Riemann Surfaces and Algebraic Curves

Abstract

If F(Z, W) is a polynomial in two variables, with complex coefficients, that is not simply a constant, its zero set

$$ C{\text{ }} = \{ (z,w) \in {{\mathbb{C}}^{{\text{2}}}}:F\left( {z,w} \right){\text{ }} = 0\} $$

is called a “complex affine plane curve.” Identifying ℂ² with ℝ⁴, C is defined by two real equations: the vanishing of the real and imaginary parts of F(z,w). We may therefore expect C to be a surface, and this expectation is generally true, except that, just as in the case of real curves, C may have singularities. We will use the construction of the preceding section to “remove” these singularities, and also add some points over them and “at infinity,” to get a compact Riemann surface. In fact, if F is not irreducible, the surface we get will be the disjoint union of the surfaces we get from the irreducible factors of F, so we assume for now that F is an irreducible polynomial, i.e., it has no nontrivial factors but constants. Write

$$ F(Z,W) = {{a}_{0}}(Z){{W}^{n}} + {{a}_{1}}(Z){{W}^{{n - 1}}} + \ldots + {{a}_{{n - 1}}}(Z)W + {{a}_{n}}(Z), $$

with a _i (Z) a polynomial in Z alone, and a₀(Z)≠0. We may also assume that n is positive, for otherwise F = bZ + c, and C is isomorphic to ℂ, given by the projection to the second factor. We will need a little piece of algebra.