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Elliptic Tales by Robert Gross, Avner Ash

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Chapter 4

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MULTIPLICITIES AND DEGREE

Road Map

The process of forcing our two definitions of degree to give the same answer is still not at an end. We have already added elements to our number systems and to the plane. Unfortunately, the line y = 0 will intersect the parabola y = x2 only at the point (0, 0), even if we use complex numbers and the projective plane. To solve this problem, we must invent an idea called multiplicity, which allows us to count certain points of intersection more than once. In particular, we will count the point (0, 0) in this example twice, forcing a solution to our problem.

1. Curves as Varieties

If we have a homogeneous ...

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