ScholarLens Back to search
2005•Proceedings of the American Mathematical SocietyOpen access

Irreducibility of the (-1)-classes on smooth rational surfaces

Mustapha Lahyane

Open full text 11 citations

Abstract

We give a characterization for a ( − 1 ) (-1) -divisor D D on a smooth rational surface X X to be irreducible under the assumption that an anticanonical divisor − K X -K_X of X X is nef. Here − K X -K_X is nef means K X . C ≤ 0 K_X . C \leq 0 for every effective divisor C C on X X , and a ( − 1 ) (-1) -divisor D D is a divisor such that the two numerical conditions D 2 = − 1 = D . K X D^2 =-1=D.K_X hold. As an application we give explicit examples of blowing up the projective plane at nine points infinitely near such that the obtained surface has an infinite number of ( − 1 ) (-1) -curves. A

Open-access reader

About this research paper

What this paper is about

We give a characterization for a ( − 1 ) (-1) -divisor D D on a smooth rational surface X X to be irreducible under the assumption that an anticanonical divisor − K X -K_X of X X is nef. Here − K X -K_X is nef means K X . C ≤ 0 K_X . C \leq 0 for every effective divisor C C on X X , and a ( − 1 ) (-1) -divisor D D is a divisor such that the two numerical conditions D 2 = − 1 = D . K X D^2 =-1=D.K_X hold. As an application we give explicit examples of blowing up the projective plane at nine points infinitely near such that the obtained surface has an infinite number of ( − 1 ) (-1) -curves. A

Why it matters

OpenAlex reports 11 citations for this work. Citation counts describe recorded attention and do not establish research quality.

Key contribution

A contribution statement is not available in the OpenAlex record.

Method / approach

Method details are not available in the OpenAlex metadata.

Main findings

Findings are not separately available in the OpenAlex metadata.

Limitations

Limitations are not available in the OpenAlex metadata.

Applications

Application details are not available in the OpenAlex metadata.

Available abstract

We give a characterization for a ( − 1 ) (-1) -divisor D D on a smooth rational surface X X to be irreducible under the assumption that an anticanonical divisor − K X -K_X of X X is nef. Here − K X -K_X is nef means K X . C ≤ 0 K_X . C \leq 0 for every effective divisor C C on X X , and a ( − 1 ) (-1) -divisor D D is a divisor such that the two numerical conditions D 2 = − 1 = D . K X D^2 =-1=D.K_X hold. As an application we give explicit examples of blowing up the projective plane at nine points infinitely near such that the obtained surface has an infinite number of ( − 1 ) (-1) -curves. A

Key concepts: Divisor (algebraic geometry), Irreducibility, Mathematics, Rational surface, Surface (topology), Plane (geometry), Pure mathematics, Intersection (aeronautics)

Related papers

Related paper 1Related paper 2
Back to paper searchBrowse research topicsOriginal source
Irreducibility of the (-1)-classes on smooth rational surfaces — Research Paper | ScholarLens