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.

The canonical volume of minimal 3-folds of general type

2018 ◽  
Vol 29 (03) ◽  
pp. 1850023
Author(s):  
Huanping Zhu
Keyword(s):  
Linear System ◽  
Lower Bounds ◽  
General Type ◽  
Rational Map ◽  
Canonical Map ◽  
Volume Formula ◽  
Complete Linear System ◽  
Minimal Formula

Let [Formula: see text] be a nonsingular projective [Formula: see text]-fold of general type. Denote by [Formula: see text] the [Formula: see text]-canonical map of [Formula: see text] which is the rational map naturally associated to the complete linear system [Formula: see text]. Suppose that [Formula: see text] be a minimal [Formula: see text]-fold of [Formula: see text] and [Formula: see text] the pluricanonical section index. In this paper, we obtain the lower bounds of the canonical volume [Formula: see text] in term of [Formula: see text] for [Formula: see text]. In addition, we also classify the weighted baskets [Formula: see text] of [Formula: see text] satisfying [Formula: see text].

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 Cuspidal quintics and surfaces with , and 5-torsion

2016 ◽  
Vol 19 (1) ◽  
pp. 42-53
Author(s):  
Carlos Rito
Keyword(s):  
Computer Algebra ◽  
General Type ◽  
Free Action ◽  
The Other ◽  
Singular Set ◽  
Canonical Map ◽  
Singular Sets ◽  
Surface Of General Type ◽  
Quintic Threefold ◽  
Hyperplane Sections

If $S$ is a quintic surface in $\mathbb{P}^{3}$ with singular set 15 3-divisible ordinary cusps, then there is a Galois triple cover ${\it\phi}:X\rightarrow S$ branched only at the cusps such that $p_{g}(X)=4$, $q(X)=0$, $K_{X}^{2}=15$ and ${\it\phi}$ is the canonical map of $X$. We use computer algebra to search for such quintics having a free action of $\mathbb{Z}_{5}$, so that $X/\mathbb{Z}_{5}$ is a smooth minimal surface of general type with $p_{g}=0$ and $K^{2}=3$. We find two different quintics, one of which is the van der Geer–Zagier quintic; the other is new.We also construct a quintic threefold passing through the 15 singular lines of the Igusa quartic, with 15 cuspidal lines there. By taking tangent hyperplane sections, we compute quintic surfaces with singular sets $17\mathsf{A}_{2}$, $16\mathsf{A}_{2}$, $15\mathsf{A}_{2}+\mathsf{A}_{3}$ and $15\mathsf{A}_{2}+\mathsf{D}_{4}$.

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).

scholarly journals Note on Galois Cohomology

1953 ◽  
Vol 5 ◽  
pp. 97-104
Author(s):  
Tadasi Nakayama
Keyword(s):  
Galois Extension ◽  
Class Group ◽  
Present Note ◽  
Galois Cohomology ◽  
Group Cohomology ◽  
Ground Field ◽  
General Group ◽  
3 Dimensional ◽  
The Relationship ◽  
American Authors

Let K/k be a Galois extension. Formerly the writer studied, in [3], [4], a certain correlation of factor sets in K/k with the norm class group of K/k, and extended it, in [5.], to 3-dimensional cocycles. The present note is to study the same relationship for general n-cocycles. As a matter of fact, the constructions which underlie the relationship have become common places in cohomology theory, through the works of French and American authors, and indeed the construction to bring certain (non-Galois) cocycles into the ground field k has been discussed by Baer [1, Theorem C] for general dimensions n under the setting of general group cohomology.

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

scholarly journals Some infinite sequences of canonical covers of degree 2

2021 ◽  
Vol 21 (1) ◽  
pp. 143-148
Author(s):  
Nguyen Bin
Keyword(s):  
General Type ◽  
Canonical System ◽  
Surfaces Of General Type ◽  
Canonical Map ◽  
Base Points ◽  
Surface Of General Type ◽  
Infinite Sequences

Abstract In this note, we construct three new infinite families of surfaces of general type with canonical map of degree 2 onto a surface of general type. For one of these families the canonical system has base points.

scholarly journals Even canonical surfaces with small K2, I

1993 ◽  
Vol 129 ◽  
pp. 115-146 ◽  
Author(s):  
Kazuhiro Konno
Keyword(s):  
Number Field ◽  
Complex Number ◽  
Algebraic Surface ◽  
General Type ◽  
Canonical Bundle ◽  
Rational Map ◽  
Birational Map ◽  
Surface Of General Type ◽  
Canonical Surface ◽  
Complex Number Field

Let S be a minimal algebraic surface of general type defined over the complex number field C, and let K denote the canonical bundle. According to [10], we call S a canonical surface if the rational map ФK associated with | K | induces a birational map of S onto the image X. We denote by Q (X) the intersection of all hyperquadrics through X.

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