10 problems

Let $(Y',B_{Y'})$ be a pair, and let $\mathfrak{M}(Y',B_{Y'})$ denote the set of b-free b-divisors that are log canonically saturated, meaning … A model has canonically confined si…

Roé-Urbinati's conjecture. The valuation $v$ is b-divisorial if and only if

Effective b-semiampleness conjecture. There is an integer $I=I(d)$ such that one can choose the diagram with $K_{X'}+\Delta'=\phi^*(K_X+\Delta)$, $I\mathbb M_{Y'}$ Cartier and free…

Let $X/Z$ be a proper morphism, let $\eta$ be the general point of $Z$, and let $S_i$ be distinct prime b-divisors of $X$. Let $\mathfrak P$ be a polyhedron in the wlc geography of…

Effective b-semiampleness conjecture. There exists a positive integer $I=I(d,\Gamma)$ such that, for every such $0$-contraction with $\dim X=d$ and $D^{\mathrm h}\in\Gamma$,…

Strictness conjecture. All the inclusions in this sequence are strict.

BPF–psef duality conjecture. A $b$-numerical class $\alpha\in\operatorname{w-N}^k(\mathcal{X})$ is psef if and only if, for every $b$-numerical class…

Let $f:(X, B)\to Y$ be a basic slc-trivial fibration, and let $\mathbf M$ denote its moduli part, a moduli $\mathbb Q$-b-divisor on $Y$. B-semi-ampleness conjecture. The moduli par…

Functorial determination conjecture. There exist a covariant functor $Y:\mathcal{C}'\to\mathcal{C}$ and a natural transformation $F:Y\to\iota$ such that:

Nef Cartier b-divisor determination conjecture. If $D\in\operatorname{\mathbf{CDiv}}_{\mathbb{Q}}X$ is nef, then there exists a determination of $D$ with center equal to the Cartie…