The Stacks project

Lemma 53.22.4. Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \mathcal{O}_ X) = k$. Assume the singularities of $X$ are at-worst-nodal. If $X$ does not have a rational tail (Example 53.22.1), then for every reduced connected closed subscheme $Y \subset X$, $Y \not= X$ of dimension $1$ we have $\deg (\omega _ X|_ Y) \geq \dim _ k H^1(Y, \mathcal{O}_ Y)$.

Proof. Let $Y \subset X$ be as in the statement. Then $k' = H^0(Y, \mathcal{O}_ Y)$ is a field and a finite extension of $k$ and $[k' : k]$ divides all numerical invariants below associated to $Y$ and coherent sheaves on $Y$, see Varieties, Lemma 33.44.10. Let $Z \subset X$ be as in Lemma 53.4.6. We will use the results of this lemma and of Lemmas 53.19.16 and 53.19.17 without further mention. Then we get a short exact sequence

\[ 0 \to \omega _ Y \to \omega _ X|_ Y \to \mathcal{O}_{Y \cap Z} \to 0 \]

See Lemma 53.4.6. We conclude that

\[ \deg (\omega _ X|_ Y) = \deg (Y \cap Z) + \deg (\omega _ Y) = \deg (Y \cap Z) - 2\chi (Y, \mathcal{O}_ Y) \]

Hence, if the lemma is false, then

\[ 2[k' : k] > \deg (Y \cap Z) + \dim _ k H^1(Y, \mathcal{O}_ Y) \]

Since $Y \cap Z$ is nonempty and by the divisiblity mentioned above, this can happen only if $Y \cap Z$ is a single $k'$-rational point of the smooth locus of $Y$ and $H^1(Y, \mathcal{O}_ Y) = 0$. If $Y$ is irreducible, then this implies $Y$ is a rational tail. If $Y$ is reducible, then since $\deg (\omega _ X|_ Y) = -[k' : k]$ we find there is some irreducible component $C$ of $Y$ such that $\deg (\omega _ X|_ C) < 0$, see Varieties, Lemma 33.44.6. Then the analysis above applied to $C$ gives that $C$ is a rational tail. $\square$

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