8 problems

Let $C$ be a Gorenstein monomial curve of Stöhr–Torres type. Stöhr–Torres non-smoothability conjecture. The curve $C$ is not smoothable. The conjecture is motivated by the example…

Let $L_r^6$ be a general homogeneous curve singularity consisting of $r$ lines through the origin in $\mathbb{A}^6$, in the interval $56<r\leq126$. Negative-weight deformation conj…

Let $(X,\mathsf{d})$ be a noncollapsed $\operatorname{RCD}(-2,3)$ space whose tangent cones are Euclidean. Smoothability conjecture. The space $(X,\mathsf{d})$ should be bi-Hölder…

Let $X$ be a smooth variety of dimension $n\geq 3$, let $p\in X$, and let $\mathrm{GHilb}^k_p(X)$ denote the punctual geometric subset of the smoothable component of…

Smoothability conjecture. $Z$ is smoothable for all $m,n\in\mathbb{N}$.

Let $\Delta$ be Freedman's contractible topological $4$-manifold whose boundary is the Poincaré homology sphere, and let $H$ be a handlebody with interior homeomorphic to…

Severi's conjecture. Every irreducible curve can be smoothed in a flat family of curves with constant degree and arithmetic genus.

Rigidity conjecture. There exist no rigid fat points, no rigid reduced curve singularities, and no rigid normal surface singularities.