# On surfaces with $p_g=0$, $q=1$, and $K^2=-3$ [closed]

I am having a trouble in understanding the Example 4.7 (pages 65-66), the genus two fibrations with $p_g=0$, $q=1$, and $K^2 = -3$, in "Surfaces fibrées en courbes de genre deux", Lecture Notes in Mathematics, 1137, by Xiao Gang(http://link.springer.com/book/10.1007%2FBFb0075351). My French is not good and the Google translator confused me even more. If you are familiar with this construction, could you please outline it?

Especially, I don't understand how the branch divisor $D$ is defined on page 66. Also, does it follow from the construction that the fibration admits $-1$ sphere section? Thanks in advance.

Note: The above fibration is obtained from $\mathbb{CP}^2\# 7(-\mathbb{CP}^2)$ by taking two fold branched cover along a certain degree six divisor $D$, where the divisor $D$ is constructed by considering a complete quadrangle in $\mathbb{CP}^2$.

• I don't think anyone will go hunting for the reference, so you should describe what the problem is if you would like to get an answer... – Igor Rivin Sep 20 '15 at 15:24
• Igor: do you want me to describe the construction in French? I don't have clear description in English. – guest12 Sep 20 '15 at 16:56
• I am saying you should describe it in SOME language, so people don't have to hunt for it. – Igor Rivin Sep 20 '15 at 17:10
• At a minimum, you should provide a link to the paper. – Felipe Voloch Sep 20 '15 at 17:15
• I have provided a link to the book. The construction is 1 page long (with a figure). Some experts might be a well aware of this fibration. – guest12 Sep 20 '15 at 17:25

Xiao Gang is taking a configuration of six lines in the plane with 4 triple points $x,z_1,z_2,z_3$ and three double points $y_1,y_2,y_3$. He considers a general quartic through the seven points which is double on the $y_i$ (the first part of the discussion is the proof that such a quartic is irreducible).
Then he blows up the seven chosen points, and consider the reducible divisor $D$ union of the strict transform of the six lines and of the quartic: this is even (being the strict transform of a curve of even degree with even multiplicity in each point) and then he takes the double cover of the blow-up branched on this curve. Note that the image of the branch curve on the plane has degree $10$, not $6$ as you claim.
Then he considers the genus 2 fibration pull-back of the pencil of lines through $x$, and then he says that he considers the relatively minimal model of this fibration: in other words, he contracts all $(-1)$-curves contained in fibres. Indeed if you consider the strict transforms of the six lines on the double cover, with the reduced strucure, they are rational curves with self-intersection $(-1)$. Xiao Gang contracts the three contained in fibres, given by the lines through $x$: the other three lines give the sections you are looking for.