Xiao's Degree-Four Fibration and the Polizzi Model

Let $S_E$ be the surface in Xiao’s degree-four family identified by Polizzi as a smooth divisor in $E(3)$. Starting from Polizzi’s branch model and Xiao’s classification of the thirteen singular fibers, we derive the local spherical braids and their Picard–Lefschetz lifts, including the colored $3+3$ partitions at the seven reducible fibers. We identify the normalization of each bisection with $E\mathrm{/}\{\pm 1\}$ and its degree-two ruling map with the quotient by the involution induced by translation by a nonzero two-torsion point. We prove ${\pi }_1(S_E)\cong {\mathbb{Z}}^2$, compute the elementary-divisor-$4$ Albanese kernel of a regular genus-two fiber, and show that the six nonseparating Picard–Lefschetz transformations represent the six cusps of $\mathrm{\Gamma }(4)$. We also prove that the natural product tori near an elliptic section are nullhomologous and cannot lower $b_1$ below $2$ by torus surgery. As a separate application of the degree-three monodromy, we give a twisted-double construction of an exotic $\mathbb{C}{P}^2\mathrm{\#}7{\overline{\mathbb{C}P}}^2$.