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edited Apr 13, 2017 at 12:58

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@Andrea: Perhaps this example would be a good motivation (see also this post 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.

@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.

@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.

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created Oct 14, 2010 at 20:35

Sándor Kovács

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@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.