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 $\mathbb{CP}^2$.



On surfaces with $p_g=0$, $q=1$, and $K^2=-3$
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.


