Principal symmetric ideals in the coordinate rings of curves

The paper investigates the arithmetic and geometric properties of principal symmetric ideals (PSIs) within the coordinate rings of symmetric affine plane curves. The primary aim is to address the combinatorial instability of minimal generators for powers of PSIs in polynomial rings by translating the problem into the arithmetic geometry of symmetric curves. This approach allows for a precise geometric understanding of PSIs, particularly focusing on their prime factorization and ramification properties.

The study is conducted within the framework of Dedekind domains, where the coordinate ring of a symmetric curve is treated as a one-dimensional domain. This setting simplifies the problem by reducing the Krull dimension and allowing the use of the Ideal Class Group to measure arithmetic obstructions. The paper demonstrates that the prime factorization of a PSI is determined by the S2-orbits of its symmetric intersection locus, with ramification corresponding to tangential intersections, which are globally detected by a Symmetric Discriminant ideal.

Key results include the finding that the ideal class of any PSI is a 2-torsion element in the Ideal Class Group, leading to a strict periodicity in the powers of a PSI. Specifically, the powers alternate between being principal and requiring exactly two generators. This periodicity resolves the generator instability problem by showing that the obstruction to a PSI being principal is bounded by the torsion of the Class Group.

The paper also introduces a Parity Criterion, which localizes the arithmetic obstruction to the axis of symmetry. By analyzing intersection multiplicities along the diagonal, the criterion provides a numerical test for principality in the hyperelliptic setting. This geometric insight is further supported by the introduction of the Symmetric Index, which quantifies the obstruction to principality based on diagonal intersections.

Limitations of the study include its restriction to two-variable cases and symmetric coordinate rings, which may not directly extend to higher dimensions or non-symmetric settings. Future work is suggested to explore the primary decomposition of PSIs in higher-dimensional symmetric quotient rings, potentially broadening the applicability of the geometric and arithmetic techniques developed in this paper.