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Enumerative Algebraic Geometry: Proceedings of the 1989 by Zeuthen Symposium, Steven L. Kleiman, H. G. Zeuthen, Anders

By Zeuthen Symposium, Steven L. Kleiman, H. G. Zeuthen, Anders Thorup

1989 marked the a hundred and fiftieth anniversary of the start of the nice Danish mathematician Hieronymus Georg Zeuthen. Zeuthen's identify is identified to each algebraic geometer as a result of his discovery of a uncomplicated invariant of surfaces. notwithstanding, he additionally did primary learn in intersection conception, enumerative geometry, and the projective geometry of curves and surfaces. Zeuthen's striking devotion to his topic, his attribute intensity, thoroughness, and readability of proposal, and his special and succinct writing type are really inspiring.

During the earlier ten years or so, algebraic geometers have reexamined Zeuthen's paintings, drawing from it notion and new instructions for improvement within the box. The 1989 Zeuthen Symposium, held in the summertime of 1989 on the Mathematical Institute of the college of Copenhagen, supplied a ancient chance for mathematicians to collect and look at these parts in modern mathematical examine that have developed from Zeuthen's fruitful principles. This quantity, containing papers offered in the course of the symposium, in addition to others encouraged through it, illuminates a few at present energetic parts of study in enumerative algebraic geometry

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Extra info for Enumerative Algebraic Geometry: Proceedings of the 1989 Zeuthen Symposium : Proceedings of a Symposium Held July 30-August 6, 1989, With Support Fro

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Rn die Abstande der Seitengeraden von 0, so gilt G(R 1 , G(r1 , •• • , Rn} ....... ••• , rn) n "",,-sec-. n - Bezeichnen wir namlich die Eckpunkte in natiirlicher Reihenfolge mit PI' ... , P n , P n + 1 = PI' den Abstand von Pi P H1 undO mit riLden Winkel Pi 0 Pi + 1 mit 2 ai und den Abstand 0 Pi mit Ri und bewegen wir die Punkte Pi und Pi + 1 auf den fest en Geraden 0 Pi bzw. OPi + 1 so, daB der Inhalt des Dreiecks 0 Pi Pi + 1 konstant bleibt, so umhiillen die Geraden Pi Pi + 1 eine Hyperbel.

Die Erfiillung dieses Kriteriums besagt, daB die Flache mit der Gleichung z = I (x, y) elliptisch gekriimmt ist. Zur Entscheidung, ob in dies em Fall I (x, y) konvex oder konkav ist, hat man etwa das Vorzeichen von lu oder 11/1/ zu untersuchen. 1st lu ~ 0 (und folglich auch 11/1/~0), so ist I(x, y) konvex. Dagegen ist im Fall lzap lu ~ 0 unsere Funktion konkav. Sind diese letzteren Bedingungen erfiillt, so ist I(x, y) auch noch im Fall lu/1/1/- 1;11 ~ 0 konvex bzw. konkav. Betrachten wir die Funktion ·z = x« yfJ; x, Y ~ 0, 0 ~ ~,P die mit Riicksicht auf so gilt xiY~ Zz zZ1/1/ - + ...

Wir haben jetzt noeh zu zeigen, daB in (1) Gleiehheit nur fUr eine Ellipse gilt. 1st Tn (I) keine Konstante, so gibt es wegen der Stetigkeit von Tn (t) sieher einen Wert to fUr den T,,(to) > T 2nn sin ~nn ausfallt. Doeh kann - wie SAS [2] gezeigt hat - T,,(t) eine Konstante sein, ohne daB e (t) selbst eine Konstante ist. Wir zeigen, daB in dies em Fall Tn (t) nieht das einbesehriebene n-Eek vom groBtmogliehen Inhalt ist. Setzen wir voraus, daB die Polygone T,,(t) bei keinem Wert von t sieh vergroBern lassen.

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