A complex algebraic plane curve is the set of complex solutions to a polynomial equation f(x, y)=0. This is a 1 complex dimensional subset of C², or in more conventional terms it is a surface living in a space of 4 real dimensions. These objects are also called Riemann surfaces, at least away from the singularities. The study of these objects and their generalizations within algebraic geometry involves a lot of technical material, and it is easy to forget that this is geometry. So I've put together a small picture gallery, with just enough theory to explain how to make them. More systematic treatments can be found in the references given below.

Gallery

• Conics
• An elliptic curve
• Nodal and cuspidal curves
• Higher genus curves

References

1. F. Kirwan, Complex algebraic curves
2. R. Miranda, Algebraic curves and Riemann surfaces
3. M. Reid, Undergraduate algebraic geometry
4. G. Springer, Riemann surfaces

Notes

P := (x,y) -> WeierstrassP(x + y*I,1,0);
PP := (x,y) -> WeierstrassPPrime(x + y*I,1,0);
IR := (theta, z) -> cos(theta)*Re(z) + sin(theta)*Im(z);
Gr1 := theta -> plot3d([Re(P(x,y)), Im(P(x,y)), 0.3*IR(theta,PP(x,y))] , x=0..3.74,y=0..3.74, color=[sin(2*Pi*x/3.74), 0.5, sin(2*Pi*y/3.74)], view=[-1..1,-1..1, -1..1], grid=[35,35]);
L1 := theta -> spacecurve([Re(P(x,1.854)), Im(P(x,1.854)), 0.3*IR(theta,PP(x,1.854))] , x=0..3.74, color=red, thickness=2);
L2 := theta -> spacecurve([Re(P(x,0)), Im(P(x,0)), 0.3*IR(theta, PP(x,0))] , x=1..2.7, color=red, thickness=2);
L3 := theta->spacecurve([Re(P(1.854,y)), Im(P(1.854,y)), 0.3*IR(theta,PP(1.854,y))] , y=0..3.74, color=yellow, thickness=2);
Gr := theta-> display({Gr1(theta) , L1(theta),L2(theta),L3(theta)});