Algebraic curve

A one-dimensional (irreducible) algebraic variety is called an algebraic curve. In particular, an algebraic curve in the two-dimensional complex space ^C2 is called a plane curve. A plane curve Γ consists of all the points that make an irreducible polynomial f(X,Y) of some two variables zero. In other words, Γ={(x,y)∈C2 ^| f(x,y)=0}, and the degree of the polynomial f is called the degree of the curve Γ.

If the point P=(x,y) of the plane curve Γ is (∂f/∂X)(x,y),
=(∂f/∂Y)(x,y)=0
If the above condition is satisfied, then point P is called a singular point of Γ. At any point that is not a singular point, only one tangent line can be drawn to the curve Γ. If Γ has singular points, there are only a finite number of them. A curve with no singular points is called a smooth curve.

In the defining polynomial f, X and Y are replaced by X/Z and Y/Z, respectively, and the denominators are eliminated to obtain homogeneous polynomials in X, Y, and Z, which determine a projective algebraic variety in two-dimensional projective space. Such a plane curve is called a projective plane curve, and is a curve with a point at infinity added to Γ, which plays an important role in the theory of algebraic functions and Riemann surfaces.

[Tsuneo Kanno]

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

一次元（既約）代数多様体を代数曲線という。とくに、二次元複素空間C²のなかの代数曲線を平面曲線という。平面曲線Γは、ある二変数の既約多項式f(X,Y)をゼロにする点全体からなっている。つまり、Γ=｛(x,y)∈C²|f(x,y)=0｝で、このとき多項式fの次数を曲線Γの次数という。

　平面曲線Γの点P=(x,y)が
　　(∂f/∂X)(x,y)
　　　=(∂f/∂Y)(x,y)=0
を満たすとき、点PをΓの特異点という。特異点でない点で、曲線Γにただ1本の接線が引ける。Γの特異点は、存在したとしても、有限個しかないが、とくに特異点をもたないΓを滑らかな曲線という。

　定義多項式fでX、Yをそれぞれ、X/Z，Y/Zで置き換え、分母をうまく払って得られるX、Y、Zの同次多項式で、二次元射影空間内に射影的代数多様体が決まる。このようなを射影的平面曲線というが、はΓに無限遠点を加えたもので、代数関数論やリーマン面論で重要な役をする。

［菅野恒雄］

出典　小学館　日本大百科全書(ニッポニカ)日本大百科全書(ニッポニカ)について　情報 | 凡例