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Special points on intersections of hypersurfaces

Claudio Gómez-Gonzáles

TL;DR

This work advances the arithmetic study of polynomial solvability by connecting classical resolvent-degree questions to modern geometric methods. It develops a unified framework based on total $j$-polar varieties, polar cones, and $\mathcal{E}$-versality to guarantee the density of level-$\ell$ points on intersections of hypersurfaces, providing explicit, computable bounds on $\mathcal{H}(r)$ and thereby improving upper bounds for $\mathrm{RD}(n)$ in several regimes. The authors apply the framework to sporadic groups, using invariant theory to obtain sharper $\mathrm{RD}(G)$ bounds for several groups, including McL, Ru, He, Fi$_{23}$, Fi$_{24}'$, B, and M, illustrating the practical impact on the resolvent-degree program. Overall, the paper offers a constructive pathway to incorporate future advances in invariants and moduli into the arithmetic of special points and the resolvent-degree problem.

Abstract

We establish lower bounds on the ambient dimension for an intersection of hypersurfaces to have a dense collection of ``level $\ell$" 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 $\operatorname{RD}(n)$ and $\operatorname{RD}(G)$ for the sporadic groups as part of outlining frameworks for incorporating future advances in the theory.

Special points on intersections of hypersurfaces

TL;DR

This work advances the arithmetic study of polynomial solvability by connecting classical resolvent-degree questions to modern geometric methods. It develops a unified framework based on total -polar varieties, polar cones, and -versality to guarantee the density of level- points on intersections of hypersurfaces, providing explicit, computable bounds on and thereby improving upper bounds for in several regimes. The authors apply the framework to sporadic groups, using invariant theory to obtain sharper bounds for several groups, including McL, Ru, He, Fi, Fi, B, and M, illustrating the practical impact on the resolvent-degree program. Overall, the paper offers a constructive pathway to incorporate future advances in invariants and moduli into the arithmetic of special points and the resolvent-degree problem.

Abstract

We establish lower bounds on the ambient dimension for an intersection of hypersurfaces to have a dense collection of ``level " 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 and for the sporadic groups as part of outlining frameworks for incorporating future advances in the theory.

Paper Structure

This paper contains 13 sections, 17 theorems, 68 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Background
  4. Polar cones and Fano varieties
  5. ed, RD, and E-versality
  6. Special points via total j-polar varieties
  7. Obliteration lemmas
  8. Bounds on flj(m)
  9. Bounds on RD(n)
  10. Known bounds
  11. New bounds
  12. Sharpened bounds on RD(G) for the sporadic groups
  13. Computing bounds on flj(m)

Key Result

Theorem 1.1

With notation as above, the following bounds hold:

Figures (2)

  • Figure 1: Application of Lemma \ref{['lemma:pts_from_planes']} to bound $f^\ell_0(\bm)$.
  • Figure 2: Implementation of a bound on $\mathop{\mathrm{RD}}\nolimits$, R_bound, in terms of its Hamilton function, H_bound.

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Proposition 2.6: GGW2025
  • ...and 39 more