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Fix a field $k$. Let $X,Y$ be normal integral $k$-schemes of finite type and $h: Y \to X$ a proper birational $k$-morphism. Moreover, assume that $X$ is Gorenstein, i.e. it admits a canonical divisor $K_X$ which is Cartier.

Let $K_X$ be a Cartier canonical divisor of $X$ and $K_Y$ a canonical divisor of $Y$, so the divisor $$K_Y-h^*K_X$$ can be considered. Its linear class is independent of the choices of $K_X$ and $K_Y$.

This in mind, the relative canonical divisor of $h$ is usually defined to be the unique divisor in $Y$ supported on the exceptional locus of $h$ which is linearly equivalent to $K_Y-h^*K_X$. My question is:

How do we prove the existence and uniqueness of such divisor?

I'm a bit stuck trying to prove this (especially uniqueness) and I can't find any reference in the literature...

I also have two side questions about the relative canonical divisor:

  1. When is it effective?
  2. Is it related to $\Omega^d_{Y/X}$? Here, $d=\dim X = \dim Y$.

Thank you in advance for your time! Any help will be appreciated!

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    $\begingroup$ As to why it is unique, notice that if if it exceptionally supported, this means that $K_Y$ and $K_X$ agree where $h$ is an isomorphism (in fact, if we pick $K_X$, there is a unique $K_Y$ with this property and visa versa, via an argument building upon the observation below). Indeed, suppose that $K'_Y = K_Y + \mathrm{Div}_Y(f)$ is another canonical divisor. Then its not difficult to see that $K'_X = K_X + \mathrm{Div}_X(f)$ is the unique canonical divisor that agrees with $K'_Y$ over where $h$ is an isomorphism. Via a direct computation, one see that $$K'_Y - h^* K'_X = K_Y - h^*K_X$$ $\endgroup$ Commented Mar 31, 2024 at 4:01
  • $\begingroup$ @KarlSchwede Oh, I see! Thank you :) Also, what about the two side questions? Could you tell me something about it? $\endgroup$ Commented Apr 1, 2024 at 8:18
  • $\begingroup$ I don't think I have a great answer for you on the last question. If $Y$ and $X$ are both smooth, the relative canonical can be computed as the Jacobian of the map (as in multivariable calculus over $\mathbb{C}$), which is I think is related to the sort of thing you want. Note, I'm not sure you want $\Omega_{Y/X}^d$ as the fibers of $Y \to X$ have dimension at most $d-1$. $\endgroup$ Commented Apr 2, 2024 at 18:59
  • $\begingroup$ For the relative canonical to be effective, that means that $X$ has "canonical singularities" (since Gorenstein, this = "rational singularities"). This isn't so helpful though, as this is basically the definition. Note, sometimes it is effective and sometimes it is not. A good example to think about is $k[x_1, ..., x_n]/(f)$ where $f$ is a homogeneous equation of degree $d$ with isolated singularity. This is canonical if and only if $d \leq n-1$. To see this, compute the relative canonical from the formula $(K_Z + Y')|_Y'$ where $Z$ is the blowup of the origin in $\mathbb{A}^2$ $\endgroup$ Commented Apr 2, 2024 at 19:02
  • $\begingroup$ @KarlSchwede Dear Karl, thank you for your comments! $\endgroup$ Commented Apr 5, 2024 at 8:55

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