14 problems

Let $X$ be a variety, let $i$ be a nonnegative integer, and write $\underline{\Omega}^{i}_{X}$ for the $i$-th graded piece of the Du Bois complex of $X$. For a sheaf or complex, le…

Suppose $X$ is embedded in a smooth variety $Y$ of dimension $d$. Choose a log resolution … of the pair $(Y,X)$ that is an isomorphism away from $X$, and let … Thus $E$ is a simple…

Let $e$ be a positive integer, and let $Z$ be a locally complete intersection variety of pure dimension over a number field $K$. Define … Here…

Let $(X, \Delta)$ be a quasi-projective, log canonical pair of dimension $n$. Let $V\subset X$ be a closed subset such that … and suppose that $V$ contains all log canonical center…

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

Du Bois singularities conjecture. $X$ has only Du Bois singularities.

Let $R$ be a ring essentially of finite type over a field of characteristic zero. Du Bois–F-injective correspondence conjecture. The scheme $\operatorname{Spec} R$ has only Du Bois…

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…

Let $X$ be a reduced scheme of finite type over an algebraically closed field of characteristic $0$. A scheme is of dense $F$-injective type if it has reductions to characteristic…

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

Deformation conjecture for Du Bois singularities. Then $X$ has only Du Bois singularities in a neighborhood of $x$.

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 $D\subset X$ be a reduced Cartier divisor, and let $x\in D$. Assume that $D$ has only Du Bois singularities in a neighborhood of $x$. Steenbrink's conject…

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…