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 them. More systematic treatments can be found in the references given below.

Gallery

• Conics
• An elliptic curve
• Nodal and cuspidal curves
• Higher genus curves
• Euler characteristic and genus
• Automorphic functions and uniformization
• Function fields

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