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.

ENRIQUES’ CLASSIFICATION IN CHARACTERISTIC 0$" height="12pt">: THE -THEOREM

2018 ◽  
Vol 235 ◽  
pp. 201-226
Author(s):  
FABRIZIO CATANESE ◽  
BINRU LI
Keyword(s):  
Positive Characteristic ◽  
Kodaira Dimension ◽  
Algebraic Surfaces ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Precise Version

The main goal of this paper is to show that Castelnuovo–Enriques’ $P_{12}$ - theorem (a precise version of the rough classification of algebraic surfaces) also holds for algebraic surfaces $S$ defined over an algebraically closed field $k$ of positive characteristic ( $\text{char}(k)=p>0$ ). The result relies on a main theorem describing the growth of the plurigenera for properly elliptic or properly quasielliptic surfaces (surfaces with Kodaira dimension equal to 1). We also discuss the limit cases, i.e., the families of surfaces which show that the result of the main theorem is sharp.

scholarly journals Fibrations with moving cuspidal singularities

1991 ◽  
Vol 122 ◽  
pp. 161-179 ◽  
Author(s):  
Yoshifumi Takeda
Keyword(s):  
Algebraic Geometry ◽  
Characteristic Zero ◽  
Positive Characteristic ◽  
Closed Field ◽  
Projective Surface ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Algebraically Closed Field ◽  
Smooth Projective Surface ◽  
Almost All

Let f: V → C be a fibration from a smooth projective surface onto a smooth projective curve over an algebraically closed field k. In the case of characteristic zero, almost all fibres of f are nonsingular. In the case of positive characteristic, it is, however, known that there exist fibrations whose general fibres have singularities. Moreover, it seems that such fibrations often have pathological phenomena of algebraic geometry in positive characteristic (see M. Raynaud [7], W. Lang [4]).

Pencils of real binary cubics

1983 ◽  
Vol 93 (3) ◽  
pp. 477-484 ◽  
Author(s):  
C. T. C. Wall
Keyword(s):  
Real Case ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
The Real

A complete and satisfying account of the classification of pencils of binary cubics over an algebraically closed field was given by Newstead (2). Extending these results to the real case is not a matter of mere routine since new questions arise, for example the separation of roots of the cubics in a pencil (as well as their reality).

HOPF ALGEBRAS OF DIMENSION 30

2010 ◽  
Vol 09 (01) ◽  
pp. 11-15 ◽  
Author(s):  
DAIJIRO FUKUDA
Keyword(s):  
Hopf Algebra ◽  
Group Algebra ◽  
Hopf Algebras ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Finite Dimensional

This paper contributes to the classification of finite dimensional Hopf algebras. It is shown that every Hopf algebra of dimension 30 over an algebraically closed field of characteristic zero is semisimple and thus isomorphic to a group algebra or the dual of a group algebra.

THE LOEWY STRUCTURE OF -VERMA MODULES OF SINGULAR HIGHEST WEIGHTS

2015 ◽  
Vol 16 (4) ◽  
pp. 887-898
Author(s):  
Noriyuki Abe ◽  
Masaharu Kaneda
Keyword(s):  
Algebraic Group ◽  
Maximal Torus ◽  
Positive Characteristic ◽  
Closed Field ◽  
Verma Modules ◽  
Reductive Algebraic Group ◽  
Algebraically Closed Field ◽  
Loewy Length ◽  
Frobenius Kernel ◽  
Field Of Positive Characteristic

Let $G$ be a reductive algebraic group over an algebraically closed field of positive characteristic, $G_{1}$ the Frobenius kernel of $G$, and $T$ a maximal torus of $G$. We show that the parabolically induced $G_{1}T$-Verma modules of singular highest weights are all rigid, determine their Loewy length, and describe their Loewy structure using the periodic Kazhdan–Lusztig $P$- and $Q$-polynomials. We assume that the characteristic of the field is sufficiently large that, in particular, Lusztig’s conjecture for the irreducible $G_{1}T$-characters holds.

scholarly journals Order 3 Elements in G2 and Idempotents in Symmetric Composition Algebras

2018 ◽  
Vol 70 (5) ◽  
pp. 1038-1075 ◽  
Author(s):  
Alberto Elduque
Keyword(s):  
Conjugacy Classes ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Exceptional Groups ◽  
Composition Algebras ◽  
Characteristic 3

AbstractOrder three elements in the exceptional groups of type G2 are classiûed up to conjugation over arbitrary fields. Their centralizers are computed, and the associated classification of idempotents in symmetric composition algebras is obtained. Idempotents have played a key role in the study and classification of these algebras.Over an algebraically closed field, there are two conjugacy classes of order three elements in G2 in characteristic not 3 and four of them in characteristic 3. The centralizers in characteristic 3 fail to be smooth for one of these classes.

Centers and Azumaya loci for finite W-algebras in positive characteristic

2021 ◽  
pp. 1-32
Author(s):  
BIN SHU ◽  
YANG ZENG
Keyword(s):  
Lie Algebra ◽  
Nilpotent Element ◽  
Positive Characteristic ◽  
Irreducible Representations ◽  
Maximal Dimension ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Simple Lie Algebra ◽  
Maximal Spectrum ◽  
Smooth Locus

Abstract In this paper, we study the center Z of the finite W-algebra $${\mathcal{T}}({\mathfrak{g}},e)$$ associated with a semi-simple Lie algebra $$\mathfrak{g}$$ over an algebraically closed field $$\mathbb{k}$$ of characteristic p≫0, and an arbitrarily given nilpotent element $$e \in{\mathfrak{g}} $$ We obtain an analogue of Veldkamp’s theorem on the center. For the maximal spectrum Specm(Z), we show that its Azumaya locus coincides with its smooth locus of smooth points. The former locus reflects irreducible representations of maximal dimension for $${\mathcal{T}}({\mathfrak{g}},e)$$ .

scholarly journals The construction problem for Hodge numbers modulo an integer in positive characteristic

2020 ◽  
Vol 8 ◽  
Author(s):  
Remy van Dobben de Bruyn ◽  
Matthias Paulsen
Keyword(s):  
Positive Characteristic ◽  
Closed Field ◽  
Construction Problem ◽  
Algebraically Closed Field ◽  
Serre Duality ◽  
Hodge Numbers ◽  
Field Of Positive Characteristic

Abstract Let k be an algebraically closed field of positive characteristic. For any integer $m\ge 2$ , we show that the Hodge numbers of a smooth projective k-variety can take on any combination of values modulo m, subject only to Serre duality. In particular, there are no non-trivial polynomial relations between the Hodge numbers.

scholarly journals GENERAL HYPERPLANE SECTIONS OF THREEFOLDS IN POSITIVE CHARACTERISTIC

2018 ◽  
Vol 19 (2) ◽  
pp. 647-661 ◽  
Author(s):  
Kenta Sato ◽  
Shunsuke Takagi
Keyword(s):  
Projective Variety ◽  
Positive Characteristic ◽  
Hyperplane Section ◽  
Three Dimensional ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Double Points ◽  
Canonical Singularities ◽  
Hyperplane Sections

In this paper, we study the singularities of a general hyperplane section $H$ of a three-dimensional quasi-projective variety $X$ over an algebraically closed field of characteristic $p>0$. We prove that if $X$ has only canonical singularities, then $H$ has only rational double points. We also prove, under the assumption that $p>5$, that if $X$ has only klt singularities, then so does $H$.

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