The degree-genus formula for nonsingular complex algebraic curves

Any nonsingular projective curve 􀀃 = 􀀅(􀀁) ⊂ ℙ􀀊(ℂ) ( 􀀁 , a nonzero homogeneous polynomial in three variables) is compact connected orientable 2- manifold. These manifolds can be obtained from spheres by adding 􀀁 handles. This number is called the genus of the manifold. These manifolds are classified by their genus. Two manifolds with the same genus are homeomorphic. Just looking at the degree of 􀀁, it is possible to calculate the genus of the curve 􀀃 and thus to classify the curve. If deg 􀀁 = 􀀐, the number of genus can easily be found by the genus formula 􀀌 = 1 2 (􀀐 − 1)(􀀐 − 2) In the study, the degree-genus formula has been proved.