2024/08/16 by Samir Canning, Canning, Samir, Dragos Oprea +3 · 2 citations
Mathematics · #14C15 #14C17 #14K10 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2408.08718
openalex publication_date 2024/08/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The tautological Chow ring of the moduli space Ag of principally polarized abelian varieties of dimension g was defined and calculated by van der Geer in 1999. By studying the Torelli pullback of algebraic cycles classes from Ag to the moduli space Mgct of genus g of curves of compact type, we prove that the product class [A1× A5]∈ CH5(A6) is non-tautological, the first construction of an interesting non-tautological algebraic class on the moduli spaces of abelian varieties. For our proof, we use the complete description of the the tautological ring R^*(M6ct) in genus 6 conjectured by Pixton and recently proven by Canning-Larson-Schmitt. The tautological ring R^*(M6ct) has a 1-dimensional Gorenstein kernel, which is geometrically explained by the Torelli pullback of [A1× A5]. More generally, the Torelli pullback of the difference between [A1× Ag-1] and its tautological projection always lies in the Gorenstein kernel of R^*(Mgct). The product map A1× Ag-1→ Ag is a Noether-Lefschetz locus with general Neron-Severi rank 2. A natural extension of van der Geer's tautological ring is obtained by including more general Noether-Lefschetz loci. Results and conjectures related to cycle classes of Noether-Lefschetz loci for all g are presented.