## $$F$$-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

### Citations:

Zbl 1183.13006; Zbl 1190.13003; Zbl 1130.13002; Zbl 1213.13014
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