Let $\pi:X\rightarrow C$ be a flat and proper morphism over $\mathbb{C}$ where $X$ is a smooth projective surface and $C$ is a smooth projective curve. Assume that all the fibers of $\pi$, except finitely many, are smooth projective genus $g$ curves.

Q1:Then what is the exact definition of the discriminant divisor of $\pi$ and more importantly, what is its geometrical content?

My first guess is that it should probably keep track of critical points of $f$, i.e., points $x\in X$ such that $d\pi_x$ has not maximal rank, but then, it is not a divisor (codimension one cycle) on $X$...

Q2: Do we have a general definition of the "discriminant divisor" associated to a sufficiently nice map between two smooth complex (varieties) manifolds?