发表机构
University of Minnesota; Harvard University; University of South Florida（明尼苏达大学; 哈佛大学; 南佛罗里达大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文从Polizzi分支模型和Xiao的奇异纤维分类出发，推导局部球面辫及其Picard-Lefschetz提升，证明基本群为Z^2，计算Albanese核，并给出奇异CP^2#7CP^2-bar的扭曲双构造。
AI 中文摘要
设$S_E$为Xiao四次族中的曲面，由Polizzi识别为$E(3)$中的光滑除子。从Polizzi的分支模型和Xiao对十三个奇异纤维的分类出发，我们推导出局部球面辫及其Picard--Lefschetz提升，包括七个可约纤维处的着色$3+3$划分。我们将每条二截线的正规化与$E/\{\pm1\}$等同，并将其二次覆盖映射与由非零二阶挠点平移诱导的对合之商等同。我们证明$\pi_1(S_E)\cong\mathbb Z^2$，计算正则亏格二纤维的初等因子为$4$的Albanese核，并证明六个非分离Picard--Lefschetz变换表示$\Gamma(4)$的六个尖点。我们还证明了椭圆截面附近的自然乘积环面是零同调的，且不能通过环面手术将$b_1$降至$2$以下。作为三次单值化的一个独立应用，我们给出了一个扭曲双构造，得到一个奇异的$\mathbb CP^2\\#7\overline{\mathbb CP}^{\\,2}$。
英文摘要
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 $E/\{\pm1\}$ and its degree-two ruling map with the quotient by the involution induced by translation by a nonzero two-torsion point. We prove $π_1(S_E)\cong\mathbb Z^2$, 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 $Γ(4)$. 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 $\mathbb CP^2\#7\overline{\mathbb CP}^{\,2}$.
Comments28 pages
