Lemma 53.22.6. Let $k$ be a field. Let $X$ be a proper scheme over $k$ of dimension $1$ with $H^0(X, \mathcal{O}_ X) = k$. Assume the singularities of $X$ are at-worst-nodal. Consider a sequence

\[ X = X_0 \to X_1 \to \ldots \to X_ n = X' \]

of contractions of rational tails (Example 53.22.1) until none are left. Then

if the genus of $X$ is $0$, then $X'$ is an irreducible plane conic,

if the genus of $X$ is $1$, then $\omega _{X'} \cong \mathcal{O}_ X$,

if the genus of $X$ is $> 1$, then $\omega _{X'}^{\otimes m}$ is globally generated for $m \geq 2$.

If the genus of $X$ is $\geq 1$, then the morphism $X \to X'$ is independent of choices and formation of this morphism commutes with base field extensions.

**Proof.**
We proceed by contracting rational tails until there are none left. Then we see that (1), (2), (3) hold by Lemma 53.22.5.

Uniqueness. To see that $f : X \to X'$ is independent of the choices made, it suffices to show: any rational tail $C \subset X$ is mapped to a point by $X \to X'$; some details omitted. If not, then we can find a section $s \in \Gamma (X', \omega _{X'}^{\otimes 2})$ which does not vanish in the generic point of the irreducible component $f(C)$. Since in each of the contractions $X_ i \to X_{i + 1}$ we have a section $X_{i + 1} \to X_ i$, there is a section $X' \to X$ of $f$. Then we have an exact sequence

\[ 0 \to \omega _{X'} \to \omega _ X \to \omega _ X|_{X''} \to 0 \]

where $X'' \subset X$ is the union of the irreducible components contracted by $f$. See Lemma 53.4.6. Thus we get a map $\omega _{X'}^{\otimes 2} \to \omega _ X^{\otimes 2}$ and we can take the image of $s$ to get a section of $\omega _ X^{\otimes 2}$ not vanishing in the generic point of $C$. This is a contradiction with the fact that the restriction of $\omega _ X$ to a rational tail has negative degree (Lemma 53.22.2).

The statement on base field extensions follows from Lemma 53.22.3. Some details omitted.
$\square$

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