Genus of plane curves

Definitions, theorems concerning fundamental group, covering space and the lifting criterion are given. Singular homology has been studied and the Euler characteristic of any space has been defined. Orientation of manifolds have been defined. Algebraic variety has been defined and some properties have been investigated. Let D be discriminant polynomial of p . In particular connectedness of any complex algebraic curve in P 2 (C) has been proved using the covering space (V(p) \ π −1 (D),C\ D,πY ). Compact connected orientable 2 -manifolds are classified by their genus. The surface, homeomorphic to the given curve, can be determined by studying the polynomial and by calculating genus of the variety defined by the polynomial. Let C ⊂ P 2 (C), p(X,Y,Z) be a nonsingular projective curve defined by the irreducible homogeneous polynomial p(X,Y,Z). It has been proved that if deg p = n , then the genus of C is g = (n−1)(n−2) 2 . Key Words: Fundamental Group, Covering Space,Singular Homology, Algebraic Variety, Genus