2 added 604 characters in body

I was told during a summer school on the MMP a nice example (which I have also mentioned here on MO) that I'm not able to figure out anymore.

The example (due, I think, to Miles Reid) is a smooth compact threefold $X$ such that the numbero number of sections of $\mathcal{O}_X(m K_X)$ grows like $m^3/4$ (if I recall correctly). The nice thing about this example is the following.

Assume $X$ is birational to a smooth variety $Y$ such that $K_Y$ is nef. Then sections of $K_Y$ grow in the same fashion, in particular $K_Y$ is big, so by Kawamata-Viehweg vanishing we have $h^0 (Y, \mathcal{O}_Y(m K_Y)) = \chi(Y, \mathcal{O}_Y(m K_Y)) \sim m^3/4$, and by Riemann-Roch we find $K_Y^3 = 3/2$, which is not possible, since that number must be integer.

So, if one wants to have a minimal model for $X$, one has to allow singular varieties into the picture.

Can anyone tell me how the variety $X$ is obtained (or another example in a similar flavour)?

EDIT: As I wrote in the comments to VA's answer, I'm looking for an elementary example, where $\mathcal{O}_X(m K_X)$ can be computed and compared to the Riemann-Roch expansion, in order to have a completely intersection-theoretic argument. In particular I'd like to avoid using the concepts of canonical and terminal singularities, since I view this example mainly as a motivation to introduce exactly those concepts. It would also be nice if one could directly find a smooth $X$, rather then using Hironaka to resolve a singular variety with fractional $K_Y^3$ (which one secretly knows is terminal).

1

# Minimal model which is necessarily singular

I was told during a summer school on the MMP a nice example (which I have also mentioned here on MO) that I'm not able to figure out anymore.

The example (due, I think, to Miles Reid) is a smooth compact threefold $X$ such that the numbero of sections of $\mathcal{O}_X(m K_X)$ grows like $m^3/4$ (if I recall correctly). The nice thing about this example is the following.

Assume $X$ is birational to a smooth variety $Y$ such that $K_Y$ is nef. Then sections of $K_Y$ grow in the same fashion, in particular $K_Y$ is big, so by Kawamata-Viehweg vanishing we have $h^0 (Y, \mathcal{O}_Y(m K_Y)) = \chi(Y, \mathcal{O}_Y(m K_Y)) \sim m^3/4$, and by Riemann-Roch we find $K_Y^3 = 3/2$, which is not possible, since that number must be integer.

So, if one wants to have a minimal model for $X$, one has to allow singular varieties into the picture.

Can anyone tell me how the variety $X$ is obtained (or another example in a similar flavour)?