8 problems

Let $(X, \mathfrak{a})$ be a log pair over an algebraically closed field $k$ of characteristic zero, and let $D$ be a normal Cartier prime divisor on $X$. Let $x \in D$ be a closed…

Let $X$ be a normal variety, $S$ a normal Cartier divisor, and $B=\sum b_iB_i$ a $\mathbb{Q}$-divisor. Assume that $K_X+S+B$ is $\mathbb{Q}$-Cartier. Inversion of Adjunction. The t…

Let $X$ be a nonsingular variety over a field of characteristic zero, and let $Y=\sum_{i=1}^k t_iY_i$ be a formal combination where $t_i>0$ are real numbers and each…

PIA conjecture. One has

Let $(X,\mathfrak{a})$ be a log pair, let $D$ be a normal Cartier prime divisor, and let $x\in D$ be a closed point. Suppose that $D$ is not contained in the cosupport of the…

Let $C=\operatorname{Spec}(\mathcal{O}_{K})$ be the spectrum of the ring of integers of a number field, let $\pi\colon\mathcal{X}\to C$ be as in the preceding notation, and let…

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 $(X,G+\Delta)$ be a pair such that $G$ is a reduced divisor with no component in the support of the effective divisor $\Delta$, and let $Z$ be a closed subset of $G$. Let…