scholarly journals Deformations of rational curves in positive characteristic

2020 ◽  
Vol 2020 (769) ◽  
pp. 55-86
Author(s):  
Kazuhiro Ito ◽  
Tetsushi Ito ◽  
Christian Liedtke
Keyword(s):  
Positive Characteristic ◽  
Rational Curves ◽  
Kodaira Dimension ◽  
Sufficient Criterion ◽  
Higher Dimensional ◽  
Negative Kodaira Dimension

AbstractWe study deformations of rational curves and their singularities in positive characteristic. We use this to prove that if a smooth and proper surface in positive characteristic p is dominated by a family of rational curves such that one member has all δ-invariants (resp. Jacobian numbers) strictly less than {\frac{1}{2}(p-1)} (resp. p), then the surface has negative Kodaira dimension. We also prove similar, but weaker results hold for higher-dimensional varieties. Moreover, we show by example that our result is in some sense optimal. On our way, we obtain a sufficient criterion in terms of Jacobian numbers for the normalization of a curve over an imperfect field to be smooth.

scholarly journals Classification of one-dimensional superattracting germs in positive characteristic

2014 ◽  
Vol 35 (7) ◽  
pp. 2242-2268 ◽  
Author(s):  
MATTEO RUGGIERO
Keyword(s):  
Positive Characteristic ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Dimensional Version

We give a classification of superattracting germs in dimension $1$ over a complete normed algebraically closed field $\mathbb{K}$ of positive characteristic up to conjugacy. In particular, we show that formal and analytic classifications coincide for these germs. We also give a higher-dimensional version of some of these results.

scholarly journals Threefolds with negative Kodaira dimension and positive irregularity

1983 ◽  
Vol 91 ◽  
pp. 163-172 ◽  
Author(s):  
Mauro Beltrametti ◽  
Paolo Francia
Keyword(s):  
Kodaira Dimension ◽  
Complex Field ◽  
Negative Kodaira Dimension

The purpose of this paper is to study threefolds X, with negative Kodaira dimension k(X) and positive irregularity q(X), defined over the complex field C.

scholarly journals The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension

2012 ◽  
Vol 22 (2) ◽  
pp. 201-248 ◽  
Author(s):  
Sébastien Boucksom ◽  
Jean-Pierre Demailly ◽  
Mihai Păun ◽  
Thomas Peternell
Keyword(s):  
Kähler Manifold ◽  
Kodaira Dimension ◽  
Kahler Manifold ◽  
Compact Kähler Manifold ◽  
Effective Cone ◽  
Negative Kodaira Dimension

ℓ∞-COHOMOLOGY AND METABOLICITY OF NEGATIVELY CURVED COMPLEXES

1999 ◽  
Vol 09 (01) ◽  
pp. 51-77 ◽  
Author(s):  
IGOR MINEYEV
Keyword(s):  
Simplicial Complex ◽  
Universal Cover ◽  
Fundamental Groups ◽  
Sufficient Criterion ◽  
Positive Dimension ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Higher Dimensional ◽  
Isoperimetric Functions ◽  
Finite Simplicial Complex

We prove the analog of de Rham's theorem for ℓ∞-cohomology of the universal cover of a finite simplicial complex. A sufficient criterion is given for linearity of isoperimetric functions for filling cycles of any positive dimension over ℝ. This implies the linear higher dimensional isoperimetric inequalities for the fundamental groups of finite negatively curved complexes and of closed negatively curved manifolds. Also, these groups are ℝ-metabolic.

THE BOGOMOLOV–MIYAOKA–YAU INEQUALITY FOR LOG CANONICAL SURFACES

2001 ◽  
Vol 64 (2) ◽  
pp. 327-343 ◽  
Author(s):  
ADRIAN LANGER
Keyword(s):  
Vector Bundles ◽  
Universal Cover ◽  
Kodaira Dimension ◽  
Projective Surface ◽  
Chern Classes ◽  
Singular Surfaces ◽  
Smooth Projective Surface ◽  
Log Canonical ◽  
Surface Singularities ◽  
Negative Kodaira Dimension

Let X be a smooth projective surface of non-negative Kodaira dimension. Bogomolov [1, Theorem 5] proved that c21 [les ] 4c2. This was improved to c21 [les ] 3c2 by Miyaoka [12, Theorem 4] and Yau [19, Theorem 4]. Equality c21 [les ] 3c2 is attained, for example, if the universal cover of X is a ball (if κ(X) = 2 then this is the only possibility). Further generalizations of inequalities for Chern classes for some singular surfaces with (fractional) boundary were obtained by Sakai [16, Theorem 7.6], Miyaoka [13, Theorem 1.1], Kobayashi [6, Theorem 2; 7, Theorem 12], Wahl [18, Main Theorem] and Megyesi [10, Theorem 10.14; 11, Theorem 0.1].In [8] we introduced Chern classes of reflexive sheaves, using Wahl's local Chern classes of vector bundles on resolutions of surface singularities. Here we apply them to obtain the following generalization of the Bogomolov–Miyaoka–Yau inequality.

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