Suppose X is a one dimension torus and L is a line bundle over X, I think one class of log canonical surface singularity comes from the contraction of one elliptic curve from the total space L. My question is: is the contraction just contraction of the zero section? And do we need the line bundle to be ample?Thanks a lot.

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    $\begingroup$ I am not sure why people are downvoting. In order to contract the zero section, you need that the normal bundle of the zero section is the dual invertible sheaf of an ample invertible sheaf. In that case, the contraction of the zero section does give an isolated surface singularity that is log canonical. A great reference is Section 4.4 of Koll'ar and Mori. $\endgroup$ May 4, 2017 at 18:54


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