Hikari Iwasaki | Searcharxiv

arXiv2026

Recursive Formula for the Equations of Hessenberg Varieties

Hessenberg varieties are subvarieties of the flag variety, defined by containment conditions on flags with respect to a linear operator. The study of these varieties lies in the intersection of algebraic geometry, combinatorics, and representation theory. In this paper, we develop an algebro-geometric procedure for determining the closed subvariety structure of a Hessenberg variety $\mathcal{H}(X,h)$ in the flag variety for any linear operator $X$ and Hessenberg function $h$, by imposing a partial order on the Hessenberg functions and analyzing the relation of the corresponding Hessenberg varieties. In particular, we give a concrete recursive formula for determining all equations cutting out a given Hessenberg variety in each Schubert cell. As an application, we provide an alternative geometric proof of Tymoczko's results on the existence of affine pavings of a given Hessenberg variety and on the dimension count of its cells.

math.AG↗

arXiv2025

Goppa Duality for Surfaces

Gale duality is an involution of point configurations in projective spaces. Goppa duality extends this concept to a duality between linear series on a Gorenstein curve passing through prescribed points. We generalize this classical result to surfaces, establishing a duality for linear series on surfaces realizing prescribed points as a complete intersection of two divisors. We present several applications, including existence and uniqueness results for Veronese surfaces satisfying conditions to pass through given points or curves. As a key example, we give an alternative proof of Coble's result on the existence of four Veronese surfaces passing through nine general points in projective 5-space.

math.AG↗