19 problems

Let $\underline{\mathcal{G}}$ be the relevant parahoric group scheme and let $\mu$ be the specified cocharacter datum, and write $\widetilde{M}_{\underline{\mathcal{G}},\mu}$ for t…

Higher rational implies higher Du Bois conjecture. In general, $k$-rational singularities should imply $k$-Du Bois singularities.

Intersection-complex vanishing conjecture.

Higher Du Bois implies lower rational conjecture. If $X$ has lci singularities and $X$ is $k$-Du Bois, then $X$ is $(k-1)$-rational.

Let $\mathcal C$ be a smooth complete irreducible curve of genus $g\geq 2$, and let $\mathcal E$ be a stable bundle on $\mathcal C$ of degree $2g-1$. Define $F_{\mathcal E}$ to be…

Let $\mathrm{S}$ be a smooth projective surface, and let $\mathcal{E}$ be a locally free sheaf on $\mathrm{S}$. For a positive integer $l$, let…

BCM-rationality conjecture. $R$ is BCM-rational if and only if $R$ has rational singularities.

Let $X$ be a smooth complex variety of dimension $n\geq 3$, let $D$ be a reduced divisor on $X$ with rational singularities, and let $F_\bullet\mathscr{O}_X(*D)$ denote Saito's Hod…

Let $Z$ be a complex variety of dimension $n\geq 2$ with rational singularities, and let $f\colon Y\to Z$ be a resolution of singularities whose reduced exceptional divisor $E$ has…

Let $b(\underline{s})$ be a several-variable $b$-function of semi-invariants on a more general prehomogeneous vector space, and suppose it has the form … where $d_i\in\mathbb{N}$,…

Let $Q$ be a quiver. For an extended Dynkin quiver, let $T_1,\dots,T_m$ be pairwise non-isomorphic indecomposables in $\operatorname{Rep}(Q)$ such that…

Modified Kollár's conjecture. Then $V$ has a projective closure $Y$ with rational singularities such that the closure of $C$ in $Y$ is Cohen–Macaulay.

Sharpness conjecture. For any wild quiver $Q$, there exist dimension vectors $\gamma$ and $\alpha$ such that

Let $(X,D)$ be a pair with $D$ a reduced Weil divisor. Let $H$ be a Cartier divisor on $X$ with no components in common with $D$. Assume that $(H,D\cap H)$ is Du Bois and that…

A normal surface singularity is a surface singularity that is normal, and a Newton non-degenerate singularity is one for which, for every $\mathbf{u}\in\mathbb{R}_{\geq 0}^n$, the…

Rationality conjecture for a Du Bois divisor. Then $X$ has rational singularities, and in particular is Cohen–Macaulay.

Let $X$ be a reduced scheme of finite type over a field of characteristic zero, and let $D$ be a Cartier divisor with Du Bois singularities. Assume that $X\setminus D$ has rational…

Let $X$ be a variety, let $Y$ be an ambient variety, and suppose the criterion referred to as the Easy Du Bois Criterion characterizes Du Bois singularities when $Y$ is smooth. A v…

Let $f:X\to Y$ be a good resolution of an isolated rational singularity $o\in Y$, and let the exceptional divisor be … with effective simple normal crossings support. Let $\Delta$…