# When does a morphism of varieties preserve codimension of points?

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?

• 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. – Lazzaro Campeotti Jun 16 '20 at 10:35
• @LazzaroCampeotti that looks like an answer to me - would you care to record it below? – KReiser Jun 16 '20 at 23:03

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.
• 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? – Luke Jun 17 '20 at 12:49