The F-different and a canonical bundle formula
2017; Linguagem: Inglês
10.2422/2036-2145.201510_012
ISSN2036-2145
Autores Tópico(s)Advanced Algebra and Geometry
ResumoWe study the structure of Frobenius splittings (and generalizations thereof) induced on compatible subvarieties W ⊆ X.In particular, if the compatible splitting comes from a compatible splitting of a divisor on some birational model E ⊆ X ′ -→ X (ie, W is a log canonical center), then we show that the divisor corresponding to the splitting on W is bounded below by the divisorial part of the different as studied by Ambro, Kawamata, Kollár, Shokurov, and others.We also show that difference between the divisor associated to the splitting and the divisorial part of the different is largely governed by the (non-)Frobenius splitting of fibers of E -→ W .In doing this analysis, we recover an F -canonical bundle formula by reinterpretting techniques common in the theory of Frobenius splittings., and hence φ corresponds to an effective divisor linearly equivalent to (1 -p e )K X .• We use D φ to denote this effective divisor corresponding to φ.• We write ∆ φ := 1 p e -1 D φ and note that ∆ φ ∼ Q -K X .It is easy to see that this process can be reversed.Given an effective divisor ∆ such that∆ with ∆ φ = ∆.The choice of φ is also unique up to pre-multiplication by units of Γ(X, O X ).One can also form the self-composition of φ =: φ e .For instanceFor additional discussion of this correspondence between maps and divisors, see [BS13, Section 4].Now suppose that L is an arbitrary line bundle on the F -finite normal integral scheme X.Then using the same argument, a map φ :Maps and non-effective Q-divisors.We now recall how to interpret non-effective ∆.The main idea is that we are still given a map from F e * L but that the image need not be in O X , it might be in some fractional ideal in K(X).Indeed, suppose we are given an O X -linear map φ : F e * L -→ K(X).Then it is easy to see (at least locally) that for some effective Cartier divisor E on X that φ(F e * L ((1 -p e )E)) ⊆ O X .This gives us an effective divisor ∆where G is some effective Weil divisor supported where ∆ φ is not effective.Indeed, simply reflexify φ(F e * L ).Lemma 2.3.With notations as above, ∆ φ is independent of the choice of E.Proof.Suppose that E 1 , E 2 are two effective Cartier divisors with φ(F e * L ((1 -p e )E i )) ⊆ O X .Without loss of generality, we may assume that E 1 ≤ E 2 .Then we have the following compositionThe map ψ yields ∆ 1 = ∆ ψ -E 1 and the composition yields ∆ 2 = ∆ µ -E 2 .On the other hand, straightforward local computation shows that ∆With notation as above, suppose that we fix an embedding L ⊆ K(X).Then our map φ = φ e : F e * L -→ K(X) yields a map φ e : F e * K(X) -→ K(X).We can then formand more generally φ ne = φ • (F e * φ (n-1)e ) : F ne * K(X) -→ K(X).We then restrict φ ne to F ne * L 1+p e +...+p (n-1)e yielding: (2.3.1)φ ne : F ne * L 1+p e +...+p (n-1)e -→ K(X).Lemma 2.4.With notations as above, ∆ φ ne = ∆ φ e .Proof.Choose E such that φ e (F e * L ((1 -p e )E)) ⊆ O X .It follows immediately that φ ne (F ne * L 1+p e +...+p (n-1)e ((1 -p ne )E)) ⊆ O X since that is how composition works for effective divisors.Define ∆ ′e and ∆ ′ne corresponding to the restricted maps φ e | F e * L ((1-p e )E) and φ ne | F ne * L 1+...+p (n-1)e ((1-p ne )E) respectively.We know ∆ ′e -E =: ∆ φ e and also that ∆ ′ne -E =: ∆ φ ne .But ∆ ′ne = ∆ ′e since they correspond to compositions of the same map (see [BS13, Lemma 4.1.2]and [Sch09, Theorem 3.11]).The conclusion follows.2.2.p -e -linear maps and base extension.The following subsection is only utilized briefly in Algorithm 5.15 and can be skipped on a first reading.We start this subsection by summarizing some results from [ST14].Suppose that f : Z -→ W is a finite dominant map of normal F -finite integral schemes and φ W : F e * L W -→ K(W ) is an O W -linear map from a line bundle L W on W . Then φ W corresponds to a divisor ∆ W as above.Let T : f * K(Z) -→ K(W ) be a nonzero map between the fraction fields of Z and W respectively.Using an argument similar to that above, the map T| f * O Z gives us a Weil divisor R T which should be thought of as a type ramification divisor.If f is separable, Proof.This is essentially in [HW02, Main Theorem] using different language, other proofs can be found in [BS13, Section 7.2].We briefly sketch the argument.Choose π : Y -→ X a proper birational map from a normal Y such that O X,E appears as the generic point of some divisor E ⊆ Y .We first extend φ to F e * K(X) = F e * K(Y ), and then considerand hence K Y + ∆ Y = π * (K X + ∆).However, the divisor ∆ Y induced from φ Y obviously agrees with ∆ X wherever π is an isomorphism.The result follows by localization.We next verify that sub-log canonical and sub-F -pure are the same for valuation rings.Lemma 2.14.Suppose that R ⊆ K(X) is a discrete valuation ring with parameter t ∈ R. R ∼ = L ⊆ K(X) and φ :* ut a ) for some unit u ∈ R and some a ∈ Z.It follows that ∆ φ = a p e -1 div(t).On the other hand, it is easy to see that Image(φ) ⊇ O X if and only if a ≤ p e -1.This proves the lemma.Combining the previous two results, we immediately have the following.This was first shown in [HW02, Main Theorem] although they assumed that ∆ is effective.Corollary 2.15.If (X, ∆) is a pair with (p e -1)(K X + ∆) Q-Cartier, and (X, ∆) is sub-F -pure, then (X, ∆) is sub-log canonical.Proof.This is easy, if φ : F e * L -→ K(X) corresponds to ∆ and φ(F e * L ) ⊇ O X , then, abusing notation and extending φ to the fraction field, obviously 1 ∈ φ(F e * L ⊗ O X O X,E ) and hence O X,E ⊆ φ(F e * L ⊗ O X O X,E ) for any divisorial discrete valuation ring O X,E lying over X.In particular, each (Spec O X,E , ∆ X,E ) is sub-F -pure (where K X,E + ∆ X,E = π * (K X + ∆)).The previous two lemmas immediately imply that the discrepancy divisor along all such valuations is ≥ -1 and the result follows.
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