Reflection Symmetry and Symplectic Monodromy in Hyperelliptic Pencils: Moving Pairs, Critical Values, and Quartic Squares

Abstract Let $A \subset \mathbb{R}$ consist of $2g \ge 4$ distinct points, and consider the moving-pair hyperelliptic pencil $$y^2 = \left(\prod_{a \in A}(z - a)\right) \left((z - \delta)^2 - t^2\right).$$ We determine its connected geometric monodromy on first cohomology: it is $\mathrm{Sp}_{2g}$ when $A \neq 2\delta - A$, and $\mathrm{Sp}_{2\lfloor g/2\rfloor} \times \mathrm{Sp}_{2\lceil g/2\rceil}$ otherwise. The symmetric case follows from two quotient pencils; in the asymmetric case, Picard–Lefschetz logarithms from isolated nodes and simultaneous two-node fibres generate the full symplectic Lie algebra. We also classify the connected monodromy on the degree-eight critical-value stratum $(2, 2, 2, 1)$ by polynomial indecomposability and establish full $\mathrm{Sp}_6$ for $$y^2 = (z^4 + z)^2 - t$$ through radial vanishing cycles. The associated integral marking has a rank-three logarithmic image with cyclic quotient of order four. All equalities concern connected Zariski closures over $\mathbb{Q}_2$.