The Stacks project

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

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

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

  3. 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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