Primitive Exterior Squares, Local Conductors, and Exceptional Monodromy in a Weighted Genus-Two Pencil
Year: 2026
Type: preprint
Abstract: Title Primitive Exterior Squares, Local Conductors, and Exceptional Monodromy in a Weighted Genus-Two Pencil Description We investigate local Artin conductors, parabolic cohomology, and exceptional geometric monodromy for the rank-five primitive exterior-square representation associated with the first cohomology of a weighted genus-two pencil. Using the Grothendieck–Ogg–Shafarevich formula, we derive an explicit parabolic cohomology dimension formula under specified local inertia representations. We construct normalization atlases for genuine finite triple collisions and the all-equal-square degeneration at infinity, analyze their compact-type stable fibres, and relate their deck transformations to tame inertia through equivariant cohomological specialization. On the hidden Möbius symmetry locus, explicit elliptic quotient families and complementary multiplicative degenerations yield a connected geometric monodromy group SL(2) × SL(2), with a quadratic component interchange. The primitive invariant dimension is zero, the total Artin conductor is fourteen, and the parabolic cohomology has dimension four. On the equal-square locus, a separate bielliptic calculation gives a one-dimensional primitive invariant space, total Artin conductor eight, and vanishing parabolic cohomology. The analysis combines explicit stable-reduction geometry, elliptic quotients, tame descent, geometric semisimplicity, and exact algebraic verification. It complements the companion study of full symplectic monodromy by determining local and cohomological invariants on exceptional geometric strata. The accompanying research and verification archive contains symbolic calculations, finite-field checks, normalization identities, and reproducibility records. Keywords Genus-two pencils; Primitive exterior square; Artin conductors; Parabolic cohomology; Stable reduction; Tame inertia; Exceptional monodromy; Elliptic quotients; Quadratic descent; Grothendieck–Ogg–Shafarevich formula; ℓ-adic cohomology; Arithmetic geometry
Author: Lee, Byoungwoo
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225247220
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