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By Ivan Soprunov

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First, (x0 , . . , xn ) ∼ (x0 , . . , xn ) if and only if (x0 , . . , xn ) = λ(x0 , . . , xn ) for some λ ∈ F∗ is an equivalence relation on An+1 \ {0}. We obtain � An+1 \ {0} � = classes of (x0 , . . , xn ) ∈ An+1 \ {0} under ∼ . ∼ We denote by (x0 : · · · : xn ) the equivalence class of (x0 , . . , xn ) and call it the homogeneous coordinate of the corresponding point in Pn . In particular, we have the projective line: Pn = P2 = {lines in A3 passing through (0, 0, 0)}. Two points (x1 , y1 , z1 ) ∼ (x2 , y2 , z2 ) if and only if (x2 , y2 , z2 ) = λ(x1 , y1 , z1 ) for some λ ∈ F∗ , in which case they define the same line in A3 through (0, 0, 0).

There exist u, v ∈ F(x)[y] such that uf + vg = 1. After clearing the denominators in u, v we get u1 f + v1 g = c(x) for some c(x) ∈ F[x] and u1 , v1 ∈ F[x, y]. If (α, β) is a common zero of f, g then c(α) = 0. Therefore there could be only finitely many such α. Furthermore, β must be a root of f (α, y), so there are only finitely many such β as well. � It is an interesting question how many common zeroes f and g can have. We will answer this question in the case of an algebraically closed field.

Let C ⊂ P2 be a projective curve. The set CA = C ∩ Uz is called the affine part of C. It is defined by the polynomial f (x, y) = F (x, y, 1). 40 2. 3. Tangent Lines and Singular Points We begin with an example of a projective curve in P2C intersecting three different lines. 46. We will find the intersection points of a conic C with equation x2 + 4y 2 − z 2 = 0 and (a) L1 given by x − y = 0, (b) L2 given by x − 2z = 0, (c) L3 given by x − z = 0 2 2 2 For C ∩ L1 substituting x= √ √ y into the equation of C we get x + 4x −√z = 0 which factors √ as ( 5x − z)( 5 + z) = 0.

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