Exercise 111.54.5. Let $k$ be a field. Let $\pi : X \to Y$ be a finite birational morphism of curves with $X$ a projective nonsingular curve over $k$. It follows from the material in the course that $Y$ is a proper curve and that $\pi $ is the normalization morphism of $Y$. We have also seen in the course that there exists a dense open $V \subset Y$ such that $U = \pi ^{-1}(V)$ is a dense open in $X$ and $\pi : U \to V$ is an isomorphism.

Show that there exists an effective Cartier divisor $D \subset X$ such that $D \subset U$ and such that $\mathcal{O}_ X(D)$ is ample on $X$.

Let $D$ be as in (1). Show that $E = \pi (D)$ is an effective Cartier divisor on $Y$.

Briefly indicate why

the map $\mathcal{O}_ Y \to \pi _*\mathcal{O}_ X$ has a coherent cokernel $Q$ which is supported in $Y \setminus V$, and

for every $n$ there is a corresponding map $\mathcal{O}_ Y(nE) \to \pi _*\mathcal{O}_ X(nD)$ whose cokernel is isomorphic to $Q$.

Show that $\dim _ k H^0(X, \mathcal{O}_ X(nD)) - \dim _ k H^0(Y, \mathcal{O}_ Y(nE))$ is bounded (by what?) and conclude that the invertible sheaf $\mathcal{O}_ Y(nE)$ has lots of sections for large $n$ (why?).

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