scholarly journals On the geometry of bifurcation currents for quadratic rational maps

2014 ◽  
Vol 35 (5) ◽  
pp. 1369-1379 ◽  
Author(s):  
FRANÇOIS BERTELOOT ◽  
THOMAS GAUTHIER
Keyword(s):  
Projective Space ◽  
Moduli Space ◽  
Complex Projective Space ◽  
The Self ◽  
Rational Maps ◽  
Two Dimensional ◽  
Lelong Numbers ◽  
Complex Projective

We describe the behaviour at infinity of the bifurcation current in the moduli space of quadratic rational maps. To this purpose, we extend it to some closed, positive $(1,1)$-current on a two-dimensional complex projective space and then compute the Lelong numbers and the self-intersection of the extended current.

scholarly journals COMPLETE INTERSECTIONS AND MOVABLE CURVES ON THE MODULI SPACE OF SIX-POINTED RATIONAL CURVES

2012 ◽  
Vol 22 (06) ◽  
pp. 1250049
Author(s):  
PAUL L. LARSEN
Keyword(s):  
Projective Space ◽  
Complete Intersection ◽  
Moduli Space ◽  
Complex Projective Space ◽  
Projective Variety ◽  
Rational Curves ◽  
Complete Intersections ◽  
Family Of Curves ◽  
Complex Projective

A curve on a projective variety is called movable if it belongs to an algebraic family of curves covering the variety. We consider when the cone of movable curves can be characterized without existence statements of covering families by studying the complete intersection cone on a family of blow-ups of complex projective space, including the moduli space of stable six-pointed rational curves and the permutohedral or Losev–Manin moduli space of four-pointed rational curves. Our main result is that the movable and complete intersection cones coincide for the toric members of this family, but differ for the non-toric member, the moduli space of six-pointed rational curves. The proof is via an algorithm that applies in greater generality. We also give an example of a projective toric threefold for which these two cones differ.

scholarly journals A Property of upper level sets of Lelong numbers of currents on ℙ2

2017 ◽  
Vol 28 (14) ◽  
pp. 1750110 ◽  
Author(s):  
James J. Heffers
Keyword(s):  
Projective Space ◽  
Level Set ◽  
Level Sets ◽  
Complex Projective Space ◽  
Lelong Number ◽  
Lelong Numbers ◽  
Closed Current ◽  
Upper Level ◽  
Complex Projective ◽  
Positive Closed Current

Let [Formula: see text] be a positive closed current of bidimension [Formula: see text] with unit mass on the complex projective space [Formula: see text]. For [Formula: see text] and [Formula: see text] we show that if [Formula: see text] has four points with Lelong number at least [Formula: see text], the upper level set [Formula: see text] of points of [Formula: see text] with Lelong number strictly larger than [Formula: see text] is contained within a conic with the exception of at most one point.

scholarly journals Primitive and decomposable elements in homology of ΩΣℂP ∞

2021 ◽  
Vol 19 (1) ◽  
pp. 1279-1289
Author(s):  
Dae-Woong Lee
Keyword(s):  
Positive Integer ◽  
Projective Space ◽  
Complex Projective Space ◽  
The Self ◽  
Complex Projective ◽  
Positive Integers

Abstract For each positive integer n n , we let φ n : Σ C P ∞ → Σ C P ∞ {\varphi }_{n}:\Sigma {\mathbb{C}}{P}^{\infty }\to \Sigma {\mathbb{C}}{P}^{\infty } be the self-maps of the suspension of the infinite complex projective space, or the localization of this space at a set of primes which may be an empty set. Furthermore, let [ φ m , φ n ] : Σ C P ∞ → Σ C P ∞ \left[{\varphi }_{m},{\varphi }_{n}]:\Sigma {\mathbb{C}}{P}^{\infty }\to \Sigma {\mathbb{C}}{P}^{\infty } be a commutator of self-maps φ m {\varphi }_{m} and φ n {\varphi }_{n} for any positive integers m m and n n . In the current study, we show that the image of the homomorphism [ φ ˆ m , φ ˆ n ] ∗ {\left[{\hat{\varphi }}_{m},{\hat{\varphi }}_{n}]}_{\ast } in homology induced by the adjoint [ φ ˆ m , φ ˆ n ] : C P ∞ → Ω Σ C P ∞ \left[{\hat{\varphi }}_{m},{\hat{\varphi }}_{n}]:{\mathbb{C}}{P}^{\infty }\to \Omega \Sigma {\mathbb{C}}{P}^{\infty } of the commutator [ φ m , φ n ] \left[{\varphi }_{m},{\varphi }_{n}] is both primitive and decomposable. As a further support of the above statement, we provide an example.

scholarly journals CR-SUBMANIFOLDS OF TWO DIMENSIONAL COMPLEX PROJECTIVE SPACE

1990 ◽  
Vol 21 (2) ◽  
pp. 171-175
Author(s):  
SHARIEF DESHMUKH ◽  
M. A. AL-GWAIZ
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Two Dimensional ◽  
Complex Projective ◽  
Cr Submanifolds

CR-SUBMANIFOLDS OF TWO DIMENSIONAL COMPLEX PROJECTIVE SPACE

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