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Let $X$ be a toric variety. A resolution of singularities $f \colon Y \rightarrow X$ is called minimal if for every resolution $g \colon Z \rightarrow X$, there is a morphism $h \colon Z \rightarrow Y$ such that $f \circ h = g$.

I know and understand that there is a minimal resolution of singularities for toric surfaces. But what about toric varieties in general? I read that minimal resolutions of toric singularities in higher dimensions might not be unique. Does anybody know an example for a toric variety that admits two different minimal resoultions?

Do minimal resolutions of singularities of toric varieties always exist? And if no, is there a nice counterexample?

Thanks in advance!

claudi
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1 Answers1

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I think it's useful to separate the toric and non-toric aspects of the question.

The minimal resolution of singularities exsits for any surface, but in general there is no resolution that is minimal (in the sense of your defintion) for higher-dimensional varieties, because of the possibility of birational maps called flops.

Now the question is whether there are toric examples of flops, and the answer is that indeed there are. In fact the simplest example of a flop, the so-called Atiyah flop, is toric. The singular affine threefold $X=\{xy-zw\}=0$ has as its fan the "square" cone $S$ in $\mathbf{R}^3$. We get two different toric resolutions $X_i \rightarrow X \ (i=1,2)$ by subdividing $S$ into simplicial cones in two different ways, namely by including the two diagonals of the square. Note that the resulting fans are isomorphic, showing that the two varities $X_i$ are abstractly isomorphic, but they are not isomorphic over $X$.

  • Thanks a lot for your answer! I'm not sure whether I understood what you explained: Is $X={xy-zw=0}$ an example for a toric surface that has no minimal resoltion? – claudi Oct 21 '13 at 17:37
  • And what do you mean by "not isomorphic over X"? – claudi Oct 21 '13 at 17:38
  • Dear @claudi: no, as you noted every surface has a minimal resolution. My $X$ is a three-dimensional hypersurface in affine space $\mathbf{A}^4$ with a double point at $(0,0,0,0)$. To say that the $X_i$ are not isomorphic over $X$ means there is no isomorphism between them that makes the diagram $X_1 \rightarrow X \leftarrow X_2$ (in which both arrows are the resolution maps) commute. –  Oct 21 '13 at 19:42
  • oh I wanted to write "an example for a toric variety that has no minimal resolution", sorry for that mistake! Thanks for your explanation, it's very helpful! – claudi Oct 22 '13 at 08:22