Xiao's Degree-Four Fibration and the Polizzi Model
Abstract
Let SE 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/{±1} and its degree-two ruling map with the quotient by the involution induced by translation by a nonzero two-torsion point. We prove π1(SE)≅ℤ2, 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 Γ(4). We also prove that the natural product tori near an elliptic section are nullhomologous and cannot lower b1 below $2$ by torus surgery. As a separate application of the degree-three monodromy, we give a twisted-double construction of an exotic ℂP2#7ℂP̅ 2.
Related
- Xiao's Genus-Two Fibration: Branched Covers and Braid Monodromy
- Simultaneous Global Convergence Failure of Newton's and Halley's Methods in Degree Four
- Four-square polynomial triples: quadratic classification and equal-degree obstructions
- The monodromy of Xiao's genus-two fibrations in degrees 3, 4 and 5
- The fibration question in rank three
BibTeX
@misc{akhmedov2026xiaos,
title = {{Xiao's Degree-Four Fibration and the Polizzi Model}},
author = {Anar Akhmedov and Sümeyra Sakallı},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08655},
eprint = {2610.08655},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08655},
note = {CC0},
}
RIS
TY - UNPB
TI - Xiao's Degree-Four Fibration and the Polizzi Model
AU - Anar Akhmedov
AU - Sümeyra Sakallı
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08655
UR - https://arxiv.org/abs/2610.08655
AB - 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/{±1} and its degree-two ruling map with the quotient by the involution induced by translation by a nonzero two-torsion point. We prove π_1(S_E)≅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 Γ(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 ℂP^2#7ℂP̅^ 2.
ER -