# Global structure of holomorphic webs on surfaces

Vincent Cavalier; Daniel Lehmann

Banach Center Publications (2008)

- Volume: 82, Issue: 1, page 35-44
- ISSN: 0137-6934

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topVincent Cavalier, and Daniel Lehmann. "Global structure of holomorphic webs on surfaces." Banach Center Publications 82.1 (2008): 35-44. <http://eudml.org/doc/282159>.

@article{VincentCavalier2008,

abstract = {The webs have been studied mainly locally, near regular points (see a short list of references on the topic in the bibliography). Let d be an integer ≥ 1. A d-web on an open set U of ℂ² is a differential equation F(x,y,y’) = 0 with $F(x,y,y^\{\prime \}) = ∑_\{i=0\}^\{d\} a_i(x,y)(y^\{\prime \})^\{d-i\}$, where the coefficients $a_i$ are holomorphic functions, a₀ being not identically zero. A regular point is a point (x,y) where the d roots in y’ are distinct (near such a point, we have locally d foliations mutually transverse to each other, and caustics appear through the points which are not regular). It happens that many concepts on local webs may be globalized, but not always in an obvious way, and under the condition that they do not depend on local coordinates. The aim of this paper is to make these facts precise and to define the tools necessary for a global study of webs on a holomorphic surface, and in particular on the complex projective plane ℙ₂. Moreover new concepts, inducing new problems, will appear, such as the dicriticality, the irreducibility or the quasi-smoothness, which have no interest locally near a regular point of the web.},

author = {Vincent Cavalier, Daniel Lehmann},

journal = {Banach Center Publications},

keywords = {global web; dicriticality; type; degree; indistinguishability, quasi-smoothness, Blaschke curvature, Chern curvature, abelian relation},

language = {eng},

number = {1},

pages = {35-44},

title = {Global structure of holomorphic webs on surfaces},

url = {http://eudml.org/doc/282159},

volume = {82},

year = {2008},

}

TY - JOUR

AU - Vincent Cavalier

AU - Daniel Lehmann

TI - Global structure of holomorphic webs on surfaces

JO - Banach Center Publications

PY - 2008

VL - 82

IS - 1

SP - 35

EP - 44

AB - The webs have been studied mainly locally, near regular points (see a short list of references on the topic in the bibliography). Let d be an integer ≥ 1. A d-web on an open set U of ℂ² is a differential equation F(x,y,y’) = 0 with $F(x,y,y^{\prime }) = ∑_{i=0}^{d} a_i(x,y)(y^{\prime })^{d-i}$, where the coefficients $a_i$ are holomorphic functions, a₀ being not identically zero. A regular point is a point (x,y) where the d roots in y’ are distinct (near such a point, we have locally d foliations mutually transverse to each other, and caustics appear through the points which are not regular). It happens that many concepts on local webs may be globalized, but not always in an obvious way, and under the condition that they do not depend on local coordinates. The aim of this paper is to make these facts precise and to define the tools necessary for a global study of webs on a holomorphic surface, and in particular on the complex projective plane ℙ₂. Moreover new concepts, inducing new problems, will appear, such as the dicriticality, the irreducibility or the quasi-smoothness, which have no interest locally near a regular point of the web.

LA - eng

KW - global web; dicriticality; type; degree; indistinguishability, quasi-smoothness, Blaschke curvature, Chern curvature, abelian relation

UR - http://eudml.org/doc/282159

ER -

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