In this work, we construct a generalization of the $G_n^k$-theory to the case of an arbitrary hypergraph. The case of spherical braids is considered separately, using the stratification of the moduli space $\mathcal{M}_n(S^2)$ and the hypergraph $\Gamma_n^{\mathrm{sph}}$ encoding projective constraints. In contrast to the original $G_{n}^{k}$ theory, where codimension-one properties are determined by exactly $k$ particles, the present work considers various cases corresponding to strata of codimension~$1$. These groups admit nice maps to free products of cyclic groups. Among unsolved problems, we emphasize the question how the above construction works for abelian varieties and, in particular, for elliptic curves.
We extend the explicit octonionic approach to groups of type $\mathrm{E}_6$ to the twisted groups ${}^2\mathrm{SE}_{6,K}(F)$ associated with a quadratic Galois extension $K/F$. These groups act on the Albert space over $K$, preserving the Dickson--Freudenthal determinant and a Hermitean form. When the field norm $K^{\t...
J. Bray, Yegor Stepanov, Robert A. Wilson· 0 citations
We introduce level structures on tropical abelian varieties and give a modular interpretation of $A_g[m]^{\mathrm{trop}}$, the tropicalization of the moduli space $\mathcal{A}_g[m]$ of principally polarized abelian varieties with level $m$ structure. We study the case of abelian surfaces in greater depth. The link of $...
Eran Assaf, Madeline Brandt, Juliette Bruce et al.· 0 citations
For $k\geq1$, let $M_k=[(z-w)^k]\subset H^2(\mathbb D^2)$. We first determine the banded Toeplitz matrices associated with the homogeneous components of $M_k$, together with explicit formulas for their determinants and the relevant algebraic cofactors. These formulas lead to a complete description of the spectrum of th...
We extend Gauss composition and Gauss genus theory over $\mathbf{Z}$ to $\mathbb{F}_{2^n}[T]$, a polynomial ring over a finite field $\mathbb{F}_{2^n}$ of characteristic 2. We find new invariants of binary quadratic forms over $\mathbb{F}_{2^n}[T]$ by using Arf invariant and introduce new definitions of proper equivale...
For $r>2$, we study the moduli space parameterising fixed degree morphisms $\mathbb P^r\to X$ where $X$ is a smooth hypersurface of low degree. More precisely, we prove the following result: let $n\geq 2, e\geq 1$ and $X\subset \mathbb P^{n-1}$ a smooth degree $d\geq 2$ hypersurface over an algebraically closed field o...
In their work on Brill-Noether theory, Eisenbud and Harris established the geometry of the universal parameter space of linear series over curves, proving that for even genus $g$ and degree $d = g/2 + 1$, the projection to the moduli space of curves is a finite cover of degree equal to the Catalan number $C_{g/2} = \fr...
Alejandro Vargas· 0 citations
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