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American Journal of Mathematics
Article . 1931 . Peer-reviewed
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Perspective Elliptic Curves

Perspective elliptic curves
Authors: Cooper, Elizabeth Morgan;

Perspective Elliptic Curves

Abstract

Zwei ebene Kurven heißen perspektiv, wenn zwischen den Punkten der ersten und den Tangenten der zweiten eine \((1,1)\)-Korrespondenz derart hergestellt werden kann, daß jede Tangente der zweiten durch den entsprechenden Punkt der ersten läuft. So sind z. B. zwei Strahlenbüschel, die eine Kurve 2, Ordnung erzeugen, zu dieser Kurve perspektiv. Allgemeiner hat es, wenn man Erzeugungsmöglichkeiten einer Kurve untersucht, ein Interesse, nach Kurven vorgeschriebener Ordnung zu fragen, die zu ihr perspektiv sind. Die vorhandenen Arbeiten über perspektive Kurven beziehen sich auf den Fall (ebener und räumlicher) rationaler Kurven. Hier wird der Fall ebener elliptischer Kurven behandelt. Die Hauptresultate lauten: Es existieren \(\infty^{2m-n}\) zu einer vorgegebenen Kurve \(n\)-ter Ordnung perspektive Kurven \(m\)-ter Ordnung. Diese gruppieren sich zu \(\infty^1\) linearen Familien, deren jede einer birationalen Transformation der Kurve entspricht. Eine Kurve \(n\)-ter Ordnung und eine zu ihr Perspektive Kurve \(m\)-ter Ordnung haben im allgemeinen \(m+n\) Berührungspunkte. Eine genauere Behandlung perspektiver elliptischer Kurven 3. Ordnung beschließt die Arbeit.

Keywords

geometry, Geometry

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popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
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