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The context of this question is my attempt to generalise the Hurwitz theorem to varieties that aren't necessarily curves. In the case of curves, we have that a closed point always maps to a closed point because we are generally only dealing with finite morphisms. So we get a morphism from a prime divisor to a prime divisor, which we can extend linearly with ramification indices.

But as soon as we leave the world of curves this seems more difficult. The situation I am in is if we have a normal variety $Y$ with a projective birational morphism from a smooth variety $\pi: X \rightarrow Y$. Now if $p \in X$ is the generic point of a codimension $1$ subset, then why should it be true that $f(p)$ also has codimension $1$?

My attempt so far is to notice that since the morphism is dominant we have an injective morphism on local rings $\mathcal{O}_{Y, f(p)} \rightarrow \mathcal{O}_{X, p}$ from which we get that $\dim \mathcal{O}_{Y, f(p)} \leq \dim \mathcal{O}_{X, p}$. But to obtain equality we would normally want the extension of local rings to be integral. Is there any way to deduce integrality from the fact that the morphism $\pi$ is proper? If it were affine I think we could, because we would have a univerally closed morphism of affine schemes. But in this case it doesn't seem so obvious. Is it even true?

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    $\begingroup$ This is false for any dimension bigger than 1: take $f: X \rightarrow Y$ to be the blowup of a smooth point, and $p$ the generic point of the exceptional divisor. $\endgroup$ – Lazzaro Campeotti Jun 16 '20 at 10:35
  • $\begingroup$ @LazzaroCampeotti that looks like an answer to me - would you care to record it below? $\endgroup$ – KReiser Jun 16 '20 at 23:03
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This is false for any dimension bigger than 1: take $f:X \rightarrow Y$ to be the blowup of a smooth point, and $p$ the generic point of the exceptional divisor.

On the other hand, you say you are interested in the situation when $f: X \rightarrow Y$ is a birational morphism to a normal variety. Understanding the relationship between the canonical divisors of $X$ and $Y$ in this case is a fundamental issue in minimal model theory. Under suitable assumptions, there is a formula

$$K_X = f^* K_Y + \sum_i a_i E_i$$ where the sum is over all prime divisors on $X$ contracted by $f$, and the coefficients $a_i$ is a rational number called the discrepancy of $E_i$. The values of the $a_i$ depend on the singularities of $Y$, and imposing various bounds on the $a_i$ defines important classes of singularities such as terminal, canonical, and log canonical.

More details can be found in various texts such as those of Koll'ar--Mori, Matsuki, Debarre...

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  • $\begingroup$ Thank you @Lazzaro, in fact my question was noticed by studying some introductory lectures for the minimal model program where I had seen this formula. I have done some thinking in the meantime and I was wondering what exactly those suitable assumptions are. Intuitively it seems like we need the morphism to be sufficiently "nice" outside of a pure codimension $1$ set. What exactly is "nice"? And am I wrong to think that formula is a kind of generalized Hurwitz formula, or am I missing the point entirely? $\endgroup$ – Luke Jun 17 '20 at 12:49
  • $\begingroup$ Dear Luke, the terminology "Riemann--Hurwitz" is usually reserved for finite morphisms. In this context there is indeed a higher-dimensional version of the formula for curves: see for example Theorem 2.14 of math.mit.edu/~mckernan/Teaching/07-08/Autumn/18.735/l_2.pdf $\endgroup$ – Lazzaro Campeotti Jun 17 '20 at 21:38
  • $\begingroup$ As for the suitable assumptions, I suggest you consult one of the references I gave above. They can give a more accurate and enlightening explanation than I can. $\endgroup$ – Lazzaro Campeotti Jun 17 '20 at 21:44

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