scholarly journals Isomorphisms between complements of projective plane curves

2019 ◽  
Vol Volume 3 ◽  
Author(s):  
Mattias Hemmig
Keyword(s):  
Singular Point ◽  
Projective Plane ◽  
Plane Curves ◽  
Complex Numbers ◽  
Closed Field ◽  
Algebraically Closed Field

In this article, we study isomorphisms between complements of irreducible curves in the projective plane $\mathbb{P}^2$, over an arbitrary algebraically closed field. Of particular interest are rational unicuspidal curves. We prove that if there exists a line that intersects a unicuspidal curve $C \subset \mathbb{P}^2$ only in its singular point, then any other curve whose complement is isomorphic to $\mathbb{P}^2 \setminus C$ must be projectively equivalent to $C$. This generalizes a result of H. Yoshihara who proved this result over the complex numbers. Moreover, we study properties of multiplicity sequences of irreducible curves that imply that any isomorphism between the complements of these curves extends to an automorphism of $\mathbb{P}^2$. Using these results, we show that two irreducible curves of degree $\leq 7$ have isomorphic complements if and only if they are projectively equivalent. Finally, we describe new examples of irreducible projectively non-equivalent curves of degree $8$ that have isomorphic complements.

Generators of Ideals Defining Certain Surfaces in Projective Space

1996 ◽  
Vol 48 (3) ◽  
pp. 585-595 ◽  
Author(s):  
Sandeep H. Holay
Keyword(s):  
Projective Space ◽  
Projective Plane ◽  
Ample Divisor ◽  
Plane Curves ◽  
Closed Field ◽  
Divisor Class ◽  
Algebraically Closed Field ◽  
Arbitrary Characteristic ◽  
Generating Sets ◽  
Blowing Up

AbstractWe consider the surface obtained from the projective plane by blowing up the points of intersection of two plane curves meeting transversely. We find minimal generating sets of the defining ideals of these surfaces embedded in projective space by the sections of a very ample divisor class. All of the results are proven over an algebraically closed field of arbitrary characteristic.

scholarly journals Elements Generating a Proper Normal Subgroup of the Cremona Group

2020 ◽  
Author(s):  
Serge Cantat ◽  
Vincent Guirardel ◽  
Anne Lonjou
Keyword(s):  
Normal Subgroup ◽  
Projective Plane ◽  
Infinite Order ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Cremona Group ◽  
Birational Transformations ◽  
Proper Normal Subgroup

Abstract Consider an algebraically closed field ${\textbf{k}}$, and let $\textsf{Cr}_2({\textbf{k}})$ be the Cremona group of all birational transformations of the projective plane over ${\textbf{k}}$. We characterize infinite order elements $g\in \textsf{Cr}_2({\textbf{k}})$ having a power $g^n$, $n\neq 0$, generating a proper normal subgroup of $\textsf{Cr}_2({\textbf{k}})$.

scholarly journals ON RATIONAL CUSPIDAL PROJECTIVE PLANE CURVES

2005 ◽  
Vol 92 (1) ◽  
pp. 99-138 ◽  
Author(s):  
J. FERNÁNDEZ DE BOBADILLA ◽  
I. LUENGO-VELASCO ◽  
A. MELLE-HERNÁNDEZ ◽  
A. NÉMETHI
Keyword(s):  
Singular Point ◽  
Projective Plane ◽  
Plane Curve ◽  
Topological Type ◽  
Homology Sphere ◽  
Plane Curves ◽  
Curve Singularity ◽  
Witten Invariant ◽  
Rational Homology ◽  
Rational Homology Sphere

In 2002, L. Nicolaescu and the fourth author formulated a very general conjecture which relates the geometric genus of a Gorenstein surface singularity with rational homology sphere link with the Seiberg--Witten invariant (or one of its candidates) of the link. Recently, the last three authors found some counterexamples using superisolated singularities. The theory of superisolated hypersurface singularities with rational homology sphere link is equivalent with the theory of rational cuspidal projective plane curves. In the case when the corresponding curve has only one singular point one knows no counterexample. In fact, in this case the above Seiberg--Witten conjecture led us to a very interesting and deep set of `compatibility properties' of these curves (generalising the Seiberg--Witten invariant conjecture, but sitting deeply in algebraic geometry) which seems to generalise some other famous conjectures and properties as well (for example, the Noether--Nagata or the log Bogomolov--Miyaoka--Yau inequalities). Namely, we provide a set of `compatibility conditions' which conjecturally is satisfied by a local embedded topological type of a germ of plane curve singularity and an integer $d$ if and only if the germ can be realized as the unique singular point of a rational unicuspidal projective plane curve of degree $d$. The conjectured compatibility properties have a weaker version too, valid for any rational cuspidal curve with more than one singular point. The goal of the present article is to formulate these conjectured properties, and to verify them in all the situations when the logarithmic Kodaira dimension of the complement of the corresponding plane curves is strictly less than 2.

Dimension of noncommutative plane curves

2019 ◽  
Vol 18 (03) ◽  
pp. 1950057
Author(s):  
Søren Jøndrup
Keyword(s):  
Proper Ideal ◽  
Plane Curves ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Noncommutative Plane

In this paper, we prove that an algebra of the form [Formula: see text] is never right (or left) artinian in case [Formula: see text] is a proper ideal and [Formula: see text] is an uncountable, algebraically closed field of characteristic [Formula: see text].

scholarly journals On a class of numbers generated by differencial equations related with algebraic groups

1994 ◽  
Vol 133 ◽  
pp. 1-55 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Number Theory ◽  
Differential Equations ◽  
Algebraic Groups ◽  
Complex Numbers ◽  
Closed Field ◽  
Algebraic Numbers ◽  
Algebraically Closed Field ◽  
Algebraic Differential Equations

In this paper we propose a new category Qcl of complex numbers which contains π, e and the set of algebraic numbers. In fact this category contains most of the numbers studied so far in number theory. An element of the category is here called a classical number. The category of the classical numbers forms an algebraically closed field and consists of countably many numbers. The definition depends on algebraic differential equations related with algebraic groups. Throughout the paper unless otherwise stated, we deal with functions of one variable and a differential equation is an ordinary differential equation. We are inspired of the Leçons de Stockholm of Painlevé [P].

Blow Up Sequences and the Module of nth Order Differentials

1976 ◽  
Vol 28 (6) ◽  
pp. 1289-1301 ◽  
Author(s):  
William C. Brown
Keyword(s):  
Singular Point ◽  
Local Ring ◽  
Algebraic Curve ◽  
Blow Up ◽  
Quotient Ring ◽  
Integral Closure ◽  
Closed Field ◽  
Quotient Field ◽  
Algebraically Closed Field

Let C denote an irreducible, algebraic curve defined over an algebraically closed field k. Let ? be a singular point of C. We shall employ the following notation throughout the rest of this paper: R will denote the local ring at P, K the quotient field of the integral closure of R in K, A the completion of R with respect to its radical topology, and Ā the integral closure of A in its total quotient ring.

scholarly journals Représentations banales de

2013 ◽  
Vol 149 (4) ◽  
pp. 679-704 ◽  
Author(s):  
Alberto Mínguez ◽  
Vincent Sécherre
Keyword(s):  
Complex Numbers ◽  
Closed Field ◽  
Locally Compact ◽  
Central Division ◽  
Algebraically Closed Field ◽  
Finite Dimensional ◽  
Compact Field ◽  
Residue Characteristic

AbstractLet${\rm F}$be a non-Archimedean locally compact field of residue characteristic$p$, let${\rm D}$be a finite-dimensional central division${\rm F}$-algebra and let${\rm R}$be an algebraically closed field of characteristic different from$p$. We definebanalirreducible${\rm R}$-representations of the group${\rm G}={\rm GL}_{m}({\rm D})$. This notion involves a condition on the cuspidal support of the representation depending on the characteristic of${\rm R}$. When this characteristic is banal with respect to${\rm G}$, in particular when${\rm R}$is the field of complex numbers, any irreducible${\rm R}$-representation of${\rm G}$is banal. In this article, we give a classification of all banal irreducible${\rm R}$-representations of${\rm G}$in terms of certain multisegments, called banal. When${\rm R}$is the field of complex numbers, our method provides a new proof, entirely local, of Tadić’s classification of irreducible complex smooth representations of${\rm G}$.

scholarly journals On the Irreducibility of Fourth Dimensional Tuba's Representation of the Pure Braid Group on Three Strands

2018 ◽  
Vol 11 (3) ◽  
pp. 682-701
Author(s):  
Hasan A. Haidar ◽  
Mohammad N. Abdulrahim
Keyword(s):  
Braid Group ◽  
Sufficient Conditions ◽  
Complex Numbers ◽  
Closed Field ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Algebraically Closed Field ◽  
Pure Braid Group ◽  
Necessary And Sufficient

We consider Tuba's representation of the pure braid group, $%P_{3} $, given by the map $\phi :P_{3}\longrightarrow GL(4,F)$, where $F$ is an algebraically closed field. After, specializing the indeterminates used in defining the representation to non- zero complex numbers, we find sufficient conditions that guarantee the irreducibility of Tuba's representation of the pure braid group $P_{3}$ with dimension $d=4$. Under further restriction for the complex specialization of the indeterminates, we get a necessary and sufficient condition for the irreducibility of $\phi

scholarly journals The strong simple connectedness of tame algebras with separating almost cyclic coherent Auslander–Reiten components

2021 ◽  
Author(s):  
Piotr Malicki
Keyword(s):  
Closed Field ◽  
Algebraically Closed Field ◽  
Main Application ◽  
Finite Dimensional ◽  
Simple Connectedness ◽  
Tame Algebras

AbstractWe study the strong simple connectedness of finite-dimensional tame algebras over an algebraically closed field, for which the Auslander–Reiten quiver admits a separating family of almost cyclic coherent components. As the main application we describe all analytically rigid algebras in this class.

scholarly journals On 3-syzygy and unexpected plane curves

2021 ◽  
Author(s):  
Grzegorz Malara ◽  
Piotr Pokora ◽  
Halszka Tutaj-Gasińska
Keyword(s):  
Projective Plane ◽  
Plane Curves ◽  
Complex Projective Plane ◽  
Geometric Properties ◽  
Complex Projective

AbstractIn this note we study curves (arrangements) in the complex projective plane which can be considered as generalizations of free curves. We construct families of arrangements which are nearly free and possess interesting geometric properties. More generally, we study 3-syzygy curve arrangements and we present examples that admit unexpected curves.

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