Boundedness of Log Canonical Surface Generalized Polarized Pairs

Filipazzi, Stefano

doi:10.11650/tjm/171204

Cited by 10 publications

(18 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In dimension 2, Filipazzi [14] answered the above questions affirmatively. He took advantage of special features of the geometry of surfaces.…”

Section: Introductionmentioning

confidence: 81%

“…Moreover, we need M to be bounded in some sense. In dimension two, to achieve these properties, [14] uses special features of surfaces such as the well-known fact that the intersection matrix of exceptional divisors of a birational morphism of surfaces is negative definite. Unfortunately the surface arguments do not work in higher dimension.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Boundedness and volume of generalised pairs

Birkar¹

2021

Preprint

View full text Add to dashboard Cite

In this paper we investigate boundedness and volumes of generalised pairs, and give applications to usual pairs especially to a class of pairs that we call stable log minimal models.Fixing the dimension and a DCC set controlling coefficients, we will show that the set of volumes of all projective generalised lc pairs (X, B + M ) under the given data, satisfies the DCC. Futhermore, we will show that in the klt case, the set of such pairs with ample KX + B + M and fixed volume, forms a bounded family.We prove a result about descent of nef divisors to bounded families. This is the key to proving the above and various other results.We will then apply the above to study projective lc pairs (X, B) with abundant KX +B of arbitrary Kodaira dimension. In particular, we show that the set of Iitaka volumes of such pairs satisfies DCC under some natural boundedness assumptions on the fibres of the Iitaka fibration.We define stable log minimal models which consist of a projective lc pair (X, B) with semi-ample KX + B together with a divisor A ≥ 0 so that KX + B + A is ample and A does not contain any non-klt centre of (X, B). This is a generalisation of both usual stable pairs of general type and stable log Calabi-Yau pairs. Fixing appropriate invariants we show that stable log minimal models form a bounded family. Then we discuss connection with moduli spaces. Caucher Birkar3.4. Bounded coefficients on generalised lc modifications 18 3.6. Bounded crepant models 21 3.8. Finiteness of generalised lc thresholds on bounded families 24 3.10. Construction of towers of crepant models 25 3.11. Boundedness of length of towers of crepant models 26 3.13. Descent of nef divisors to bounded models 27 4. DCC of volume of generalised pairs 30 4.1. DCC of volumes of pairs with fixed birational model 31 4.3. DCC of volumes for G glc (d, Φ) 34 5. Bounded birational models for G glc (d, Φ, v) 36 6. Boundedness of generalised pairs 39 6.1. Decrease of volume 39 6.4. Log discrepancies of F glc (d, Φ, v) 41 6.6. Boundedness of F gklt (d, Φ, v) 46 7. DCC of Iitaka volumes 48 7.1. Adjunction formula for I lc (d, Φ, u) 48 7.5. DCC of Iitaka volumes 52 8. Boundedness of stable log minimal models 53 8.1. Log discrepancies on stable log minimal models 53 8.4. Boundedness of S lc (d, Φ, u, v, σ) and S klt (d, Φ, u, v,

show abstract

“…In dimension 2, Filipazzi [14] answered the above questions affirmatively. He took advantage of special features of the geometry of surfaces.…”

Section: Introductionmentioning

confidence: 81%

Section: Introductionmentioning

confidence: 99%

Boundedness and volume of generalised pairs

Birkar¹

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…This latter setup seems very hard to achieve in general, as given a generalized pair (Z → Z, B Z , M Z ) it is hard to characterize how to optimally choose Z and how many blow-ups over Z are required for such optimal choice. In this direction, there are partial results just in dimension 2 [13].…”

Section: Generalized Pairsmentioning

confidence: 99%

“…Then, M Z is bounded up to Q-linear equivalence as H Z − (K Z + B Z ). An approach of this flavor is carried out in [13] in the case dim(Z) = 2, but it seems harder in general.…”

Section: Generalized Pairsmentioning

confidence: 99%

Invariance of Plurigenera and boundedness for Generalized Pairs

Filipazzi¹,

Svaldi²

2020

RMC

Self Cite

View full text Add to dashboard Cite

In this note, we survey some recent developments in birational geometry concerning the boundedness of algebraic varieties. We delineate a strategy to extend some of these results to the case of generalized pairs, first introduced by Birkar and Zhang, when the associated log canonical divisor is ample, and the volume is fixed. In this context, we show a version of deformation invariance of plurigenera for generalized pairs. We conclude by discussing an application to the boundedness of varieties of Kodaira dimension κ(X) = dim(X) − 1.

show abstract

“…[Kol14, Theorem 4], [Bir12a, Theorem 1.1], [HX13, Theorem 1.6], [Has19, Theorem 1.1]) has a negative answer for generalized pairs even in dimension 1 by considering the projective generalized pair (X, 0, M) where X is an elliptic curve and M X ≡ 0 is a non-torsion divisor (cf. [Fil18a,6.2]).…”

Section: Introductionmentioning

confidence: 99%

Existence of flips for generalized lc pairs

Hacon¹,

Liu²

2021

Preprint

View full text Add to dashboard Cite

We prove the existence of flips for Q-factorial NQC generalized lc pairs, and the cone and contraction theorems for NQC generalized lc pairs. This answers a question of C. Birkar. As an immediate application, we show that we can run the minimal model program for Q-factorial NQC generalized lc pairs. In particular, we complete the minimal model program for Q-factorial NQC generalized lc pairs in dimension ≤ 3 and pseudo-effective Q-factorial NQC generalized lc pairs in dimension 4.

show abstract

Boundedness of Log Canonical Surface Generalized Polarized Pairs

Cited by 10 publications

References 18 publications

Boundedness and volume of generalised pairs

Boundedness and volume of generalised pairs

Invariance of Plurigenera and boundedness for Generalized Pairs

Existence of flips for generalized lc pairs

Contact Info

Product

Resources

About