B2.10

Algebraic Curves | Part II, 2001

Let f:P2P2f: \mathbb{P}^{2} \rightarrow \mathbb{P}^{2} be the rational map given by f(X0:X1:X2)=(X1X2:X0X2f\left(X_{0}: X_{1}: X_{2}\right)=\left(X_{1} X_{2}: X_{0} X_{2}\right. : X0X1)\left.X_{0} X_{1}\right). Determine whether ff is defined at the following points: (1:1:1),(0:1:1),(0:(1: 1: 1),(0: 1: 1),(0: 0:1)0: 1).

Let CP2C \subset \mathbb{P}^{2} be the curve defined by X12X2X03=0X_{1}^{2} X_{2}-X_{0}^{3}=0. Define a bijective morphism α:P1C\alpha: \mathbb{P}^{1} \rightarrow C. Prove that α\alpha is not an isomorphism.

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