I am trying to read about the genus of an algebraic curve. I have been told that there is a connection between topological genus and genus defined for an algebraic curve. Since an algebraic curve gives rise to a $2$ dimensional compact manifold(I am plotting the curve in $\mathbb{CP}^2$) I understand that it has to be connected sums of torus.
But I cannot get the genus this way. I am aware that one can find the genus by applying Quadratic Transformation but I am interested in knowing if we can find the genus by blowing up i.e by blowing up I can get a nonsingular curve which is birational to the given curve which is not necessarily a plane curve. Is there some formula to calculate genus from here?
Also say I have a projective curve then can I dehomogenise it , then blow up to make a non singular plane curve and finally homogenise it and find the genus.