Algebraic Geometry

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Example text

0 n zm0 ... 23. Das Bild der Segre-Abbildung ist die projektive Variet¨at, definiert durch das Verschwinden der 2 × 2-Minoren der Matrix (zij ). Die Segre-Abbildung ist injektiv.   x0  ..  Beweis. Das Bild von smn erh¨alt man durch Multiplikation der Matrizen  .  und xm   x0  ..  y0 ... yn . Die Spalten der Produktmatrix sind Vielfache von  . h. (zij ) hat xm Rang 1. Somit verschwinden alle 2 × 2-Minoren von (zij ). (m+1)(n+1)−1 Angenommen, die Koordinaten eines Punktes z in PK erf¨ ullen die Gleichungen zij zkl − zij zkj = 0 f¨ ur alle 0 ≤ i, k ≤ m und 0 ≤ j, l ≤ n.

Die Abbildung d : I(P ) → (K n ) (1) f → fP surjektiv, weil d(Xi ) = Xi . Der Kern von d ist I(P )2 : ∂f (P ) = 0 f¨ ur alle i ∂xi ⇐⇒ der Grad jedes Monoms in f ist ≥ 2 ⇐⇒ f ∈ I(P )2 . f ∈ ker(d) ⇐⇒ 45 Die Inklusion T pV → K n liefert eine surjektive Abbildung D : (K n ) → (T pV ) . Die Komposition I(P ) → (K n ) → (T pV ) ist surjektiv. Es gilt ker(D) = I(P )2 + I(V ), denn (1) f ∈ ker(D) ⇐⇒ fP |T pV = 0 (1) ci gi,P f¨ ur geeignete gi ∈ I(V ), ci ∈ K( TpV ist definiert als gP = 0 f¨ ur g ∈ (1) ci gp(1) = 0 (1) ci g i ⇐⇒ fP = ⇐⇒ fP − ⇐⇒ fP − (1) (1) (1) ∈ I(P )2 ⇐⇒ f ∈ I(P )2 + I(V ).

7. Sei C eine glatte projektive Kurve und f ∈ K(C)∗ . Dann gibt es nur endlich viele Punkte mit νP (f ) = 0. Beweis. Sei C ⊂ PnK . , Xn ] homogen und grad(g) = grad(h). Ist νP (f ) = 0, so ist νP (g) = 0 oder νP (h) = 0. Wir zeigen, dass die Menge {P ∈ C|νP (g) = 0} ¨ endlich ist: Sei C = C0 ∪ ... ∪ Cn eine Uberdeckung in affine Variet¨aten. h. eine glatte affine Variet¨at der Dimension 1. Dann ist g ∈ K[Ci ] Die Nullstellenmenge von g auf Ci zerf¨allt in endlich viele Komponenten. Diese sind affine Variet¨aten der Dimension 0 und somit endlich.

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