What is the discriminant divisor of a surface fibered over a curve?

Question

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?

Answer 1

If $X$ is a minimal elliptic surface, then (as you're probably aware) there's a standard way of defining the discriminant divisor on $C$, and in addition one has the beautiful formula of Ogg which states that for points $t\in C$, $$ f_t = \text{ord}_t(\mathcal{D}_{X/C}) + 1 - m_t, $$ where $f_t$ is the valuation of the conductor (which in this case is 0,1,2), and $m_t$ is the number of components of the fiber $X_t$. Then you get the full discriminant as the sum $$ \mathcal{D}_{X/C} = \sum_{t\in C} \text{ord}_t(\mathcal{D}_{X/C})(t). $$

Ogg's local formula is valid for elliptic surfaces over $\text{Spec}(R)$ for a DVR, and he proved the formula for all residue characteristics except in the case that $R$ has characteristic $0$ and its residue field $R/\mathfrak{p}$ has characteristic 2. (The hard cases are characteristic 2 and 3.)

A number of years later, in a beautiful paper, T. Saito gave a new proof that completed the elliptic surface case and generalized it to arbitrary genus fibrations. His article is

T. Saito, Conductor, discriminant, and the Noether formula of arithmetic surfaces, Duke Math J 57(1) (1988), 151-173.

Saito's article will have a formula for the discriminant of your $X\to C$ with genus $g$ generically smooth fibers, computed locally above each point of $C$.

Answer 2

I actually spent some time thinking about what this should be, not sure if it gives the right point of view: Let $f: X \to S$ be a flat, projective, local complete intersection, with $S$ normal, of (constant) relative dimension $n$. Denote by $p: H \to s$ the $f$-singular locus over $s = f(H)$. Then I would define $\mathcal D_{X/S} = p_* c_{n+1}^H(\Omega_{X/S})$ in the appropriate Chow group. Here $c_{n+1}^H(\Omega_{X/S})$ denotes a/the localized Chern class, which is a refinement of the usual Chern class. One could probably weaken the definition to assume that $H$ is only proper over $s$.

This has the virtue of coinciding with the discriminant divisor in the cases where I know there is a notion of discriminant, possibly up to some silly factor which can potentially be baked into the definition of the discriminant in that context, see Proposition 1.1 and Theorem 1.2 in http://www.math.chalmers.se/~dener/Multiplicity-of-discriminants-revision-Oct-7-2013.pdf (which has since appeared in Crelle).