16 problems

Let $(x \in X)$ be a singularity in characteristic zero. A singularity is of dense $F$-injective type if, for a model over a finitely generated $\mathbb{Z}$-subalgebra, the reducti…

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

Let $X$ be a variety over a field of characteristic $0$. Say that $X$ has dense $F$-injective type if, for a model of $X$ over a finitely generated $\a0\mathbb{Z}$-algebra, its red…

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.

Let $X$ be a variety over a field of characteristic $0$. The weak ordinarity conjecture is a conjecture concerning the behavior of Frobenius after reduction of characteristic-$0$ v…

Du Bois singularities are singularities of varieties over characteristic $0$, while $F$-injective singularities are defined in characteristic $p>0$ by injectivity of Frobenius on l…

Our example has a ${\mathbb Q}$-Cartier canonical divisor of index $4$. One may ask whether there also exist examples of potentially Du Bois singularities whose canonical divisor h…

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…

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…