Definitions of Genera are Equivalent
Theorem
In appropriate circumstances, the following definitions of genus are equivalent:
Genus of Manifold
The genus of a compact topological manifold is the number of handles it has.
Genus of Riemann Surface
The genus of a Riemann surface $R$ is the number of linearly independent holomorphic $1$-forms that are defined on $R$.
Genus of Plane Algebraic Curve
Let $\CC$ be a plane algebraic curve with no singular points.
The genus of $\CC$ is defined as:
- $\dbinom {d - 1} 2$
where $d$ denotes the degree of $\CC$.
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Proof
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Theorem: Let $C$ be a non-singular projective algebraic curve of degree $n$. Then the genus of $C$ as a Riemann surface is
- $
g=\frac{(n-1)(n-2)}{2} . $
Proof: Take the map $f=y / x$ from $C$ to $\mathbf{P}^1$.
By Ramification points of y/x, the ramification points occur where the tangents pass through $[0,0,1]$ and are therefore given by the equation
- $\frac{\partial P}{\partial z}=0$
The multiplicity is bigger than 2 only if $\map{I_p}{C, T_p}>2$, i.e. if $p$ is an inflection point, but there are only finitely many of these, so by a projective transformations we can assume that $[0,0,1]$ does not lie on the tangent to any one of them.
This means that each $m_k$ in the Riemann-Hurwitz Formula is 2, and it remains to calculate the number of ramification points.
This is the number of points of intersection of $P=0$, the curve $C$ of degree $n$, and $\partial P / \partial z=0$, a curve $D$ which is of degree $n-1$.
Since $C$ is nonsingular it is irreducible, and so $C$ and $D$ can have no common component.
We will use Bézout's Theorem, so we need to check that $[a]=\left[a_0, a_1, a_2\right] \in C \cap D$ is a nonsingular point of $D$ and that the tangent lines are distinct.
Now $\paren{P_{z x}, P_{z y}, P_{z z}}$ is not identically zero at $\left[a_0, a_1, a_2\right]$ because this would make the Hessian of $C$ vanish and we know that $\left[a_0, a_1, a_2\right]$ is not an inflection point.
This shows that $D$ is nonsingular here.
Suppose that the tangents of $C$ and $D$ coincide then $\paren{P_{z x}, P_{z y}, P_{z z}}$ is a multiple of $\paren{P_x, P_y, P_z}$. As in our discussion of inflection points we use the symmetric bilinear form $B$ defined by the matrix of partial derivatives $P_{i j}$. Then $\map B{a, a}=0=\map B{a, \alpha}$ where the tangent line joins $[a]$ and $[\alpha]$. Put $v=(0,0,1)$.
By Euler's Homogeneous Function Theorem
- $
a_0 P_{z x}+a_1 P_{z y}+a_2 P_{z z}=(n-1) P_z=0 $ since $P_z(a)=0$.
This gives $B(a, v)=0$.
Moreover since $P_{z z}(a)=\lambda P_z(a)=0$, we have $B(v, v)=0$.
Since $[a]$ is not an inflection point, $\map\det B \neq 0$ so from
- $
0=\map B{a, a}=\map B{a, \alpha}=\map B{a, v} $ we deduce $v=\mu a+\nu \alpha$. But then
- $
0=\map B{v, v}=\nu^2\map B{\alpha, \alpha} $ and, by Quadric containing a line is singular, this gives $\map\det B=0$ unless $\nu=0$. But then $[a]=[0,0,1]$ which we have specifically excluded.
We conclude that the tangents are distinct and it follows that the conditions for Bézout's Theorem hold, so the number of ramification points is exactly $n(n-1)$.
From the Riemann-Hurwitz Formula we obtain
- $
2-2 g=2 n-n(n-1) $ and so
- $
g=\frac{1}{2}(n-1)(n-2) . $
$\blacksquare$
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): genus
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): genus
- 2009: Nigel Hitchin: Algebraic Curves (B3b course notes) Section 4.3 The degree-genus formula, Theorem 23, page 49