Exceptional Symmetries and Full Symplectic Monodromy in a Weighted Genus-Two Pencil

Title Exceptional Symmetries and Full Symplectic Monodromy in a Weighted Genus-Two Pencil Description We establish a computer-assisted classification of full connected symplectic monodromy for a weighted genus-two pencil whose six hyperelliptic branch points move in three paired families, over algebraically closed fields of characteristic zero or at least seven. The main theorem gives an explicit necessary and sufficient condition for the connected geometric monodromy group to be Sp(4). Precisely three exceptional weight loci admit elliptic involutions and force proper connected monodromy. Outside these loci, labelled ordinary-node Picard–Lefschetz operators generate the full ten-dimensional symplectic Lie algebra. The proof combines explicit geometric models of nodal degenerations, tame monodromy on marked Jacobian two-torsion, integral Lie-algebra lifting, and exhaustive exact stratification of the projective weight plane. The finite certificate covers 67 locally closed strata in characteristic zero and every characteristic at least eleven, and 51 strata in characteristic seven. Explicit binary Lie-algebra witnesses and involution-matching determinants support the classification. The full connected-monodromy criterion is established independently of stable reduction at infinity, Artin conductor calculations, and the determination of disconnected monodromy groups on the exceptional loci. The accompanying research and verification archive provides deterministic programs, exact symbolic certificates, and reproducibility records. Keywords Genus-two curves; Geometric monodromy; Symplectic groups; Picard–Lefschetz theory; Hyperelliptic curves; Elliptic involutions; Jacobian two-torsion; Lie-algebra certification; Algebraic stratification; Exceptional symmetries; Computer-assisted proof; Arithmetic geometry