B2.10

Algebraic Curves | Part II, 2004

For each of the following curves CC

(i) C={(x,y)A2x3x=y2}C=\left\{(x, y) \in \mathbb{A}^{2} \mid x^{3}-x=y^{2}\right\} \quad (ii) C={(x,y)A2x2y+xy2=x4+y4}C=\left\{(x, y) \in \mathbb{A}^{2} \mid x^{2} y+x y^{2}=x^{4}+y^{4}\right\}

compute the points at infinity of CˉP2\bar{C} \subset \mathbb{P}^{2} (i.e. describe Cˉ\C\bar{C} \backslash C ), and find the singular points of the projective curve Cˉ\bar{C}.

At which points of Cˉ\bar{C} is the rational map CˉP1\bar{C} \rightarrow \mathbb{P}^{1}, given by (X:Y:Z)(X:Y)(X: Y: Z) \mapsto(X: Y), not defined? Justify your answer.

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