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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].

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