On the Hyperplane Sections of Blow-Ups of Complex Projective Plane

1989 ◽  
Vol 41 (6) ◽  
pp. 1005-1020 ◽  
Author(s):  
Aldo Biancofiore
Keyword(s):  
Line Bundle ◽  
Projective Plane ◽  
Algebraic Surfaces ◽  
Projective Surface ◽  
Complex Projective Plane ◽  
Complex Projective ◽  
Hyperplane Sections

Let L be a line bundle on a connected, smooth, algebraic, projective surface X. In this paper we have studied the following questions:1) Under which conditions is L spanned by global sections? I.e., if ɸL : X →PN denotes the map associated to the space Г(L) of the sections of L, when is ɸL a morphism?2) Under which conditions is L very ample? I.e., when does ɸL give an embedding?These problems arise naturally in the study, and in particular in the classification, of algebraic surfaces (see [8], [3], [5]).

scholarly journals Real Hypersurfaces of Nonflat Complex Projective Planes Whose Jacobi Structure Operator Satisfies a Generalized Commutative Condition

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Theocharis Theofanidis
Keyword(s):  
Projective Plane ◽  
Projective Planes ◽  
Additional Restriction ◽  
Real Hypersurfaces ◽  
Specific Function ◽  
Complex Projective Plane ◽  
Jacobi Structure ◽  
Complex Projective

Real hypersurfaces satisfying the conditionϕl=lϕ(l=R(·,ξ)ξ)have been studied by many authors under at least one more condition, since the class of these hypersurfaces is quite tough to be classified. The aim of the present paper is the classification of real hypersurfaces in complex projective planeCP2satisfying a generalization ofϕl=lϕunder an additional restriction on a specific function.

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.

Spectral Sequences

2020 ◽  
pp. 45-56
Author(s):  
Loring W. Tu
Keyword(s):  
Projective Plane ◽  
Spectral Sequence ◽  
Fiber Bundles ◽  
Spectral Sequences ◽  
Complex Projective Plane ◽  
Computational Tool ◽  
Short Introduction ◽  
Complex Projective

This chapter focuses on spectral sequences. The spectral sequence is a powerful computational tool in the theory of fiber bundles. First introduced by Jean Leray in the 1940s, it was further refined by Jean-Louis Koszul, Henri Cartan, Jean-Pierre Serre, and many others. The chapter provides a short introduction, without proofs, to spectral sequences. As an example, it computes the cohomology of the complex projective plane. The chapter then details Leray's theorem. A spectral sequence is like a book with many pages. Each time one turns a page, one obtains a new page that is the cohomology of the previous page.

scholarly journals SMOOTH, NONSYMPLECTIC EMBEDDINGS OF RATIONAL BALLS IN THE COMPLEX PROJECTIVE PLANE

2020 ◽  
Vol 71 (3) ◽  
pp. 997-1007
Author(s):  
Brendan Owens
Keyword(s):  
Projective Plane ◽  
Infinite Family ◽  
Complex Projective Plane ◽  
Rational Homology ◽  
Complex Projective

Abstract We exhibit an infinite family of rational homology balls, which embed smoothly but not symplectically in the complex projective plane. We also obtain a new lattice embedding obstruction from Donaldson’s diagonalization theorem and use this to show that no two of our examples may be embedded disjointly.

The Space of Harmonic Maps from the 2-Sphere to the Complex Projective Plane

1997 ◽  
Vol 40 (3) ◽  
pp. 285-295 ◽  
Author(s):  
T. Arleigh Crawford
Keyword(s):  
Projective Plane ◽  
Harmonic Maps ◽  
Complex Manifolds ◽  
Complex Projective Plane ◽  
Fixed Degree ◽  
Complex Projective ◽  
Path Connected

AbstractIn this paper we study the topology of the space of harmonic maps from S2 to ℂℙ2.We prove that the subspaces consisting of maps of a fixed degree and energy are path connected. By a result of Guest and Ohnita it follows that the same is true for the space of harmonic maps to ℂℙn for n ≥ 2. We show that the components of maps to ℂℙ2 are complex manifolds.

scholarly journals Towards the weighted Bounded Negativity Conjecture for blow-ups of algebraic surfaces

2019 ◽  
Vol 163 (3-4) ◽  
pp. 361-373
Author(s):  
Roberto Laface ◽  
Piotr Pokora
Keyword(s):  
Line Bundle ◽  
Characteristic Zero ◽  
Algebraic Surface ◽  
Intersection Number ◽  
Algebraic Surfaces ◽  
The Self ◽  
Projective Surface ◽  
Intersection Numbers ◽  
Weighted Version ◽  
Smooth Projective Surface

AbstractIn the present paper we focus on a weighted version of the Bounded Negativity Conjecture, which predicts that for every smooth projective surface in characteristic zero the self-intersection numbers of reduced and irreducible curves are bounded from below by a function depending on the intesection of curve with an arbitrary big and nef line bundle that is positive on the curve. We gather evidence for this conjecture by showing various bounds on the self-intersection number of curves in an algebraic surface. We focus our attention on blow-ups of algebraic surfaces, which have so far been neglected.

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