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Extra info for Conics and Cubics: A Concrete Introduction to Algebraic Curves (Undergraduate Texts in Mathematics)

Example text

Then there is a transformation that fixes N and every point of l, maps A to A0 , and sends each point X to a point X 0 collinear with X and N. (a) Prove the theorem when N is the origin and l is the line at infinity by considering the transformations x 0 ¼ rx, y 0 ¼ ry, z 0 ¼ z for nonzero numbers r. (b) Prove the theorem when l is the line at infinity and N is the point at infinity on vertical lines by considering the transformations x 0 ¼ x, y 0 ¼ y þ kz, z 0 ¼ z for nonzero numbers k. 4. 20. 6): Desargues’ Theorem Let A, C, E, A0 , C 0 , E0 be distinct points such that no two of the lines AC, CE, AE, A0 C 0 , C 0 E0 , A0 E0 , AA0 , CC 0 , EE0 are equal.

Then the total number of intersections in the projective plane is n minus the degree of r(x), as the theorem asserts. 54 I. Intersections of Curves To prove the claim, we count the intersections of y ¼ 0 and G ¼ 0 at infinity. The only possible point of intersection is (1, 0, 0), since this is the only point at infinity on the line y ¼ 0. To send (1, 0, 0) to the origin (0, 0, 1), we interchange x and z with a transformation (as in (8) of §3). 5). 1). 11. That degree is the least exponent on z in a nonzero term of G(x, 0, z).

If P lies at infinity, find the slope of the lines in the Euclidean plane that contain P. (a) x þ 2y À 6z ¼ 0 and 3x þ 4y À 15z ¼ 0. (b) À2x þ 4y À z ¼ 0 and x À 2y þ 3z ¼ 0. (c) 3x þ y þ 5z ¼ 0 and z ¼ 0. (d) 2x þ 3y À 6z ¼ 0 and Àx þ y þ 3z ¼ 0. (e) 6x À 2y þ 4z ¼ 0 and 3x À z ¼ 0. (f ) 3x þ y À 2z ¼ 0 and 6x þ 2y þ 5z ¼ 0. (g) 4x þ 3y þ 16z ¼ 0 and 3x þ 2y þ 10z ¼ 0. 6. In each part of this exercise, homogeneous coordinates are given for two points in the projective plane. 2). Find an equation for l in homogeneous coordinates in the form of (2).

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