(Reference Request) Desingularization of Fibrations

Question

I am interested in morphisms from a projective surface to a projective curve $p: X \longrightarrow C$ such that the fibres are singular curves on $X$. In the best case scenario, I'd like to replace $X$, say, birationally such that the fibres become smooth.

I don't expect something like this to always exist, but I'd like to read up on some results in this direction (e.g., what happens if $X$ is smooth, all the fibres are of a given genus and only certain singularities appear). I couldn't come up with much in the classical literature that I know, so any pointers are highly appreciated.

Answer 1

@Paul The most simple example of a fibration with smooth fibres, which degenerate to singular curves, is an elliptic fibration $p: X \longrightarrow C$. The general fibre of $p$ is a smooth elliptic curve. Kodaira has classified the finitely many types of singular fibres in elliptic fibrations (Barth, W.; Hulek, K.; Peters, Ch.; van de Ven, A.: Compact Complex Surfaces. Chap. V.7). The reference gives an example for each type by constructing an elliptic fibration over the unit disc with the singular fibre over the origin.

Each Enriques surface has an elliptic fibration $p: X \longrightarrow \mathbb P^1$ with exactly 2 singular fibres. Both have a nilpotent strucure $2F$, see Chap. VIII, 17.

Answer 2

For a modern treatment of families of curves you can try taking a look at Chapter 8.3 of Liu's book on Algebraic Geometry.

To get a good feeling for these matters I suggest working with complex algebraic surfaces over curves and trying out some examples. For instance, consider the projective family of curves $X_n := \{x^n+y^n= tz^n\}$ over $\mathbb A^1$, where $n>0$ is an integer.

Check that $X_1$ is a smooth family of projective lines.

Check that $X_2$ is a family of conics with a singular fibre over $t=0$.

More generally, check that $X_n$ is a family of degree $n$ curves in $\mathbb P^2$ which (only) degenerates at $t=0$. (In particular, the genus of the smooth fibres of $X_n$ is $(n-1)(n-2)/2$.)

You can show that $X_n\to \mathbb A^1$ is "minimal" in the sense that $X_n$ is a nonsingular variety and the fibres of $X_n \to \mathbb A^1$ contain no (-1)-curves (this is easy in this case as the fibres are irreducible).

But you can improve the situation after a base-change. In fact, you can always achieve "semi-stable fibres" after a finite ramified base-change. In the example $X_2$ above, you just need to adjoint a square root of $t$ to $\mathbb A^1$ (this corresponds to the finite ramified morphism $\mathbb A^1 \to \mathbb A^1$ sending $z$ to $z^2$).

In general, after adjoining $t^{1/n}$ to $\mathbb A^1 = $Spec $\mathbb C[t]$, you see that $X_n$ is isomorphic to $x^n+y^n = (z^\prime)^n$, where $z^\prime = ( t^{1/n}z)$. This is a smooth family of curves, so the situation is now nicer. (Not every semi-stable family of curves is smooth; in the examples above the family of curves $X_n\to \mathbb A^1$ has "potentially good reduction".)

If I may suggest another example: consider the family of projective curves over $\mathbb A^1$ given by $$y^2z = x(x-z)(x-tz).$$ What are its singular fibres? Are they semi-stable?