Uniform bounds for strongly 𝐹-regular surfaces

Cascini, Paolo; Gongyo, Yoshinori; Schwede, Karl

doi:10.1090/tran/6515

Cited by 9 publications

(15 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let θ(X, B, M) be the number of those components of M which are not components of ⌊B⌋ (such θ functions were defined in 8.1 in a more general setting). By (7), S is not a component of ⌊B⌋, hence θ(X, B, M) > 0 otherwise K X + B is ample and R(K X + B) is finitely generated, a contradiction. Define…”

Section: 2mentioning

confidence: 97%

“…We will derive a contradiction. By replacing A with 1 m S where m is sufficiently divisible and S is a general member of |mA|, and changing M, B accordingly, we can assume that (7) S := Supp A is irreducible and K X + S + ∆ is ample for any boundary ∆ supported on Supp(B) − S.…”

Section: 2mentioning

confidence: 99%

“…We reduce the theorem to the case when X is projective, B has standard coefficients, and some component of ⌊B⌋ is negative on the extremal ray: this case is [13,Theorem 4.12] which is one of the main results of that paper. A different approach is taken in [7] to prove 1.1 when B has hyperstandard coefficients and p ≫ 0 (these coefficients are of the form n−1 n + l i b i n where n ∈ N ∪ {∞}, l i ∈ Z ≥0 and b i are in some fixed DCC set).…”

mentioning

confidence: 99%

See 2 more Smart Citations

Existence of flips and minimal models for 3-folds in char $p$

Birkar¹

2016

Ann. Sci. École Norm. Sup.

140

View full text Add to dashboard Cite

We will prove the following results for 3-fold pairs (X, B) over an algebraically closed field k of characteristic p > 5: log flips exist for Qfactorial dlt pairs (X, B); log minimal models exist for projective klt pairs (X, B) with pseudo-effective K X + B; the log canonical ring R(K X + B) is finitely generated for projective klt pairs (X, B) when K X + B is a big Qdivisor; semi-ampleness holds for a nef and big Q-divisor D if D − (K X + B) is nef and big and (X, B) is projective klt; Q-factorial dlt models exist for lc pairs (X, B); terminal models exist for klt pairs (X, B); ACC holds for lc thresholds; etc. Contents 1 45 12. Non-big log divisors: proof of 1.11 45 References 47Existence of flips and minimal models for 3-folds in char p 5 our results on flips and minimal models.Dlt and terminal models. The next two results are standard consequences of the LMMP (more precisely, of special termination). They are proved in Section 7.Theorem 1.6. Let (X, B) be an lc pair of dimension 3 over k of char p > 5. Then (X, B) has a (crepant) Q-factorial dlt model. In particular, if (X, B) is klt, then X has a Q-factorialization by a small morphism.The theorem was proved in [13, Theorem 6.1] for pairs with standard coefficients.Theorem 1.7. Let (X, B) be a klt pair of dimension 3 over k of char p > 5. Then (X, B) has a (crepant) Q-factorial terminal model.The theorem was proved in [13, Theorem 6.1] for pairs with standard coefficients and canonical singularities.The connectedness principle with applications to semi-ampleness. The next result concerns the Kollár-Shokurov connectedness principle. In characteristic 0, the surface case was proved by Shokurov by taking a resolution and then calculating intersection numbers [26, Lemma 5.7] but the higher dimensional case was proved by Kollár by deriving it from the Kawamata-Viehweg vanishing theorem [20, Theorem 17.4].Theorem 1.8. Let (X, B) be a projective Q-factorial pair of dimension 3 over k of char p > 5. Let f : X → Z be a birational contraction such that −(K X +B) is ample/Z. Then for any closed point z ∈ Z, the non-klt locus of (X, B) is connected in any neighborhood of the fibre X z .The theorem is proved in Section 9. To prove it we use the LMMP rather than vanishing theorems. When dim X = 2, the theorem holds in a stronger form (see 9.3).We will use the connectedness principle on surfaces to prove some semiampleness results on surfaces and 3-folds. Here is one of them: Theorem 1.9. Let (X, B + A) be a projective Q-factorial dlt pair of dimension 3 over k of char p > 5. Assume that A, B ≥ 0 are Q-divisors such that A is ample andNote that if one could show that ⌊B⌋ is semi-lc, then the result would follow from Tanaka [29]. In order to show that ⌊B⌋ is semi-lc one needs to check that it satisfies the Serre condition S 2 . In characteristic 0 this is a consequence of Kawamata-Viehweg vanishing (see Kollár [20, Corollary 17.5]). The S 2 condition can be used to glue sections on the various irreducible components of ⌊B⌋. To prove the above semi-ampleness we instead use a ...

show abstract

Section: 2mentioning

confidence: 97%

Section: 2mentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Existence of flips and minimal models for 3-folds in char $p$

Birkar¹

2016

Ann. Sci. École Norm. Sup.

140

View full text Add to dashboard Cite

show abstract

“…The proof is essentially the same as in the case where k is algebraically closed. When k is algebraically closed, the first assertion is nothing but [5,Theorem 4.1] and the second one follows from [31,Theorem 4.2]. Definition 5.4.…”

Section: Klt Singularitiesmentioning

confidence: 99%

General Hyperplane Sections of Threefolds in Positive Characteristic

Sato¹,

Takagi

2018

J. Inst. Math. Jussieu

View full text Add to dashboard Cite

show abstract

“…Later on, the first author, together with Gongyo and Schwede, showed that a two-dimensional klt singularity (X, ∆) is strongly F -regular for big enough characteristic depending only on the coefficients of ∆ [CGS14].…”

Section: Introductionmentioning

confidence: 99%