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I have a question about several related notions in algebraic geometry. I am mainly interested in the question "what is the standard notion?" (if there is such). But I also will be happy to know what is the "natural boundary" of the definition.

  1. The canonical divisor: I saw a definition for a normal variety of  the canonical divisor as a Weil  divisor (closure of the canonical divisor on the regular set). Is this notion defined in the non-Normal case? Up to what this notion well defined?
  2. Quasi-Gorenstein/$\mathbb Q$-Gorenstein variety: I saw a definition for a normal variety that is based on the above notion of canonical divisor. Do those notions require Normality? This looks weird since it would mean that Gorenstein is not necessarily Quasi-Gorenstein or $\mathbb Q$-Gorenstein (as it is not necessarily normal)
  3. (log-)canonical/terminal, Fano, Calabi-Yaw, General type variety: I saw a definition for a normal variety that is based on the above notion of canonical divisor. However it uses the word "ample" which I do not understand when it refers to a general Weil divisor . Does this mean that these notions are defined only for $\mathbb Q$-Gorenstein varieties?
  4. Do I understand correctly that though the notions of (log-)canonical/terminal and rational singularities have anlaoges for positive characteristic, non of these analogs are considered standard. Thanks a lot
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  • $\begingroup$ Definitely the Mori program does make use of the dualizing sheaf in the non-normal case, since in many instances (e.g., inversion of adjunction), the proofs proceed by induction where the induction step passes to a simple normal crossings divisor (or something weaker, e.g., semi-log-canonical divisor) that need not be normal. In most instances, the reflexive hull of some positive tensor power of the dualizing sheaf is an invertible sheaf, so that we can treat the dualizing sheaf as a Weil divisor that is $\mathbb{Q}$-Cartier. $\endgroup$
    – Jason Starr
    Commented Oct 24 at 11:28
  • $\begingroup$ I did not understand the last sentence. Can you elaborate? $\endgroup$
    – Rami
    Commented Oct 25 at 2:50
  • $\begingroup$ The Mori program (as originally conceived) applies to projective schemes that are Cohen-Macaulay (so that they have a coherent dualizing sheaf $\omega$) such that, for some positive integer $n$, the double dual of $\omega^{\otimes n}$ is an invertible sheaf. Then this dualizing sheaf plays the role of a canonical divisor class. When the scheme is also normal, there is an associated Weil divisor class of the dualizing sheaf, and $n$ times that Weil divisor class is the Weil divisor class of a Cartier divisor. $\endgroup$
    – Jason Starr
    Commented Oct 25 at 11:25
  • $\begingroup$ Thank you for the explanations. I have more followup questions: What is double dual of a sheaf? What is "Weil divisor class"? Are you proposing the following definition to Q-Gorenstein for Cohen-Macaulay variety: "the double dual of a power of the dualizing sheaf is invertible"? Does this definition appear in the literature? So, is the notion of Q-Gorenstein defined either for normal varieties or for Cohen-Macaulay varieties? Is there a natural joint generalization of these 2 frameworks? $\endgroup$
    – Rami
    Commented Oct 25 at 12:43
  • $\begingroup$ Double dual means applying $\mathcal{H}\mathrm{om}(-, \mathcal{O}_X)$ twice. Weil divisor class the class of Weil divisors up to linear equivalence. Yes, that definition appears in the literature for normal schemes, and even when the scheme is not Cohen-Macaulay. I would say it's the standard definition. Q-Gorenstein is typically defined for normal varieties regardless of whether they are Cohen-Macaulay, however some people have an additional Cohen-Macaulay assumption. I don't think we have a framework outside of the normal case (or at least outside of the G1 + S2 case). $\endgroup$
    – Karl Schwede
    Commented Nov 1 at 14:08

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