We just work over $\mathbb C$. Let $X \to X'$ be a resolution of singularities of a normal variety $X'$. Here $X'$ is singular and $X$ is nonsingular. We assume that $K_{X'}$ is $\mathbb Q$-Cartier so that the plurigenera of $X'$ can be defined.

I am wondering that what kind of singularities $X'$ can have such that the plurigenera of $X$ and $X'$ are equal? For example, what if $X'$ only has terminal singularities? What about any other types of singularities?

  • $\begingroup$ Could you please clarify your question. Are you asking for examples of types of singularities such that the plurigenera of $X'$ are equal to the plurigenera of some resolution of singularities $X$? $\endgroup$ – Jason Starr Mar 16 '13 at 14:15
  • $\begingroup$ Thanks Jason. That is just what I want to know. I have revised my question according to your suggestion. $\endgroup$ – Tong Mar 16 '13 at 17:28

This is actually a characterization of canonical singularities.

In fact, one has the following

Proposition. A normal variety $X'$ has only canonical singularities if and only if $X'$ is $\mathbf{Q}$-Cartier and for any proper birational morphism $f \colon X \to X'$ from a nonsingular variety $X$, the natural inclusion $$f_* \mathcal{O}_X(m K_X) \hookrightarrow \mathcal{O}_{X'}(m K_{X'})$$ is an isomorphism for any $m \in \mathbf{N}$.

See for instance [K. Matsuki, Introduction to the Mori Program, p. 168] or [M. Reid, Young Person's Guide to canonical Singularities, p. 355].


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