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Download e-book for kindle: Algebraische Geometrie [Lecture notes] by Scheithauer

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Analog folgt, dass {P ∈ C|νP (h) = 0} endlich ist. 50 Sei f ∈ K(C), f = 0. Dann ist der Divisor von f gegeben durch νP (f )P ∈ Div(C). (f ) = P ∈C Es gilt (f g) = (f ) + (g) und (1/f ) = −(f ) und (f ) = 0, falls f ∈ K. Ein Divisor D ∈ Div(C) heißt Hauptdivisor, falls D = (f ) f¨ ur ein f ∈ K(C). Die Menge P (C) der Hauptdivisoren von C bildet eine Untergruppe von Div(C). Der Quotient Div(C)/P (C) wird als Divisorklassengruppe bezeichnet. Beispiel. Sei C = P1K . Dann ist D ∈ P (C) ⇐⇒ grad(D) = 0.

M heißt Transzendenzbasis von L/K, wenn L/K(M ) algebraisch ist. 6. Zwei Transzendenzbasen haben die gleiche M¨achtigkeit. Sei L/K eine K¨orpererweiterung. Dann hat L eine Transzendenzbasis u ¨ber K und wir definieren den Transzendenzgrad von L/K trgradK L als die M¨achtigkeit dieser Transzendenzbasis. 44 Beispiel. , Xn ]/I(V ). , ym ) ist. , ym } ist also eine Transzendenzbasis von K(V )/K und trgradK K(V ) = m. 3 Algebraische Charakterisierung der Dimension Sei V eine irreduzible affine Variet¨at.

Sei γ ein Weg, der die Null- und Polstellen genau einmal durchl¨auft. Dann ist νP (f ) = P ∈C Res df f = 1 2πi 53 γ df 1 =− f 2πi −γ df = 0. f Sei C eine glatte projektive Kurve. Da die Hauptdivisoren Grad 0 haben, erhalten wir eine Abbildung Cl(C) → Z Die Jacobische von C ist definiert als Jac0 (C) = Cl0 (C) = {D ∈ Cl(C)| grad(D) = 0}. Wir haben eine kurze Sequenz 0 → Cl0 (C) → Cl(C) → Z → 0. 13. Sei C eine glatte projektive Kurve. h. ist birational, also isomorph ) zuP1K . Beweis. ⇐“ Sei C ∼ = P1K .

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Algebraische Geometrie [Lecture notes] by Scheithauer


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