Algebraic Curve

An algebraic curve over a Field $K$ is an equation $f(X,Y)=0$, where $f(X,Y)$ is a Polynomial in $X$ and $Y$ with Coefficients in $K$. A nonsingular algebraic curve is an algebraic curve over $K$ which has no Singular Points over $K$. A point on an algebraic curve is simply a solution of the equation of the curve. A $K$-Rational Point is a point $(X,Y)$ on the curve, where $X$ and $Y$ are in the Field $K$.

See also Algebraic Geometry, Algebraic Variety, Curve

References

Griffiths, P. A. Introduction to Algebraic Curves. Providence, RI: Amer. Math. Soc., 1989.