50 problems

Global threshold conjecture. For some such divisor $D$, one has

Let $X$ be a variety of dimension $n$ and let $Y\subset X$ be a subvariety. The log canonical threshold of the pair $(X,Y)$ is denoted by $\operatorname{lc}(X,Y)$. Shokurov's conje…

Generalized Eckardt point conjecture. If the total log canonical threshold of $X$ is $\frac{n-1}{n}$, then a wild tiger $S$ for $X$ is a cone in $\mathbb{P}^{n-1}$ over a smooth hy…

McKernan's ACC conjecture. There exists an ACC set $\Gamma\subset [0,1]$ depending only on $d$ such that, for every such $\mathcal{F}$ on $X$, the threshold…

Jonsson–Mustaţă's weighted conjecture. The weak version asserts that there exists a quasi-monomial valuation $v\in\mathrm{Val}_X^*$ computing…

Algebraic version of Jonsson–Mustaţă's conjecture. There exists a quasi-monomial valuation $v\in\mathrm{Val}_X^*$ that computes…

Tian's strong stabilization conjecture. If $R(X,L)$ is generated by

Folklore monotonicity conjecture. There exists $k_0\in\mathbb{N}_+$ such that, whenever $k_0\leq k_1\leq k_2$,

Tian's stabilization conjecture. There exists $k_0\in\mathbb{N}_+$ such that, for every $k\geq k_0$,

Let $X$ be a normal variety of dimension $d$, and let $\mathcal{F}\subset\mathcal{G}\subset T_X$ be foliations. McKernan's nested-foliation ACC conjecture. For every positive integ…

Let $(X,\mathcal{F},B,t)$ be an adjoint foliated structure, with $B=B^{\operatorname{ninv}}+B^{\operatorname{inv}}$, and let $D$ be an $\mathbb{R}$-divisor. The ACC conjecture for…

Let $(X,B)$ be a pair of dimension $d$, let $\mathcal{F}$ be a foliation on $X$, and write $B=B^{\operatorname{ninv}}+B^{\operatorname{inv}}$ so that…

ACC conjecture for log canonical thresholds. The set

For every $n\geq 2$, let $m_n$ be the integer defined by the construction in Theorem. Let $(X,(1-b)S)$ be a complex klt Calabi–Yau pair of dimension $n$, where $S$ is a nonzero eff…

Let $X$ be a klt variety and let $\mathfrak{a}_\bullet$ be a graded sequence of ideals on $X$ with finite log canonical threshold $\operatorname{lct}(\mathfrak{a}_\bullet)$. A valu…

Threshold-equality conjecture. There are infinitely many primes $p$ for which

Facet-defining conjecture. For every rigid ray $\tau\in \operatorname{Trop}(X)\cap\mathbb{R}_{\geq 0}^p$,

Let $f\in\mathbb{Q}[x_1,\ldots,x_n]$, and for each prime number $p$ let $f_p\in\mathbb{F}_p[x_1,\ldots,x_n]$ be the polynomial obtained by reducing each coefficient of $f$ modulo…

Let $X$ be a smooth complex $n$-dimensional algebraic variety, let $f\in\operatorname{\mathcal O}_X(X)$ be nonzero, and let $P\in X$ be a point in the zero-locus of $f$ such that…

Let $X$ be a smooth complex $n$-dimensional algebraic variety, let $f\in\operatorname{\mathcal O}_X(X)$ be nonzero, and let $P\in X$ be a point in the zero-locus of $f$ such that…

Let $X$ be an $n$-dimensional variety that is locally a complete intersection with log canonical singularities. For a closed point $x$ of $X$, let $\mathrm{emb}(X,x)$ be the embedd…

Let $d$ be a positive integer, $a$ a non-negative real number, and $\Gamma$ a DCC set. For an lc germ $(X\ni x,B)$ and an $\mathbb{R}$-Cartier divisor $D$, let…

Let $I\subseteq[0,1]$ and $J\subseteq\mathbb{R}_{>0}$ be sets satisfying the descending chain condition. For a log pair $(X,\Delta)$ of dimension $n$ over an arbitrary field, with…

Cluckers–Veys conjecture. There exists a constant $c\in\mathbf{R}_{>0}$ such that the bound

Affine-space Abhyankar valuation conjecture. The following versions should hold: