Download PDF by Shafarevich I.R.: Algebraic geometry

By Shafarevich I.R.

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Let d be a nonnegative integer. A homogeneous polynomial F(x, y, z) of degree d in variables x, y, z is an expression X F(x, y, z) ¼ eij xi y j z dÀiÀj , (1) where the sigma represents summation, the coefficients eij are real numbers that are not all zero, and i and j vary over pairs of nonnegative integers whose sum is at most d. In short, a homogeneous polynomial of degree d is a nonzero polynomial such that the exponents of the variables in every term sum to d. We use capital letters to designate homogeneous polynomials.

We use transformations in two key ways. First, we compute the intersection multiplicity of two curves at any point in the projective plane by transforming that point to the origin and using the techniques of Section 1. Second, we use transformations to simplify the equations of curves. 3 A transformation is a map from the projective plane to itself that takes any point (x, y, z) to the point (x 0 , y 0 , z 0 ) determined by the equations x 0 ¼ ax þ by þ cz, y 0 ¼ dx þ ey þ fz, 0 z ¼ gx þ hy þ iz, (5) 36 I.

4. 42 I. Intersections of Curves We can now generalize the intersection properties in Section 1 from intersections at the origin to intersections at any point. 4 and the fact that transformations preserve intersection multiplicities to transform any point of intersection of two curves to the origin. 6 In the projective plane, let F(x, y, z) ¼ 0, G(x, y, z) ¼ 0, and H(x, y, z) ¼ 0 be curves, and let P be a point. Then the following results hold: (i) (ii) (iii) (iv) (v) (vi) IP (F, G) is a nonnegative integer or y.

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Algebraic geometry by Shafarevich I.R.

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