Singular points and Riemann surfaces

A plane algebraic curve is a curve given as the solution set in the complex projective plane of a complex polynomial equation F(x,y) = 0. Singular points are points on such a curve, where the partial derivatives of F are zero. At such points the curve is not smooth, but instead exhibits atypical, often unwanted behavior. In this thesis, we will describe a way to remove such singular points and replace the curve with a smooth curve, which in some sense is as close as possible to the original one. We will work with projective plane curves in the complex projective plane rather than the affine ones described above.