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When I have a projective surface $X$, for simplicity smooth, I can find a simpler smooth surface on its binational class. In this way we find in a finite number of steps the simplest surface $Y$, i.e. a smooth surface birational equivalent to $X$ with no exceptional curves in the negative part of the Mori cone with respect the canonical divisor $K_Y$. This surface $Y$ is a ruled surface or its canonical divisor $K_Y$ is nef.

In the last case we define $Y$ as the minimal model of $X$ and one can prove this is the unique smooth surface on the birational class of $X$ with canonic divisor nef, up to isomorphism of surfaces.

By other side one can consider the canonical model of $X$, i.e

$$\operatorname{Proj}( \oplus_{n}H^0(X, K_X^n)) $$

and we define the canonical dimension, or Kodaira dimension of $X$ as the dimension of its canonical model.

If I understood in the right way, the canonical dimension of $X$ is always greater or equal than the canonical dimension. When the two dimensions are the same, i.e. the canonical dimension is equal to $2$, then $X$ is called surface of general type, right?

In this case I have two new surfaces, the minimal model $Y$ of $X$ and the canonical model. Is there a relationship between them?

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    $\begingroup$ Yes. The canonical model, say $\mathscr{X}$, is the same for all surfaces in the birational class of $X$, in particular for $Y$. Therefore there is a morphism $Y\rightarrow \mathscr{X}$; it is birational and contracts the $(-2)$-curves. $\endgroup$
    – abx
    Commented Sep 7, 2020 at 10:30
  • $\begingroup$ But $Y$ is the minimal model and $\mathscr{X}$ is smooth, so this would be $Y\cong \mathscr{X}$, right? $\endgroup$
    – Federico Fallucca
    Commented Sep 7, 2020 at 10:33
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    $\begingroup$ No, $\mathscr{X}$ is not smooth. It has rational singularities, which are resolved in $Y$. $\endgroup$
    – abx
    Commented Sep 7, 2020 at 10:34
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    $\begingroup$ And what is the reasoning for which one would consider a singular surface rather than the beautiful minimal smooth model $Y$? $\endgroup$
    – Federico Fallucca
    Commented Sep 7, 2020 at 10:36
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    $\begingroup$ Several possible reasons: 1) Rational singularities are not at all "bad" (by at least a great number of definitions of "bad") 2) The canonical ring is (at least sometimes) easier to compute. Useful if you want to have an explicit model for your surface. 3) Beauty is subjective :) (and so on...) $\endgroup$
    – Enrico
    Commented Sep 7, 2020 at 11:23

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