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Journal ArticleDOI

Ueber Riemann'sche Flächen mit gegebenen Verzweigungspunkten

01 Mar 1891-Mathematische Annalen (Springer-Verlag)-Vol. 39, Iss: 1, pp 1-60
TL;DR: The Riemannsche Theorie der algebraischen Funktionen as discussed by the authors is a Theorie von der graphisch uber der komplexen Zahlenebene.
Abstract: Die grundlegende Bedeutung des vorliegenden Themas fur die Riemann’sche Theorie der algebraischen Funktionen brauche ich wohl kaum hervorzuheben. Geht doch diese Theorie von der graphisch uber der komplexen Zahlenebene konstruierten Riemann’schen Flache aus, um erst sodann die Funktionen, welche durch diese Flache bestimmt sind, zu untersuchen.

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TL;DR: When X is smooth, this turns out to be equivalent to the a priori weaker condition that two general points can be joined by a chain of rational curves and also to the stronger condition that for any finite subset Γ ⊂ X, there is a map g : P → X whose image contains Γ and such that gTX is an ample bundle as discussed by the authors.
Abstract: Recall that a proper variety X is said to be rationally connected if two general points p, q ∈ X are contained in the image of a map g : P → X. This is clearly a birationally invariant property. When X is smooth, this turns out to be equivalent to the a priori weaker condition that two general points can be joined by a chain of rational curves and also to the a priori stronger condition that for any finite subset Γ ⊂ X, there is a map g : P → X whose image contains Γ and such that gTX is an ample bundle.

495 citations

Journal ArticleDOI
TL;DR: In this article, Teke et al. presented a survey of the state-of-the-art mathematics departments in Sweden, including the Department of Mathematics, University of Stockholm, S-10691, Stockholm, Sweden (e.g.
Abstract: 1 Department of Mathematics, University of Stockholm, S-10691, Stockholm, Sweden (e-mail: teke@matematik.su.se) 2 Higher College of Mathematics, Independent University of Moscow and Institute for System Research RAS, Moscow, Russia (e-mail: lando@mccme.ru) 3 Department of Mathematics, Royal Institute of Technology, S-10044, Stockholm, Sweden (e-mail: mshapiro@math.kth.se) 4 Department of Mathematics and Department of Computer Science, University of Haifa, Haifa 31905, Israel (e-mail: alek@mathcs.haifa.ac.il)

364 citations

Posted Content
TL;DR: In this article, it was shown that a one-parameter family of rationally connected (over an algebraically closed field of characteristic 0) always has a section, and that such a family can be expressed as follows:
Abstract: We prove that a one-parameter family of rationally connected varieties (over an algebraically closed field of characteristic 0) always has a section.

360 citations

Journal ArticleDOI
TL;DR: In this article, the authors extend Schaeffer's bijection between rooted quadrangulations and well-labeled trees to the general case of Eulerian planar maps with prescribed face valences to obtain a new class of labeled trees, which they call mobiles.
Abstract: We extend Schaeffer's bijection between rooted quadrangulations and well-labeled trees to the general case of Eulerian planar maps with prescribed face valences to obtain a bijection with a new class of labeled trees, which we call mobiles. Our bijection covers all the classes of maps previously enumerated by either the two-matrix model used by physicists or by the bijection with blossom trees used by combinatorists. Our bijection reduces the enumeration of maps to that, much simpler, of mobiles and moreover keeps track of the geodesic distance within the initial maps via the mobiles' labels. Generating functions for mobiles are shown to obey systems of algebraic recursion relations.

319 citations

References
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TL;DR: In den nachfolgenden Zeilen beschaftige ich mich namentlich mit der Aufgabe: alle irreduzibeln algebraischen Gleichungen as discussed by the authors.
Abstract: In den nachfolgenden Zeilen beschaftige ich mich namentlich mit der Aufgabe: alle irreduzibeln algebraischen Gleichungen $$f = (s,z) = 0$$ zu bestimmen, welche durch eine rationale eindeutig umkehrbare Transformation $$\left\{ \begin{gathered}s' = \varphi (s,z) \hfill \\z' = \psi (s,z) \hfill \\\end{gathered} \right.$$ in sich ubergefuhrt werden konnen, oder — was offenbar auf dasselbe hinauskommt — alle diejenigen Riemann’schen Flachen (algebraischen Gebilde) anzugeben, auf welchen eine ein-eindeutige algebraische Korrespondenz (s, z; s′, z′) existiert. Der Fall, in welchem das Geschlecht p des Gebildes gleich Null oder Eins ist, bildet bei dieser Untersuchung einen leicht fur sich zu behandelnden, ubrigens seit langem erledigten Ausnahmefall. Ich setze deshalb im Folgenden, wenn ich nicht ausdrucklich das Gegenteil bemerke, stets voraus, dass das Geschlecht der zu betrachtenden Gebilde grosser ist als Eins.

16 citations