Download PDF by C. Herbert Clemens: A scrapbook of complex curve theory

By C. Herbert Clemens

ISBN-10: 0821833073

ISBN-13: 9780821833070

ISBN-10: 5432108070

ISBN-13: 9785432108074

This high-quality e-book through Herb Clemens quick turned a favourite of many algebraic geometers whilst it used to be first released in 1980. it's been well-liked by newcomers and specialists ever on account that. it really is written as a publication of 'impressions' of a trip during the idea of advanced algebraic curves. Many themes of compelling good looks take place alongside the best way. A cursory look on the topics visited unearths a superbly eclectic choice, from conics and cubics to theta capabilities, Jacobians, and questions of moduli. by means of the top of the ebook, the topic of theta services turns into transparent, culminating within the Schottky challenge. The author's cause was once to inspire extra research and to stimulate mathematical task. The attentive reader will study a lot approximately complicated algebraic curves and the instruments used to check them. The e-book could be in particular invaluable to an individual getting ready a direction relating to advanced curves or an individual drawn to supplementing his/her interpreting

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Extra resources for A scrapbook of complex curve theory

Sample text

Fu>) irgendein zu a zugehoriger Zweig von C. Setzt man cr = f<" — K = a t + . M so folgt aus /(£<'>, £">, f<») = / ( l , a t + a, f) = 0 u n d S ' W ^ ' . <(ft°, 3 ° , fix)). a* = «,+ (--^/j£) *+.... >. >. 3 ° • #>) (» = 3, . , «). (1) Da f ein Zweig ist, so ist a =)= 0 wegen der Bedingung in der Definition eines Zweiges. 3 £(I) zu $ Equivalent ist. S a t z B. f(l), | ( 2 ) £(r) seien die r zugehorigen Zweige von a. in reduzierter Form. , x) schneidet die Kurve C in a r dann und nur dann s-fach, wenn JJ Fl («, f (0 ) durch rs teilbar ist.

Daraus folgt, daB T* (A, /<) durch Z Aj Fj (6>w) fur i = 2, . . , n teilbar ist, mithin auch durch j h i = 2 (zijFj(&% j Folglich. ist fliZhFM*)) eine Linearform in A mit Koeffizienten aus K (<5). i), und folglich auch ZXjQj, nicht identisch verschwindcn. Also ist das Verschwinden von T* (A, /u) fur eine von p, unabhangige Wahl der A gleichbedeutend mit ZXjQj = 0. <) = 0 mit der Teilbarkeit von T* (A, w) durch ZdtVi gleichbedeutend ist, ist damit der Satz bewiesen. Es folgt aus diesem Satz: 1.

Die Verbindungslinie von | CI) und | ( 2 ) schneide die Hyperebene xQ = 0 in einem Punkt T, der bei der relationstreuen Spezialisierung (A, &l\ f(2)) -*• (fi, rj, rj) in den Punkt co hineinriicken moge. Wir konnen T, = w, = 1 annehmen, dann ist f<2) = £(1) + (|<2) - £<») T. ,- (A, |<2>) = Qt (A, £<*> + (f<2> - £«) r) nach den Potenzen von (fi2) — fi1*)) s o haben wir GW, f<2>) = (f<,2) - *<»>)'* J5f^ (A; £<», T) + (£<2) - I™)"' + ' Hf. oder. durch (^ s) + , (A; £<», T) + . . ,( + , (A; £u>, T ) + ..

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A scrapbook of complex curve theory by C. Herbert Clemens

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