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\(F\)-adjunction. (English) Zbl 1209.13013

Let \(k\) be a field of positive characteristic \(p\) such that \([k^{1/p} : k] < \infty\). Let \(X\) be an integral separated normal \(F\)-finite Noetherian scheme essentially of finite type and \(\Delta\) an effective \(\mathbb{Q}\)-divisor such that \(K_X +\Delta\) is \(\mathbb{Q}\)-Cartier with index not divisible by \(p\). Let \(W \subset X\) be a closed subscheme, integral and normal. The author develops an adjunction theory in positive characteristic that is analogous to that of Kawamata and Shokurov which relates the singularities of \(X\) near a center of log canonicity \(W\subset X\) to the singularities of \(W\). The analog of a center of log canonicity in positive characteristic \(p\) is a center of sharp \(F\)-purity introduced by the author in 2008 [Math. Z. 265, No. 3, 687–714 (2010; Zbl 1213.13014)]. The adjunction theory, called here \(F\)-adjunction, states that under certain natural conditions there exists a canonically defined \(\mathbb{Q}\)-divisor \(\Delta_W\) on \(W\) such that \((K_X+\Delta)_{{|}_{W}} \sim_{\mathbb{Q}} K_W + \Delta_W\) and the singularities of \(X\) near \(W\) are the same as the singularities of the pair \((W, \Delta_W)\) (and the paper makes this statement precise).
The conditions of \(W\) and \((X, \Delta)\) that ensure this phenomenon are: (a) \((X, \Delta)\) is sharply \(F\)-pure at the generic point of \(W\); (b) the ideal sheaf of \(W\) is locally a center of sharp \(F\)-purity for \((X, \Delta)\).
Among other notable results, we mention that for a pair \((X=\text{Spec}(R), \Delta)\) and a graded system of ideals \(\mathfrak{a}_{\bullet}\), if the triple \((R, \Delta, \mathfrak{a}_{\bullet})\) is sharply \(F\)-pure, then there are finitely many centers of sharp \(F\)-purity. In particular for a globally \(F\)-split variety, there are at most finitely many subschemes that are compatibly split with any given splitting. This result was independently obtained by V. B. Mehta and S. Kumar in [Int. Math. Res. Not. 2009, No. 19, 3595–3597 (2009; Zbl 1183.13006)]. In the local case, results of this nature were obtained by F. Enescu and M. Hochster [Algebra Number Theory 2, No. 7, 721–754 (2008; Zbl 1190.13003)], R. Y. Sharp [Trans. Am. Math. Soc. 359, No. 9, 4237–4258 (2007; Zbl 1130.13002)].

MSC:

13A35 Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
14B05 Singularities in algebraic geometry
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