scholarly journals MMP for co-rank one foliations on threefolds

2021 ◽  
Author(s):  
Paolo Cascini ◽  
Calum Spicer
Keyword(s):  
General Type ◽  
Base Point ◽  
Minimal Models ◽  
Rank One ◽  
Dicritical Singularities

AbstractWe prove existence of flips, special termination, the base point free theorem and, in the case of log general type, the existence of minimal models for F-dlt foliated pairs of co-rank one on a $${\mathbb {Q}}$$ Q -factorial projective threefold. As applications, we show the existence of F-dlt modifications and F-terminalisations for foliated pairs and we show that foliations with canonical or F-dlt singularities admit non-dicritical singularities. Finally, we show abundance in the case of numerically trivial foliated pairs.

scholarly journals Equality in the Bogomolov–Miyaoka–Yau inequality in the non-general type case

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Feng Hao ◽  
Stefan Schreieder
Keyword(s):  
General Type ◽  
Universal Constant ◽  
Kodaira Dimension ◽  
Chern Number ◽  
Minimal Models

Abstract We classify all minimal models X of dimension n, Kodaira dimension n - 1 {n-1} and with vanishing Chern number c 1 n - 2 ⁢ c 2 ⁢ ( X ) = 0 {c_{1}^{n-2}c_{2}(X)=0} . This solves a problem of Kollár. Completing previous work of Kollár and Grassi, we also show that there is a universal constant ϵ > 0 {\epsilon>0} such that any minimal threefold satisfies either c 1 ⁢ c 2 = 0 {c_{1}c_{2}=0} or - c 1 ⁢ c 2 > ϵ {-c_{1}c_{2}>\epsilon} . This settles completely a conjecture of Kollár.

scholarly journals Finiteness Properties of Pseudo-Hyperbolic Varieties

2020 ◽  
Author(s):  
Ariyan Javanpeykar ◽  
Junyi Xie
Keyword(s):  
Dynamical Systems ◽  
Number Field ◽  
Minimal Model ◽  
General Type ◽  
Infinite Order ◽  
Minimal Models ◽  
Finiteness Result ◽  
Geometric Setting ◽  
Dynamical Degrees ◽  
Finiteness Properties

Abstract Motivated by Lang–Vojta’s conjecture, we show that the set of dominant rational self-maps of an algebraic variety over a number field with only finitely many rational points in any given number field is finite by combining Amerik’s theorem for dynamical systems of infinite order with properties of Prokhorov–Shramov’s notion of quasi-minimal models. We also prove a similar result in the geometric setting by using again not only Amerik’s theorem and Prokhorov–Shramov’s notion of quasi-minimal model but also Weil’s regularization theorem for birational self-maps and properties of dynamical degrees. Furthermore, in the geometric setting, we obtain an analogue of Kobayashi–Ochiai’s finiteness result for varieties of general type and thereby generalize Noguchi’s theorem (formerly Lang’s conjecture).

Some Results on Surfaces of General Type

2005 ◽  
Vol 57 (4) ◽  
pp. 724-749 ◽  
Author(s):  
B. P. Purnaprajna
Keyword(s):  
Linear Systems ◽  
General Type ◽  
Base Point ◽  
Linear Series ◽  
Ample Divisor ◽  
Surfaces Of General Type ◽  
Projective Normality

AbstractIn this article we prove some new results on projective normality, normal presentation and higher syzygies for surfaces of general type, not necessarily smooth, embedded by adjoint linear series. Some of the corollaries of more general results include: results on property Np associated to KS ⊗B⊗n where B is base-point free and ample divisor with B⊗K* nef, results for pluricanonical linear systems and results giving effective bounds for adjoint linear series associated to ample bundles. Examples in the last section show that the results are optimal.

scholarly journals Existence of minimal models for varieties of log general type II

2009 ◽  
Vol 23 (2) ◽  
pp. 469-490 ◽  
Author(s):  
Christopher D. Hacon ◽  
James M$^{\textup{c}}$Kernan
Keyword(s):  
General Type ◽  
Type Ii ◽  
Minimal Models

The number of the minimal models for a 3-fold of general type is finite

1987 ◽  
Vol 276 (4) ◽  
pp. 595-598 ◽  
Author(s):  
Yujiro Kawamata ◽  
Kenji Matsuki
Keyword(s):  
General Type ◽  
Minimal Models

On the number and boundedness of log minimal models of general type

2020 ◽  
Vol 53 (5) ◽  
pp. 1183-1207
Author(s):  
Diletta MARTINELLI ◽  
Stefan SCHREIEDER ◽  
Luca TASIN
Keyword(s):  
General Type ◽  
Minimal Models

scholarly journals Existence of minimal models for varieties of log general type

2009 ◽  
Vol 23 (2) ◽  
pp. 405-468 ◽  
Author(s):  
Caucher Birkar ◽  
Paolo Cascini ◽  
Christopher D. Hacon ◽  
James M$^{\textup{c}}$Kernan
Keyword(s):  
General Type ◽  
Minimal Models

scholarly journals Canonical stability in terms of singularity index for algebraic threefolds

2001 ◽  
Vol 131 (2) ◽  
pp. 241-264 ◽  
Author(s):  
MENG CHEN
Keyword(s):  
General Type ◽  
Classification Theory ◽  
Minimal Models ◽  
Ground Field ◽  
Rational Map ◽  
Singularity Index ◽  
3 Dimensional ◽  
Canonical Map ◽  
Complete Linear System ◽  
Birational Automorphism

Throughout the ground field is always supposed to be algebraically closed of characteristic zero. Let X be a smooth projective threefold of general type, denote by ϕm the m-canonical map of X which is nothing but the rational map naturally associated with the complete linear system [mid ]mKX[mid ]. Since, once given such a 3-fold X, ϕm is birational whenever m [Gt ] 0, quite an interesting thing to find is the optimal bound for such an m. This bound is important because it is not only crucial to the classification theory, but also strongly related to other problems. For example, it can be applied to determine the order of the birational automorphism group of X [21, remark in section 1]. To fix the terminology we say that ϕm is stably birational if ϕt is birational onto its image for all t [ges ] m. It is well known that the parallel problem in the surface case was solved by Bombieri [1] and others. In the 3-dimensional case, many authors have studied the problem, in quite different ways. Because, in this paper, we are interested in the results obtained by Hanamura [7], we do not plan to mention more references here. According to 3-dimensional MMP, X has a minimal model which is a normal projective 3-fold with only ℚ-factorial terminal singularities. Though X may have many minimal models, the singularity index (namely the canonical index) of any of its minimal models is uniquely determined by X. Denote by r the canonical index of minimal models of X. When r = 1 we know that ϕ6 is stably birational by virtue of [3, 6, 13 and 14]. When r [ges ] 2, Hanamura proved the following theorem.

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