Is there a description of smooth complex projective surfaces without sections?

While working on a problem a surface $X$ showed up with the following property: it is a non-ruled surface that has an elliptic fibration $X \longrightarrow \mathbb{P}^1$ with only two singular fibers, $F_1$ and $F_2$, that are not multiples of a smooth curve and at least one multiple fiber (by Kodaira dimension reasons there are at least $3$ singular fibers).

From this I could conclude that $p_g(X) = q(X) =0$ and there is no section, working out some inequalities relating the number of singular fibers with null geometric genus and other invariants.

I'm aware of Dolgachev's work on Algebraic Surfaces with $p_g(X) = q(X) =0$ but I hope there are more restrictions when it has no sections.

Almost everywhere I've read is assumed that the fibration has a section but I cannot make this assumption. The fibration shows up, it is not an object defined a priori.

Any hint or reference will be appreciated.


This might not be worth an answer but since you haven't gotten one yet, here a few references.

One can get an elliptic surface with multiple fibers from a surface without multiple fibers by a process called "logarithmic transformation". It changes a single fiber into a fiber of the same type but it will be a multiple fiber. References for this process are the book of Griffiths and Harris, the book "Compact Complex Surfaces" by Barth et al. and it is probably best explained in "Smooth Four Manifolds" by Friedman. You will find some warnings about this logarithmic transform, it is not as nice as one might hope.

The following papers might be of interest to you as well (for a different construction):

  1. On Rational Elliptic Surfaces with Multiple Fibers - Fujimoto
  2. On explicit constructions of rational elliptic surfaces with multiple fibers - Fujimoto
  3. Multiple fibers on Rational Elliptic Surfaes - Harbourne, Lang

and reference therin. All three are available online for free. Chapter 5 of the book by Dolgachev and Cossec might help as well, though it is a rather technical text.

  • $\begingroup$ Thanks for the references. I've read a few pages about Logarithmic transformations. The book from "Barth and friends" haha is my best reference. I got stuck working on a problem of counting these multiple fibers. And I was using the Kodaira dimension in the problem, hence I discarded the logarithmic transformations. $\endgroup$ – Alan Muniz Jan 31 '16 at 23:52

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