Special points on intersections of hypersurfaces

2025/10/11 by Gómez-Gonzáles, Claudio
#14G05 (Primary) #14L30 #14M15 #20C34 #Algebraic Geometry (math.AG) #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2510.10272

Abstract

We establish lower bounds on the ambient dimension for an intersection of hypersurfaces to have a dense collection of ``level ℓ" points, in the sense introduced by Arnold-Shimura, given as a polynomial in the numbers of hypersurfaces of each degree. Our method builds upon the framework for solvable points of Gómez-Gonzáles-Wolfson to include other classes of accessory irrationality, towards the problem of understanding the arithmetic of "special points." We deduce improved upper bounds on resolvent degree RD(n) and RD(G) for the sporadic groups as part of outlining frameworks for incorporating future advances in the theory.

Citations

• Solvable points on intersections of quadrics, cubics, and quartics
• A Summary of Known Bounds on the Essential Dimension and Resolvent Degree of Finite Groups
• Generalized Versality, Special Points, and Resolvent Degree for the Sporadic Groups
• Upper Bounds on Resolvent Degree via Sylvester's Obliteration Algorithm
• Upper Bounds on Resolvent Degree and Its Growth Rate
• Anders Wiman's "On the Application of Tschirnhaus Transformations to the Reduction of Algebraic Equations''
• Tschirnhaus transformations after Hilbert
• Resolvent degree, Hilbert's 13th Problem and geometry
• Versality of algebraic group actions and rational points on twisted\n varieties

Related

• Solvable points on intersections of quadrics, cubics, and quartics
• Irrational Complete Intersections
• Counting rational points on smooth hypersurfaces with high degree
• Degrees of points on irreducible hypersurfaces
• Low degree subvarieties of universal hypersurfaces