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Subadjunction of log canonical divisors for a subvariety of codimension 2. (English) Zbl 0901.14004
Kawamata, Yujiro (ed.) et al., Birational algebraic geometry. A conference on algebraic geometry in memory of Wei-Liang Chow (1911–1995), Baltimore, MD, USA, April 11–14, 1996. Providence, RI: American Mathematical Society. Contemp. Math. 207, 79-88 (1997).
The classical adjunction formula: $$(K_X+S)|_S\sim K_S$$ relates the canonical divisor of a smooth variety $$X$$ with that of a smooth divisor $$S$$ on $$X$$. If $$X$$ has singularities, then the formula is in general no longer true, as the example of a generator on a quadric cone shows. A subadjunction theorem has been given by the author [Y. Kawamata, “On Fujita’s freeness conjecture for 3-folds and 4-folds”, Math. Ann. 308, No. 3, 491-505 (1997)].
Here the following subadjunction theorem for codimension $$2$$ subvarieties is proved: If $$X$$ is a normal variety, $$D$$ an effective $$\mathbb{Q}$$-Cartier divisor such that $$(X,D)$$ is log canonical, $$W$$ an element of the set of the centres of log canonical singularities, of codimension $$2$$ in $$X$$, then there exist canonically determined effective $$\mathbb{Q}$$-divisors $$M_W$$ and $$D_W$$ on the normalization $$\widetilde W$$ of $$W$$, such that $$\nu^*(K_X+D)\sim_\mathbb{Q} K_{\widetilde W}+M_W+D_W$$ (here $$\nu$$ denotes the normalization map $$\widetilde W\to W$$). The divisor $$D_W$$ is the local contribution coming from the singularities of $$(X,D)$$, while $$M_W$$ is the global contribution coming from the moduli space of curves. Some relationships between the singularities of $$(X,D)$$ and those of the restricted pair are given.
For the entire collection see [Zbl 0869.00030].

##### MSC:
 14C20 Divisors, linear systems, invertible sheaves 14M07 Low codimension problems in algebraic geometry
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