Tootfinder

Several sufficient conditions for projective hypersurfaces to be GIT (semi)stable
Xuancong He
https://arxiv.org/abs/2510.03695 https://arxiv.org/pdf/2510.0…

Several sufficient conditions for projective hypersurfaces to be GIT (semi)stable
In this paper, I present some sufficient conditions for projective hypersurfaces to be GIT (semi-)stable. These conditions will be presented in terms of dimension and degree of the hypersurfaces, dimension of the singular locus and multiplicities of the singular points. When singularities of the hypersurface are isolated and all have multiplicity 2, we can judge its stability via the ranks of Hessian matrices at these singular points.

Weyl-invariant subspaces are (usually) not generic
Riku Kurama, Ruoxi Li, Henry Talbott, Rachel Webb
https://arxiv.org/abs/2510.03963 https://arxiv.org/pdf…

Weyl-invariant subspaces are (usually) not generic
Let $V$ be a linear representation of a connected complex reductive group $G$. Given a choice of character $θ$ of $G$, Geometric Invariant Theory defines a locus $V^{ss}_θ(G) \subseteq V$ of semistable points. We give necessary, sufficient, and in some cases equivalent conditions for the existence of $θ$ such that a maximal torus $T$ of $G$ acts on $V^{ss}_θ(T)$ with finite stabilizers. In such cases, the stack quotient $[V^{ss}_θ(G)/G]$ is is known to be Deligne-Mumford. Our proof uses th…

Critical Points and Syzygies for Feynman Integrals
Ben Page, Qian Song
https://arxiv.org/abs/2509.17681 https://arxiv.org/pdf/2509.17681

Critical Points and Syzygies for Feynman Integrals
We investigate a novel theoretical structure underlying the computation of integration-by-parts relations between Feynman integrals via syzygy-based methods. Building on insights from intersection theory, we analyze the large-$ε$ limit of dimensional regularization on the maximal cut, showing that total derivatives vanish on the critical locus of the logarithm of the Baikov polynomial--the locus known to govern the number of master integrals. We introduce "critical syzygies" as a distinguished…

Torsion in the Braid Monodromy of Elliptic Fibrations
Faye Jackson
https://arxiv.org/abs/2509.14454 https://arxiv.org/pdf/2509.14454

Torsion in the Braid Monodromy of Elliptic Fibrations
Given an elliptic fibration $π: M \to S^2$ with singular locus $Δ\subseteq S^2$, let $\operatorname{Br}(π) < \operatorname{Mod}(S^2,Δ)$ be the subgroup of the spherical braid group consisting of those braids that lift to a fiber-preserving diffeomorphism of $M$. We classify the order $n = |Δ|$ elements of $\operatorname{Br}(π)$ up to conjugacy in $\operatorname{Br}(π)$. To do so, we relate these conjugacy classes to special points on the $\operatorname{SL}_2$-character variety …

In search of maximal branes on hyper-K\"ahler manifolds
Simone Billi, Lie Fu, Annalisa Grossi, Viatcheslav Kharlamov
https://arxiv.org/abs/2509.17939 https://

In search of maximal branes on hyper-Kähler manifolds
Given a holomorphic or anti-holomorphic involution on a complex variety, the Smith inequality says that the total $\mathbb{F}_2$-Betti number of the fixed locus is no greater than the total $\mathbb{F}_2$-Betti number of the ambient variety. The involution is called maximal when the equality is achieved. In this paper, we investigate the existence problem for maximal involutions on higher-dimensional compact hyper-Kähler manifolds and on Hilbert schemes of points on surfaces. We show that fo…