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…