# Degree of finite morphisms of curves, and of their reduction “modulo p”

Let $K$ be a number field. Suppose that $$f: X \rightarrow Y$$ is a generically finite morphism of two-dimensional, normal schemes, which are projective and flat over $\operatorname{Spec} \mathcal{O}_K$. Assume further that the generic fibers $X_0,Y_0$ are smooth.

The morphism $$f_0: X_0 \rightarrow Y_0$$ is a finite morphism of smooth, projective curves, say of degree $n$. Furthermore, we can consider the base change $f_p: X_p \rightarrow Y_p$ for a prime $p \in \operatorname{Spec} \mathcal{O}_K \setminus \{0\}$. For all but finitely many primes $p$, this is also a finite morphism of smooth, projective curves. (right?)

Now to my question.
Is it true that for all but finitely many primes $p \in \operatorname{Spec} \mathcal{O}_K$, the morphism $f_p$ has degree $n$ as well?

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If you identify the degree with the rank of $f_*\mathcal{O}_X$ as an $\mathcal{O}_Y$-module, then it is not difficult to see that your question(s) has (or have) a positive answer. –  Donu Arapura May 14 '13 at 15:04
Thank you very much Donu! Your comment was very enlightening, not only in answering my question, but also in gaining a better overall perspective. –  Nils Matthes May 15 '13 at 18:29
You're welcome. –  Donu Arapura May 15 '13 at 19:16