}) \\ & = \sum\nolimits_j \left(-(D_j \cdot C_{i_j}) +  \chi(C_{i_j}, \mathcal{O}_{C_{i_j}})\right) \\ & = - \sum\nolimits_j (C_{i_1} + C_{i_2} + \ldots + C_{i_{j - 1}} \cdot C_{i_j}) + \sum\nolimits_j \chi(C_{i_j}, \mathcal{O}_{C_{i_j}}) \\ & = -\frac{1}{2}\sum\nolimits_{j \not = j'} (C_{i_{j'}} \cdot C_{i_j}) + \sum m_i \chi(C_i, \mathcal{O}_{C_i}) \\ & = \frac{1}{2} \sum m_i(C_i \cdot C_i) + \sum m_i \chi(C_i, \mathcal{O}_{C_i}) \end{align*} Perhaps the last equality deserves some explanation. Namely, since $\sum_j C_{i_j} = \sum m_i C_i$ we have $(\sum_j C_{i_j} \cdot \sum_j C_{i_j}) = 0$ by Lemma \ref{lemma-properties-form}. Thus we see that $$ 0 = \sum\nolimits_{j \not = j'} (C_{i_{j'}} \cdot C_{i_j}) + \sum m_i(C_i \cdot C_i) $$ by splitting this product into ``nondiagonal'' and ``diagonal'' terms. Note that $\kappa_i$ is a field finite over $k$ by Varieties, Lemma \ref{varieties-lemma-regular-functions-proper-variety}. Hence the genus of $C_i$ is defined and we have $\chi(C_i, \mathcal{O}_{C_i}) = [\kappa_i : k](1 - g_i)$. Putting everything together and rearranging terms we get $$ g_C = - \frac{1}{2}\sum m_i(C_i \cdot C_i) + \sum m_i[\kappa_i : k](g_i - 1) + 1 $$ which is what the lemma says too.
(In the proof of Lemma \ref{lemma-genus-formula} we have seen that the quantities used in the statement of the lemma are well defined.) We have to verify the conditions (1) -- (5) of Definition \ref{definition-type}. \medskip\noindent Condition (1) is immediate. \medskip\noindent Condition (2). Symmetry of the matrix $(C_i \cdot C_j)$ follows from Equation (\ref{equation-form}) and Lemma \ref{lemma-intersection-pairing}. Nonnegativity of $(C_i \cdot C_j)$ for $i \not = j$ is part (1) of Lemma \ref{lemma-properties-form}. \medskip\noindent Condition (3) is part (3) of Lemma \ref{lemma-properties-form}. \medskip\noindent Condition (4) is part (2) of Lemma \ref{lemma-properties-form}. \medskip\noindent Condition (5) follows from the fact that $(C_i \cdot C_j)$ is the degree of an invertible module on $C_i$ which is divisible by $[\kappa_i : k]$, see Varieties, Lemma \ref{varieties-lemma-divisible}. \medskip\noindent The genus formula proved in Lemma \ref{lemma-genus-formula} tells us that the numerical type has the genus as stated, see Definition \ref{definition-genus}.
If the numerical type is minimal, then there is no $i$ with $g_i = 0$ and $(C_i \cdot C_i) = -[\kappa_i: k]$, see Definition \ref{definition-type-minimal}. Certainly, this implies that none of the curves $C_i$ are exceptional curves of the first kind. \medskip\noindent Conversely, suppose that the numerical type is not minimal. Then there exists an $i$ such that $g_i = 0$ and $(C_i \cdot C_i) = -[\kappa_i: k]$. We claim this implies that $C_i$ is an exceptional curve of the first kind. Namely, the invertible sheaf $\mathcal{O}_X(-C_i)|_{C_i}$ has degree $-(C_i \cdot C_i) = [\kappa_i : k]$ when $C_i$ is viewed as a proper curve over $k$, hence has degree $1$ when $C_i$ is viewed as a proper curve over $\kappa_i$. Applying Algebraic Curves, Proposition \ref{curves-proposition-projective-line} we conclude that $C_i \cong \mathbf{P}^1_{\kappa_i}$ as schemes over $\kappa_i$. Since the Picard group of $\mathbf{P}^1$ over a field is $\mathbf{Z}$, we see that the normal sheaf of $C_i$ in $X$ is isomorphic to $\mathcal{O}_{\mathbf{P}_{\kappa_i}}(-1)$ and the proof is complete.
Since $X \to \Spec(R)$ is proper, the $K$-rational point extends to a morphism $a : \Spec(R) \to X$ by the valuative criterion of properness (Morphisms, Lemma \ref{morphisms-lemma-characterize-proper}). Let $x \in X$ be the image under $a$ of the closed point of $\Spec(R)$. Then $a$ corresponds to an $R$-algebra homomorphism $\psi : \mathcal{O}_{X, x} \to R$ (see Schemes, Section \ref{schemes-section-points}). It follows that $\pi \not \in \mathfrak m_x^2$ (since the image of $\pi$ in $R$ is not in $\mathfrak m_R^2$). Hence $\mathcal{O}_{X_k, x} = \mathcal{O}_{X, x}/\pi \mathcal{O}_{X, x}$ is regular (Algebra, Lemma \ref{algebra-lemma-regular-ring-CM}). Then $X_k \to \Spec(k)$ is smooth at $x$ by Algebra, Lemma \ref{algebra-lemma-separable-smooth}. It follows that $x$ is contained in a unique irreducible component $C_i$ of $X_k$, that $\mathcal{O}_{C_i, x} = \mathcal{O}_{X_k, x}$, and that $m_i = 1$. The fact that $C_i$ has a $k$-rational point implies that the field $\kappa_i = H^0(C_i, \mathcal{O}_{C_i})$ (Varieties, Lemma \ref{varieties-lemma-regular-functions-proper-variety}) is equal to $k$. This proves (1). We have $H^0(X_k, \mathcal{O}_{X_k}) = k$ because $H^0(X_k, \mathcal{O}_{X_k})$ is a field extension of $k$ (Lemma \ref{lemma-regular-model-field}) which maps to $H^0(C_i, \mathcal{O}_{C_i}) = k$. The genus equality follows from Lemma \ref{lemma-regular-model-genus}.
By vanishing of cohomology in degrees $\geq 2$ over $X_k$ (Cohomology, Proposition \ref{cohomology-proposition-vanishing-Noetherian}) any surjection of abelian sheaves on $X_k$ induces a surjection on $H^1$. Consider the sequence $$ (X_k)_{red} = Z_0 \subset Z_1 \subset \ldots \subset Z_m = X_k $$ of Lemma \ref{lemma-regular-model-field}. Since the field maps $H^0(Z_j, \mathcal{O}_{Z_j}) \to H^0((X_k)_{red}, \mathcal{O}_{(X_k)_{red}}) = k$ are injective we conclude that $H^0(Z_j, \mathcal{O}_{Z_j}) = k$ for $j = 0, \ldots, m$. It follows that $H^0(X_k, \mathcal{O}_{X_k}) \to H^0(Z_{m - 1}, \mathcal{O}_{Z_{m - 1}})$ is surjective. Let $C = C_{i_m}$. Then $X_k = Z_{m - 1} + C$. Let $\mathcal{L} = \mathcal{O}_X(-Z_{m - 1})|_C$. Then $\mathcal{L}$ is an invertible $\mathcal{O}_C$-module. As in the proof of Lemma \ref{lemma-regular-model-field} there is an exact sequence $$ 0 \to \mathcal{L} \to \mathcal{O}_{X_k} \to \mathcal{O}_{Z_{m - 1}} \to 0 $$ of coherent sheaves on $X_k$. We conclude that we get a short exact sequence $$ 0 \to H^1(C, \mathcal{L}) \to H^1(X_k, \mathcal{O}_{X_k}) \to H^1(Z_{m - 1}, \mathcal{O}_{Z_{m - 1}}) \to 0 $$ The degree of $\mathcal{L}$ on $C$ over $k$ is $$ (C \cdot -Z_{m - 1}) = (C \cdot C - X_k) = (C \cdot C) $$ Set $\kappa = H^0(C, \mathcal{O}_C)$ and $w = [\kappa : k]$. By definition of the degree of an invertible sheaf we see that $$ \chi(C, \mathcal{L}) = \chi(C, \mathcal{O}_C) + (C \cdot C) = w(1 - g_C) + (C \cdot C) $$ where $g_C$ is the genus of $C$. This expression is $< 0$ as $X$ is minimal and hence $C$ is not an exceptional curve of the first kind (see proof of Lemma \ref{lemma-numerical-type-minimal-model}). Thus $\dim_k H^1(C, \mathcal{L}) > 0$ which finishes the proof.
Looking over the proof of Lemma \ref{lemma-genus-reduction-bigger-than} we see that in order to get equality, the inequalities (\ref{equation-genus-change-special-component}), (\ref{equation-genus-change}), and (\ref{equation-change-intersections}) have to be equalities. \medskip\noindent Let $C_i$ be the irreducible component containing $x$. Equality in (\ref{equation-genus-change-special-component}) shows via Algebraic Curves, Lemma \ref{curves-lemma-genus-normalization} that $C_{i, \overline{k}}^\nu \to C_{i, \overline{k}}$ is an isomorphism. Hence $C_{i, \overline{k}}$ is smooth and part (1) holds. \medskip\noindent Next, let $C_i \subset X_k$ be another irreducible component. Then we may assume we have $D =  D' + C_i$ as in the induction step in the proof of Lemma \ref{lemma-genus-reduction-bigger-than}. Equality in (\ref{equation-genus-change}) immediately implies that $\kappa_i/k$ is finite separable. Equality in (\ref{equation-change-intersections}) implies either $a_{ij} = 1$ for some $j$ or that there is a unique $C_j \subset D'$ meeting $C_i$ and $a_{ij} = w_i$. In both cases we find that $C_i$ has a $\kappa_i$-rational point $c$ and $c = C_i \cap C_j$ scheme theoretically. Since $\mathcal{O}_{X, c}$ is a regular local ring, this implies that the local equations of $C_i$ and $C_j$ form a regular system of parameters in the local ring $\mathcal{O}_{X, c}$. Then $\mathcal{O}_{C_i, c}$ is regular by (Algebra, Lemma \ref{algebra-lemma-regular-ring-CM}). We conclude that $C_i \to \Spec(\kappa_i)$ is smooth at $c$ (Algebra, Lemma \ref{algebra-lemma-separable-smooth}). It follows that $C_i$ is geometrically integral over $\kappa_i$ (Varieties, Lemma \ref{varieties-lemma-variety-with-smooth-rational-point}). To finish we have to show that $C_i$ is smooth over $\kappa_i$. Observe that $$ C_{i, \overline{k}} = C_i \times_{\Spec(k)} \Spec(\overline{k}) = \coprod\nolimits_{\kappa_i \to \overline{k}} C_i \times_{\Spec(\kappa_i)} \Spec(\overline{k}) $$ where there are $[\kappa_i : k]$-summands. Thus if $C_i$ is not smooth over $\kappa_i$, then each of these curves is not smooth, then these curves are not normal and the normalization morphism drops the genus (Algebraic Curves, Lemma \ref{curves-lemma-genus-normalization}) which is disallowed because it would drop the geometric genus of $C_i/k$ contradicting $[\kappa_i : k] g_i = g_{geom}(C_i/k)$.
By Resolution of Surfaces, Lemma \ref{resolve-lemma-contract-ample} we can contract $C_n$ by a morphism $f : X \to X'$ such that $X'$ is regular and is projective over $R$. Thus we see that $X'$ is as in Situation \ref{situation-regular-model}. Let $x \in X'$ be the image of $C_n$. Since $f$ defines an isomorphism $X \setminus C_n \to X' \setminus \{x\}$ it is clear that $m'_i = m_i$ for $i < n$. \medskip\noindent Part (2) of the lemma is immediately clear from the existence of the morphisms $C_i \to C'_i$ and $C_n \to x \to C'_i$. \medskip\noindent By Divisors, Lemma \ref{divisors-lemma-blow-up-pullback-effective-Cartier} the pullback $f^{-1}C'_i$ is defined. By Divisors, Lemma \ref{divisors-lemma-effective-Cartier-divisor-is-a-sum} we see that $f^{-1}C'_i = C_i + e_i C_n$ for some $e_i \geq 0$. Since $\mathcal{O}_X(C_i + e_i C_n) = \mathcal{O}_X(f^{-1}C'_i) = f^*\mathcal{O}_{X'}(C'_i)$ (Divisors, Lemma \ref{divisors-lemma-pullback-effective-Cartier-divisors}) and since the pullback of an invertible sheaf restricts to the trivial invertible sheaf on $C_n$ we see that $$ 0 = \deg_{C_n}(\mathcal{O}_X(C_i + e_i C_n)) = (C_i + e_i C_n \cdot C_n) = (C_i \cdot C_n) + e_i(C_n \cdot C_n) $$ As $f_j = f|_{C_j} : C_j \to C_j$ is a proper birational morphism of proper curves over $k$, we see that $\deg_{C'_j}(\mathcal{O}_{X'}(C'_i)|_{C'_j})$ is the same as $\deg_{C_j}(f_j^*\mathcal{O}_{X'}(C'_i)|_{C'_j})$ (Varieties, Lemma \ref{varieties-lemma-degree-birational-pullback}). Looking at the commutative diagram $$ \xymatrix{ C_j \ar[r] \ar[d]_{f_j} & X \ar[d]^f \\ C'_j \ar[r] & X' } $$ and using Divisors, Lemma \ref{divisors-lemma-pullback-effective-Cartier-divisors} we see that $$ (C'_i \cdot C'_j) = \deg_{C'_j}(\mathcal{O}_{X'}(C'_i)|_{C'_j}) = \deg_{C_j}(\mathcal{O}_X(C_i + e_i C_n)) = (C_i + e_i C_n \cdot C_j) $$ Plugging in the formula for $e_i$ found above we see that (1) holds.
By Lemma \ref{lemma-regular-model-pic} we have $\mathcal{L} \cong \mathcal{O}_X(\sum a_i C_i)$ for some $a_i \in \mathbf{Z}$. The degree of $\mathcal{L}|_{C_j}$ is $\sum_j a_j(C_i \cdot C_j)$. In particular $(\sum a_i C_i \cdot \sum a_i C_i) = 0$. Hence we see from Lemma \ref{lemma-properties-form} that $(a_1, \ldots, a_n) = q(m_1, \ldots, m_n)$ for some $q \in \mathbf{Q}$. Thus $\mathcal{L} = \mathcal{O}_X(lD)$ for some $l \in \mathbf{Z}$ where $D = \sum (m_i/d) C_i$ is as in Lemma \ref{lemma-multiple-fibre-normal-bundle} and we conclude.
Recall that $w_i = [\kappa_i : k]$ where $\kappa_i = H^0(C_i, \mathcal{O}_{C_i)})$ and recall that the degree of any invertible module on $C_i$ is divisible by $w_i$ (Varieties, Lemma \ref{varieties-lemma-divisible}). Thus we can consider the map $$ \frac{\deg}{w} : \Pic(X) \to \mathbf{Z}^{\oplus n}, \quad \mathcal{L} \mapsto (\frac{\deg(\mathcal{L}|_{C_1})}{w_1}, \ldots, \frac{\deg(\mathcal{L}|_{C_n})}{w_n}) $$ The image of $\mathcal{O}_X(C_j)$ under this map is $$ ((C_j \cdot C_1)/w_1, \ldots, (C_j \cdot C_n)/w_n) = (a_{1j}/w_1, \ldots, a_{nj}/w_n) $$ which is exactly the image of the $j$th basis vector under the map $(a_{ij}/w_i) : \mathbf{Z}^{\oplus n} \to \mathbf{Z}^{\oplus n}$ defining the Picard group of $T$, see Definition \ref{definition-picard-group}. Thus the canonical map of the lemma comes from the commutative diagram $$ \xymatrix{ \mathbf{Z}^{\oplus n} \ar[r] \ar[d]_{\text{id}} & \Pic(X) \ar[r] \ar[d]^{\frac{\deg}{w}} & \Pic(C) \ar[r] \ar[d] & 0 \\ \mathbf{Z}^{\oplus n} \ar[r]^{(a_{ij}/w_i)} & \mathbf{Z}^{\oplus n} \ar[r] & \Pic(T) \ar[r] & 0 } $$ with exact rows (top row by Lemma \ref{lemma-regular-model-pic}). The description of the kernel is clear.
Taking $h$-torsion in the exact sequence of Lemma \ref{lemma-regular-model-pic} we obtain the exactness of $0 \to \Pic(X)[h] \to \Pic(C)[h]$ because $h$ is prime to $d$. Using the map Lemma \ref{lemma-characterize-trivial} we get a map $\Pic(C)[h] \to \Pic(T)[h]$ which annihilates elements of $\Pic(X)[h]$. Conversely, if $\xi \in \Pic(C)[h]$ maps to zero in $\Pic(T)[h]$, then we can find an invertible $\mathcal{O}_X$-module $\mathcal{L}$ with $\deg(\mathcal{L}|_{C_i}) = 0$ for all $i$ whose restriction to $C$ is $\xi$. Then $\mathcal{L}^{\otimes h}$ is $d$-torsion by Lemma \ref{lemma-characterize-trivial}. Let $d'$ be an integer such that $dd' \equiv 1 \bmod h$. Such an integer exists because $h$ and $d$ are coprime. Then $\mathcal{L}^{\otimes dd'}$ is an $h$-torsion invertible sheaf on $X$ whose restriction to $C$ is $\xi$.
Observe that $X \times_{\Spec(R)} \Spec(R/\pi^n)$ is a finite order thickening of $(X_k)_{red}$ (this follows for example from Cohomology of Schemes, Lemma \ref{coherent-lemma-power-ideal-kills-sheaf}). Thus the canonical map $\Pic(X \times_{\Spec(R)} \Spec(R/\pi^n)) \to \Pic((X_k)_{red})$ identifies $h$ torsion by More on Morphisms, Lemma \ref{more-morphisms-lemma-torsion-pic-thickening} and our assumption on $h$. Thus if $\mathcal{L}$ is an $h$-torsion invertible sheaf on $X$ which restricts to the trivial sheaf on $(X_k)_{red}$ then $\mathcal{L}$ restricts to the trivial sheaf on $X \times_{\Spec(R)} \Spec(R/\pi^n)$ for all $n$. We find \begin{align*} H^0(X, \mathcal{L})^\wedge & = \lim H^0(X \times_{\Spec(R)} \Spec(R/\pi^n), \mathcal{L}|_{X \times_{\Spec(R)} \Spec(R/\pi^n)}) \\ & \cong \lim H^0(X \times_{\Spec(R)} \Spec(R/\pi^n), \mathcal{O}_{X \times_{\Spec(R)} \Spec(R/\pi^n)}) \\ & = R^\wedge \end{align*} using the theorem on formal functions (Cohomology of Schemes, Theorem \ref{coherent-theorem-formal-functions}) for the first and last equality and for example More on Algebra, Lemma \ref{more-algebra-lemma-isomorphic-completions} for the middle isomorphism. Since $H^0(X, \mathcal{L})$ is a finite $R$-module and $R$ is a discrete valuation ring, this means that $H^0(X, \mathcal{L})$ is free of rank $1$ as an $R$-module. Let $s \in H^0(X, \mathcal{L})$ be a basis element. Then tracing back through the isomorphisms above we see that $s|_{X \times_{\Spec(R)} \Spec(R/\pi^n)}$ is a trivialization for all $n$. Since the vanishing locus of $s$ is closed in $X$ and $X \to \Spec(R)$ is proper we conclude that the vanishing locus of $s$ is empty as desired.
We have already proved this lemma in much greater generality, see Algebraic Curves, Lemma \ref{curves-lemma-etale-local-structure-nodal-family}. All we have to do here is to translate the statement given there into the statement given above. \medskip\noindent First, if the morphism $X \to \Spec(R)$ is smooth at $x$, then we can find an \'etale morphism $U \to \mathbf{A}^1_R = \Spec(R[u])$ for some affine open neighbourhood $U \subset X$ of $x$. This is Morphisms, Lemma \ref{morphisms-lemma-smooth-etale-over-affine-space}. After replacing the coordinate $u$ by $u + 1$ if necessary, we may assume that $x$ maps to a point in the standard open $D(u) \subset \mathbf{A}^1_R$. Then $D(u) = \Spec(A)$ with $A = R[u, v]/(uv - 1)$ and we see that the result is true in this case. \medskip\noindent Next, assume that $x$ is a singular point of the fibre. Then we may apply Algebraic Curves, Lemma \ref{curves-lemma-etale-local-structure-nodal-family} to get a diagram $$ \xymatrix{ X \ar[d] & U \ar[rr] \ar[l] \ar[rd] & & W \ar[r] \ar[ld] & \Spec(\mathbf{Z}[u, v, a]/(uv - a)) \ar[d] \\ \Spec(R) & & V \ar[ll] \ar[rr] & & \Spec(\mathbf{Z}[a]) } $$ with all the properties mentioned in the statement of the cited lemma. Let $x' \in U$ be the point mapping to $x$ promised by the lemma. First we shrink $V$ to an affine neighbourhood of the image of $x'$. Say $V = \Spec(R')$. Then $R \to R'$ is \'etale. Since $R$ is a discrete valuation ring, we see that $R'$ is a finite product of quasi-local Dedekind domains (use More on Algebra, Lemma \ref{more-algebra-lemma-Dedekind-etale-extension}). Hence (for example using prime avoidance) we find a standard open $D(f) \subset V = \Spec(R')$ containing the image of $x'$ such that $R'_f$ is a discrete valuation ring. Replacing $R'$ by $R'_f$ we reach the situation where $V = \Spec(R')$ with $R \subset R'$ an \'etale extension of discrete valuation rings (extensions of discrete valuation rings are defined in More on Algebra, Definition \ref{more-algebra-definition-extension-discrete-valuation-rings}). \medskip\noindent The morphism $V \to \Spec(\mathbf{Z}[a])$ is determined by the image $h$ of $a$ in $R'$. Then $W = \Spec(R'[u, v]/(uv - h))$. Thus the lemma holds with $A = R'[u, v]/(uv - h)$. If $h = 0$ then we clearly obtain the first case mentioned in the lemma. If $h \not = 0$ then we may write $h = \epsilon \pi^n$ for some $n \geq 0$ where $\epsilon$ is a unit of $R'$. Changing coordinates $u_{new} = \epsilon u$ and $v_{new} = v$ we obtain the second isomorphism type of $A$ listed in the lemma.
Namely, let $x' \in X'$ be the image of $E$. Then the only issue is to see that $X' \to \Spec(R)$ is at-worst-nodal of relative dimension $1$ in a neighbourhood of $x'$. The closed fibre of $X \to \Spec(R)$ is reduced, hence $\pi \in R$ vanishes to order $1$ on $E$. This immediately implies that $\pi$ viewed as an element of $\mathfrak m_{x'} \subset \mathcal{O}_{X', x'}$ but is not in $\mathfrak m_{x'}^2$. Since $\mathcal{O}_{X', x'}$ is regular of dimension $2$ (by definition of contractions in Resolution of Surfaces, Section \ref{resolve-section-minus-one}), this implies that $\mathcal{O}_{X'_k, x'}$ is regular of dimension $1$ (Algebra, Lemma \ref{algebra-lemma-regular-ring-CM}). On the other hand, the curve $E$ has to meet at least one other component, say $C$ of the closed fibre $X_k$. Say $x \in E \cap C$. Then $x$ is a node of the special fibre $X_k$ and hence $\kappa(x)/k$ is finite separable, see Algebraic Curves, Lemma \ref{curves-lemma-nodal}. Since $x \mapsto x'$ we conclude that $\kappa(x')/k$ is finite separable. By Algebra, Lemma \ref{algebra-lemma-separable-smooth} we conclude that $X'_k \to \Spec(k)$ is smooth in an open neighbourhood of $x'$. Combined with flatness, this proves that $X' \to \Spec(R)$ is smooth in a neighbourhood of $x'$ (Morphisms, Lemma \ref{morphisms-lemma-smooth-at-point}). This finishes the proof as a smooth morphism of relative dimension $1$ is at-worst-nodal of relative dimension $1$ (Algebraic Curves, Lemma \ref{curves-lemma-smooth-relative-dimension-1}).
To make sense out of this statement, recall that a minimal model is defined as a regular proper model without exceptional curves of the first kind (Definition \ref{definition-minimal-model}), that minimal models exist (Proposition \ref{proposition-exists-minimal-model}), and that minimal models are unique if the genus of $C$ is $> 0$ (Lemma \ref{lemma-minimal-model-unique}). Keeping this in mind the implications (2) $\Rightarrow$ (1) and (3) $\Rightarrow$ (2) are clear. \medskip\noindent Assume (1). Let $X$ be a proper model of $C$ which is at-worst-nodal of relative dimension $1$ over $R$. Applying Lemma \ref{lemma-blowup-at-worst-nodal} we see that we may assume $X$ is regular as well. Let $$ X = X_m \to X_{m - 1} \to \ldots \to X_1 \to X_0 $$ be as in Lemma \ref{lemma-pre-exists-minimal-model}. By Lemma \ref{lemma-blowdown-at-worst-nodal} and induction this implies $X_0$ is at-worst-nodal of relative dimension $1$ over $R$. \medskip\noindent To finish the proof we have to show that (2) implies (3). This is clear if the genus of $C$ is $> 0$, since then the minimal model is unique (see discussion above). On the other hand, if the minimal model is not unique, then the morphism $X \to \Spec(R)$ is smooth for any minimal model as its special fibre will be isomorphic to $\mathbf{P}^1_k$ by Lemma \ref{lemma-nonuniqueness}.
If $X$ is a smooth proper model, then the special fibre is connected (Lemma \ref{lemma-regular-model-connected}) and smooth, hence irreducible. This immediately implies that it is minimal. Thus (1) implies (2).  To finish the proof we have to show that (2) implies (3). This is clear if the genus of $C$ is $> 0$, since then the minimal model is unique (Lemma \ref{lemma-minimal-model-unique}). On the other hand, if the minimal model is not unique, then the morphism $X \to \Spec(R)$ is smooth for any minimal model as its special fibre will be isomorphic to $\mathbf{P}^1_k$ by Lemma \ref{lemma-nonuniqueness}.
Say $T$ is given by $n, m_i, a_{ij}, w_i, g_i$. If $T$ is not minimal, then there exists a $(-1)$-index. After replacing $T$ by an equivalent type we may assume $n$ is a $(-1)$-index. Applying Lemma \ref{lemma-contract-picard-group} we find $\Pic(T) \subset \Pic(T')$ where $T'$ is a numerical type of genus $g$ (Lemma \ref{lemma-contract}) with $n - 1$ indices. Thus we conclude by induction on $n$ provided we prove the lemma for minimal numerical types. \medskip\noindent Assume that $T$ is a minimal numerical type of genus $\geq 2$. Observe that $g_{top} \leq g$ by Lemma \ref{lemma-genus-nonnegative}. If $A = (a_{ij})$ then since $\Pic(T) \subset \Coker(A)$ by Lemma \ref{lemma-picard-T-and-A}. Thus it suffices to prove the lemma for $\Coker(A)$. By Lemma \ref{lemma-bound-wm} we see that $m_i|a_{ij}| \leq 768g$ for all $i, j$. Hence the result by Lemma \ref{lemma-recurring-symmetric-integer}.
Choose a closed immersion $C \to \mathbf{P}^n_K$. Let $X$ be the scheme theoretic image of $C \to \mathbf{P}^n_R$. Then $X \to \Spec(R)$ is a projective model of $C$ by Lemma \ref{lemma-closure-is-model}. By Lemma \ref{lemma-regular} there exists a resolution of singularities $X' \to X$ and $X'$ is a model for $C$. Then $X' \to \Spec(R)$ is proper as a composition of proper morphisms. Then we may apply Lemma \ref{lemma-pre-exists-minimal-model} to obtain a minimal model.
We recall that by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-section-immersion} we have that $e$ is a closed immersion (resp.\ quasi-compact, resp.\ an immersion) if $G \to B$ is separated (resp.\ quasi-separated, resp.\ locally separated). For the converse, consider the diagram $$ \xymatrix{ G \ar[r]_-{\Delta_{G/B}} \ar[d] & G \times_B G \ar[d]^{(g, g') \mapsto m(i(g), g')} \\ B \ar[r]^e & G } $$ It is an exercise in the functorial point of view in algebraic geometry to show that this diagram is cartesian. In other words, we see that $\Delta_{G/B}$ is a base change of $e$. Hence if $e$ is a closed immersion (resp.\ quasi-compact, resp.\ an immersion) so is $\Delta_{G/B}$, see Spaces, Lemma \ref{spaces-lemma-base-change-immersions} (resp.\ Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-quasi-compact}, resp.\ Spaces, Lemma \ref{spaces-lemma-base-change-immersions}).