Exceptional Symmetries, Geometric Monodromy, and Local Conductors in a Weighted Genus-Two Pencil
Year: 2026
Type: preprint
Abstract: We classify geometric monodromy and local conductors for the genus-two pencil $$y^2 = (z^2 - a^2s)((z - 1)^2 - b^2s)((z - 2)^2 - c^2s)$$ over algebraically closed fields of characteristic zero or at least seven. The connected geometric monodromy is $\mathrm{Sp}_4$ away from three explicit elliptic-involution loci; on their union it is $\mathrm{SL}_2 \times \mathrm{SL}_2$, with a factor-swapping component on the two hidden loci. The proof uses labelled Picard–Lefschetz meridians, a finite exact Lie-algebra certificate, and an alternative elliptic argument on the exceptional loci. For the rank-five primitive exterior square we determine the geometric Artin conductor and the parabolic cohomology dimension throughout the weight plane. On nonexceptional weights the formulas are $6 + 2m + 2\epsilon_{\mathrm{tr}}$ and $2m - 4 + 2\epsilon_{\mathrm{tr}}$, where $m$ counts distinct nonzero finite bad parameters and $\epsilon_{\mathrm{tr}}$ detects a genuine triple collision. On the hidden locus the corresponding values are $14$ and $4$; on the reflection locus they are $(12, 4)$, $(10, 2)$, or $(8, 0)$, according to the weight. Explicit elliptic quotients, tame stable reduction, and equivariant specialization provide the local inertia calculations. Reproducible exact certificates are supplied separately from the geometric arguments.
Author: Lee, Byoungwoo
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225348380