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Singularities of the Minimal Model Program by Sándor Kovács, János Kollár

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3.4 More examples

In Section 3.1 we used a very simple recipe to construct log canonical singularities: take a cone over a variety with lc singularities whose canonical class is either negative or trivial.

Here we introduce a slight twist as in Kollár (2011b). We start with a reducible variety Z with snc singularities and construct a smooth (or mildly singular) variety Y that contains Z as a Cartier divisor with negative normal bundle L−1. A theorem of Artin (1970) says that Z can be contracted (analytically or as an algebraic space) and we get a singular point (0 X). One can also think of (0 X) as a deformation of the affine cone Ca(X,L). ...

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