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

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?

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If $X\to Y$ is a flat projective morphism, we obtain a morphism from $Y$ to some Hilbert scheme. One should presumably take the discriminant divisor to be the pullback of the singular locus from the Hilbert scheme, as long as that is actually a divisor. –  Will Sawin May 31 at 1:52
Hi @Will this seems to be a good first guiding principle for what the discriminant divisor should be. Thanks. –  Hugo Chapdelaine May 31 at 2:52

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 $C$.

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Hi Joe, can you think of $\mathcal{D}_{X/C}$? as the push forward of the relative dualizing sheaf $\Omega_{X/C}$? –  Hugo Chapdelaine May 31 at 3:01