Let $f:X \rightarrow Y$ be a dominant morphisms between two irreducible algebraic affine complex varieties. If $X$ and $Y$ have the same dimension, it is known that in this case we have two very nice properties: (i) the map $y \mapsto |f^{-1}(y)|$ is constant over a non-empty open (with respect to the Zariski topology on $Y$) subset of $Y$, and (ii) every regular value of $Y$ is a local minimum of the map $y \mapsto |f^{-1}(y)|$.
My question is the following: What other types of properties exist that behave similarly for dominant morphisms?
For a particular example I am interested in, now allow $\dim X \geq \dim Y$ in our original set up. Assuming $y \in Y$ is a regular value of $f$, we can ask what the genus of the algebraic set $f^{-1}(y)$ is. Is it true in this case that the map $y \mapsto \text{genus of } f^{-1}(y)$ is (i) constant over a non-empty subset of $Y$, and (ii) locally minimised/maximised at every regular value of $Y$? What other similar such properties are there (e.g., degree, number of irreducible components, etc.)?
Any references to back up any claims will be particularly helpful. Since my knowledge regarding algebraic geometry is rather limited to the basics, any answering utilising schemes will most likely be useless for my purposes.
