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Sorry for my poor English.

Let $X$ be a reducible projective variety.

My question is:

  1. How can I compute the dualizing sheaf of $X$ and express it in an explicit way?
  2. Is there a method to get dualizing sheaf of whole reducible variety $X$ from the information of dualizing sheaves of its irreducible components?

Currently I'm not concern the general case, but I want a few accessible concrete examples such as:

  1. reducible hypersurfaces,
  2. union of toric varieties glued at isomorphic orbits(Alexeev calls it stable toric variety).

The reason why I concern is to understand the limit in moduli spaces of stable pairs.

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2 Answers 2

up vote 10 down vote accepted

With regards part 2.

Let's assume that you have two components $X_1$ and $X_2$ (or even unions of components) such that $X_1 \cup X_2 = X$=. Let $I_1$ and $I_2$ denote the ideal sheaves of $X_1$ and $X_2$ in $X$.

Set $Z$ to be the scheme $X_1 \cap X_2$, in other words, the ideal sheaf of $Z$ is $I_1 + I_2$.

It is easy to see you have a short exact sequence $$0 \to I_1 \cap I_2 \to I_1 \oplus I_2 \to (I_1 + I_2) \to 0$$ where the third map sends $(a,b)$ to $a-b$.

The nine-lemma should imply that you have a short exact sequence

$$0 \to O_X \to O_{X_1} \oplus O_{X_2} \to O_Z \to 0$$

If you Hom this sequence into the dualizing complex of $X$, you get a triangle $$\omega_Z^. \to \omega_{X_1}^. \oplus \omega_{X_2}^. \to \omega_{X}^. \to \omega_Z^.[1]$$

You can then take cohomology and, depending on how things intersect (and what you understand about the intersection), possibly answer your question.

If $X_1$ and $X_2$ are hypersurfaces with no common components (which should imply everything in sight is Cohen-Macualay) then these dualizing complexes are all just sheaves (with various shifts), and you just get a short exact sequence $$0 \to \omega_{X_1} \oplus \omega_{X_2} \to \omega_{X} \to \omega_{Z} \to 0$$

Technically speaking, I should also probably push all these sheaves forward onto $X$ via inclusion maps.

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For hypersurfaces the answer is as follows. Assume that you have components $X_i$ of a reducible hypersurface $X$. Then $\omega_X|_{X_i} \cong \omega_{X_i}(\sum_{i \neq j} X_j ) := \omega_{X_i} \otimes \mathcal{O}(\sum_{i \neq j} X_j )$.

As for the proof: let $X':= \sum_{i \neq j} X_j$. Then we have a few equations:

$\omega_{X_i} \cong \omega_{P^n}(X_i)|_{X_i}$, $\omega_{X'} \cong \omega_{P^n}(X')|_{X'}$


$\omega_{X_i}(X') \cong \omega_{P^n}(X_i+ X')|_{X_i} \cong \left( \omega_{P^n} (X) |_X \right) $ $|_{X_i} \cong {\omega_X} |_{X_i}$

Concerning how you can get generally the canonical sheaf of a reduced scheme from its component, here is something very similar: if your scheme is semi-normal and $S_2$ (which is true for stable pairs), then when you take its normalization $\pi : \tilde{X} \to x$, the support of the points where $\pi$ is not an isomorphism, is a divisor, say $B$. Then you have the formula $\pi^* \omega_X \cong \omega_{\tilde{X}}(B)$. There is a little bit about this in the 4th section of http://arxiv.org/abs/0801.1541 .

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Thank you! I'll check the paper. –  Moon Apr 12 '10 at 1:58

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