The natural way of defining an algebraic curve in the complex projective plane is clearly:
Definition 7.2.1 By an algebraic curve is meant a subsetC⊆P2(C)of the com- plex projective plane which, in some system of homogeneous coordinates, can be described as the set of those points whose coordinates satisfy an equation
P (X, Y, Z)=0
whereP (X, Y, Z)is a non-zero homogeneous polynomial.
Of course, Definition7.2.1does not depend on the choice of the system of ho- mogeneous coordinates:
Lemma 7.2.2 In every system of homogeneous coordinates, an algebraic curve can be described as the set of those points whose coordinates satisfy an equation
P (X, Y, Z)=0
whereP (X, Y, Z)is a non-zero homogeneous polynomial.
Proof A change of homogeneous coordinates is expressed by linear formulas. These formulas transform a non-zero homogeneous polynomial into another non-zero ho-
mogeneous polynomial.
However, if we define an algebraic curve as in Definition7.2.1, we should pay attention to which properties depend only on the subsetC⊆P2(C), and which prop- erties depend explicitly on the polynomialP (X, Y, Z)used to describeC. Does this make a difference?
In other words, the following question arises:
Is the equation of an algebraic curve, in a given system of homogeneous coordinates, nec- essarily unique?
Of course the answer is “no”, as we discovered already in the study of quadrics (see Sect.2.26)! The two equations
P (X, Y, Z)=0, k P (X, Y, Z)=0, 0=k∈C trivially determine the same curve.
Thus a more sensible question would rather be:
Are two equations of an algebraic curve in the same system of homogeneous coordinates necessarily proportional?
Once more the answer is trivially “no”! The two equations
P (X, Y, Z)=0, P (X, Y, Z)n=0, 0=n∈N still determine the same subsetC⊆P2(C).
This last case is worth more attention. By TheoremB.4.9, we know that the poly- nomialP (X, Y, Z)factors uniquely (up to multiplicative constants) as a product of irreducible polynomials and by PropositionC.1.3, all these factors remain homoge- neous. Clearly, if we replace one of these irreducible factors by a power of it, we do not change the roots ofP (X, Y, Z). So clearly, if we want to reach sensible results, it would be wise to work only with equationsP (X, Y, Z)=0 whereP is a non-zero homogeneous polynomial without any multiple irreducible factors.
We therefore adopt the following definition:
272 7 Algebraic Curves Definition 7.2.3 By a simple equation of an algebraic curve inP2(C)we mean an equation
P (X, Y, Z)=0
whereP (X, Y, Z)a non-zero homogeneous polynomial without any multiple fac- tors.
Notice that determining whether an equation of a curve is simple is a rather easy task: it suffices to apply CorollaryD.1.5and compute a resultant.
This time, due to the fact that the field of complex numbers is algebraically closed, we get the following important result:
Theorem 7.2.4 Two simple equations of an algebraic curve in a given system of homogeneous coordinates are necessarily proportional.
Proof Consider two simple equations
P (X, Y, Z)=0, Q(X, Y, Z)=0
of the algebraic curve C in a given system of homogeneous coordinates. Given an irreducible factor R(X, Y, Z), all its roots are points of C, thus are roots of Q(X, Y, Z). By Proposition D.2.4, R(X, Y, Z) divides Q(X, Y, Z). Since all the irreducible factors of P (X, Y, Z) are simple, this immediately implies that P (X, Y, Z)itself dividesQ(X, Y, Z). Analogously, the polynomialQ(X, Y, Z)di- videsP (X, Y, Z), proving that both polynomials are proportional.
Of course a special case of interest is:
Definition 7.2.5 By an irreducible algebraic curve we mean an algebraic curve whose simple equation is an irreducible polynomial.
Clearly:
Lemma 7.2.6 Every algebraic curve is in a unique way the union of finitely many irreducible curves.
Proof By TheoremB.4.9, factor the simple equation of the curve into its irreducible
factors.
Lemma7.2.6 reduces—for many purposes—the study of an arbitrary curve to that of irreducible curves. The Eisenstein criterionB.4.11is certainly a useful tool to check the irreducibility of a curve. On the other hand, finding the irreducible factors of a given polynomialP (X, Y, Z)is generally quite a difficult task.
We shall also use the following terminology:
Definition 7.2.7 An algebraic curveCis called a component of an algebraic curve DwhenC⊆D.
Proposition 7.2.8 Consider two algebraic curvesCandDwith respective simple equationsP (X, Y, Z)=0 andQ(X, Y, Z)=0. The following conditions are equiv- alent:
1. Cis a component ofD;
2. P (X, Y, Z)dividesQ(X, Y, Z).
Proof (2⇒1)is obvious. Conversely, by PropositionD.2.4, each irreducible factor ofP (X, Y, Z)is an irreducible factor ofQ(X, Y, Z).
Let us conclude this section by mentioning that, at some places, we will find it convenient to use possibly non-simple equations. For example in some arguments, given a curve with simple equationP (X, Y, Z)=0, we shall want to consider the curve (when it turns out to be one) with equation ∂P∂X(X, Y, Z)=0. Of course this partial derivative can very well admit a multiple irreducible factor.
Counterexample 7.2.9 The equation X3+Y2Z=0 is simple, but its partial derivative with respect toXis a non-simple equation of a line.
Proof One has
P (X, Y, Z)=X3+Y2Z, ∂P
∂X(X, Y, Z)=3X2.
By the Eisenstein criterionB.4.11, the polynomialP, viewed as a polynomial in Xwith coefficients inC[Y, Z], is at once irreducible, thus a fortiori simple; in the Eisenstein criterion, simply choosed=Z. Of course the partial derivative is a non-
simple equation of the lineX=0.