Riemann Surfaces and Algebraic Curves
Riemann Surfaces and Algebraic Curves
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黎曼曲面和代数曲线
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发表时间：
1995
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通讯作者：
W. Fulton
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作者：
W. Fulton
如果 F(Z, W) 是两个变量的多项式，具有复系数，那么它不仅仅是一个常数，它的零集
$$ C{\text{ }} = \{ (z,w) \in {{\mathbb{C}}^{{\text{2}}}}:F\left( {z,w} \right){\text{ }} = 0\} $$
称为“复仿射平面曲线”。将 ℂ2 与 ℝ4 等同，C 由两个实数方程定义：F(z,w) 实部和虚部的消失。因此，我们可以期望 C 是一个曲面，并且这种期望通常是正确的，只是与真实曲线的情况一样，C 可能具有奇点。我们将使用上一节的构造来“删除”这些奇点，并在它们上方和“无穷远处”添加一些点，以获得紧凑的黎曼曲面。事实上，如果 F 不是不可约的，我们得到的曲面将是我们从 F 的不可约因子得到的曲面的不相交并，所以我们现在假设 F 是一个不可约多项式，即它没有非平凡因子，而是常数。写
$$ F(Z,W) = {{a}_{0}}(Z){{W}^{n}} + {{a}_{1}}(Z){{W}^{{n - 1}}} + \ldots + {{a}_{{n - 1}}}(Z)W + {{a}_{n}}(Z)，$$
其中 a i (Z) 是仅 Z 的多项式，且 a0(Z)≠0。我们还可以假设 n 为正，否则 F = bZ + c，并且 C 与 ℂ 同构，由第二个因子的投影给出。我们需要一点代数知识。
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.