Is it true that if $X$ and $Y$ are two irreducible algebraic curves over $\mathbb{C}$ and $f:X\dashrightarrow Y$ is a rational map that is generically n:1 for some $n\in \mathbb{Z}^{+}$ then the geometric genus of $Y$ is less than or equal to the geometric genus of $X$?



The geometric genus under a generically finite to one rational map
Mohan' answer in the comments is correct: this is more a comment to his answer.


If you are only interested in the inequality you do not need Riemann-Hurwitz formula as I will explain in the following. The advantage is that it generalizes to higher dimension.


Replace first, as Mohan suggests, the irreducible curves with their normalizations. 
Then you can pull-back holomorphic 1-forms. Since a finite map is a local diffeomorphism on a (dense) open set, the pull-back of a 1-form not identically zero is not the zero form. In other word the pull-back is an injective linear map among the respective vector space of holomorphic 1-forms whose dimensions are by definition the respective geometric genera. This proves the inequality.


