Let C: F(X,Y,Z) = 0 be a projective curve given by a homogeneous poly- nomial F...

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Let C: F(X,Y,Z) = 0 be a projective curve given by a homogeneous poly- nomial F € C[X, Y, Z], and let P = P2 be a point. (a) Prove that P is a singular point of C if and only if OF Əx (P) = 3F (P) = ƏF ax OF az (P) = 0. ᎧᏃ (b) If P is a non-singular point of C, prove that by the equation OF (P)X + ay (P)Y+ the tangent line to C at P is given OF az (P)Z = 0. Let C: F(X,Y,Z) = 0 be a projective curve given by a homogeneous poly- nomial F € C[X, Y, Z], and let P = P2 be a point. (a) Prove that P is a singular point of C if and only if OF Əx (P) = 3F (P) = ƏF ax OF az (P) = 0. ᎧᏃ (b) If P is a non-singular point of C, prove that by the equation OF (P)X + ay (P)Y+ the tangent line to C at P is given OF az (P)Z = 0.

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Answer a Let P xyz P2 By definition P is a singular point of C if and only if there e... View the full answer