1. Degree Genus Formula
In this note, P2 denotes the 2-dimensional complex projective space. Recall that a com- plex homogeneous polynomial F (X, Y, Z) of degree d is a complex polynomial such that
F (λX, λY, λZ) = λdF (X, Y, Z).
The zero set of a family of homogeneous polynomials S ⊂ C[X, Y, Z] defined an projective algebraic set or a projective curve:
V (S) = {(ζ0: ζ1: ζ2) : F (ζ0, ζ1, ζ2) = 0, F ∈ S}.
When V (S) is irreducible (as a topological space), we call V (S) a projective variety or a projective irreducible curve. When S = {F (X, Y, Z)} consists of a single homogeneous polynomial, we denote V (S) by V (F ) or simply by F (X, Y, Z) = 0.
Let C be a projective curve F (X, Y, Z) = 0. A point p(ζ0 : ζ1 : ζ2) ∈ C is singular if FX(ζ0, ζ1, ζ2) = FY(ζ0, ζ1, ζ2) = FZ(ζ0, ζ1, ζ2) = 0
for any representative (ζ0, ζ1, ζ2) of p. The set of singular points of C is denoted by Sing(C).
The curve C is nonsingular if Sing(C) = ∅.
Theorem 1.1. Let F (X, Y, Z) be a complex homogeneous polynomial of degree d. Suppose the curve defined by
C = {(ζ0: ζ1: ζ2) ∈ P2 : F (ζ0, ζ1, ζ2) = 0}
is a nonsingular curve in P2. The genus g of C is given by g = (d − 1)(d − 2)
2 .
Remark. The degree of a projective (or affine plane) curve is defined to be the degree of its defining polynomial.
Let f (x, y) = y2− 4x3+ g2x + g3 be a complex polynomial in x, y. Assume that the plane curve
C = {(a, b) ∈ C2 : f (a, b) = 0}
is nonsingular. The curve C can be embedded into a smooth curve in P2.
On P2, we set Uj = {(ζ0 : ζ1 : ζ2) : ζj 6= 0} for j = 0, 1, 2. Then {U0, U1, U2} is a open covering of P2. We define ψj : Uj → C2,
ψ0(ζ0 : ζ1: ζ2) = ζ1 ζ0,ζ2
ζ0
, ψ1(ζ0 : ζ1: ζ2) = ζ0
ζ1
,ζ2
ζ1
, ψ2(ζ0 : ζ1: ζ2) = ζ0
ζ2
,ζ1 ζ2
.
Lemma 1.1. These maps ψj : Uj → C2 are homeomorphisms in Zariski topology (also in complex analytic topology). Hence the two dimensional complex projective space is a finite union of affine varieties.
If (a, b) ∈ C, assume that a = ζ0/ζ2 and b = ζ1/ζ2. Then f (a, b) = 0 implies that ζ2ζ12= 4ζ03− g2ζ0ζ22− g3ζ33. Let us denote F (X, Y, Z) = ZY2− 4X3− g2XZ2− g3Z3. Then F (X, Y, Z) ∈ C[X, Y, Z] is a homogeneous polynomial of degree 3. The variety defined by F (X, Y, Z) = 0 is denoted by E = {(ζ0 : ζ1 : ζ2) : F (ζ0, ζ1, ζ2) = 0}. Then the map
1
2
ϕ : C → E defined by ψ(a, b) = (a : b : 1) is an embedding. (It is also a morphism of varieties)
Definition 1.1. The projective curve E in P2 is called an elliptic curve.
Since the projective curve E is defined by a homogeneous polynomial of degree 3, by genus degree formula, the genus of E is g = (3 − 1)(3 − 2)/2 = 1.
Simiarly, every plane curve can be embedded into a projective curve. Suppose C is a plane curve defined by f (x, y) = 0 with deg f (x, y) = d. Set
F (X, Y, Z) = Zdf X Z,Y
Z
.
Then the map {(a, b) ∈ C2 : f (a, b) = 0} → {(ζ0 : ζ1 : ζ2) : F (ζ0, ζ1, ζ2) = 0} sending (a, b) to (a : b : 1) is an embedding.
Example 1.1. The Fermat curve Xn+ Yn+ Zn= 0 is a nonsingular curve in P2 of genus (n − 1)(n − 2)/2.
Question: How to compute or define the genus of projective curve with singularities?
What is the meaning of genus of a projective curve?
If we consider the complex analytic topology on P2, every projective curve is closed in P2. Since P2 is compact Hausdorff in complex analytic topology and every closed subset of a compact Hausdorff space is again compact, every projective curve is compact. If the projective curve is nonsingular, the projective curve is an one dimensional compact complex manifold. This implies that every nonsingular projective curve is a compact two dimensional smooth orientable real manifold (compact orientable surface). We know C is diffeomorphic to Σg where g is the genus of C and Σg is the connected sum of g-tori. We obtain certain information about the integer singular (or simplicial) homology H∗(C; Z) of C and the complex de Rham cohomology HdR∗ (C; C) of C. We will discuss these in another note.
