Minimal model which is necessarily singular

Question

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 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).

Answer 1

@Andrea: Perhaps this example would be a good motivation (see also this post). I am referring to your comment above to VA's answer.

I think this is example is due to Iitaka, or someone from his school, in any case it predates Miles Reid. Take a $3$-dimensional abelian variety $A$ and mod out by the involution $(−1)$. Resolve the resulting $64$ double points and call the result X. Then it is relatively easy to prove that $X$ is not birational to a smooth projective variety with a nef canonical bundle (the point is that you have to blow down the exceptional divisors over these $64$ double points, so the minimal model will be $A$ mod $(-1)$). I believe that at the time this example was thought of as proof that minimal models did not exist in higher dimensions, but then Reid and Mori realized that it only means that minimal models need not be smooth.

I think the right way to think about this is that minimal models have no worse than terminal singularities. It turns out that terminal singularities are smooth in codimension 2, so in particular a 2-dimensional terminal singularity is actually smooth. So, one could argue that even minimal models of surfaces have terminal singularities, that is, that's the natural class of singularities for a minimal model. It just so happens that in dimension 2, these singularities are indistinguishable from smooth points.

Answer 2

Minimal models of smooth varieties of general type differ by flops, and there are finitely many of them. In dimension 3, a flop does not change the type of the singularity. So if one minimal model is singular then all of them are.

So take any singular 3-fold with terminal singularities and ample $K_X$, and that would be an example of a minimal model which is always singular. E.g., how about starting with $X'=$ the hypersurface $x_1x_2-x_3x_4=0$ in $\mathbb P^4$, and take $X$ to be the double cover of $X'$ ramified in a smooth divisor $D$ of degree $>6$. Let $Y$ be a smooth model of $X$, for example its desingularization.

To obtain an example with fractional $K_X^3$, start with a singular 3-fold $X'$ with terminal singularities with fractional $K_{X'}^3$ and let $X$ be a double cover, as the above. (You want $2K_{X'}^3$ to be fractional in this case, since $K_X^3 = 2(K_{X'}+\frac12 D)^3$.)

There are many choices for $X'$. For example, an appropriate weighted projective space works. If you are lazy to do it yourself, search for Reid's list of 95 Fano 3-folds which are complete intersections in weighted projective spaces. Most of them have fractional $K_{X'}^3$.