597 problems

Classification conjecture. The only numerically elementary $(-1)$-curves satisfying $\delta<2\mu_1$ are the lines through two points.

Iitaka–Severi conjecture. The number of birational equivalence classes of such manifolds $Y$ is finite. This conjecture generalizes the de Franchis–Severi finiteness theorem for ma…

Mori structures conjecture. The Mori structures of $U$ are precisely $U$ itself and

Kawamata–Morrison–Totaro cone conjecture. The group $\operatorname{Aut}(X/S,\Delta)$ acts on $\mathcal{A}^e(X/S)$ with finitely many orbits of Mori faces, while…

Let $(0,0)$ be an isolated singular point of an algebraic curve over a number field $k$, with local ring $\mathcal{O}=k[[x,y]]/(f)$ and multiplicity $m$. Let $C$ be the nonsingular…

Variant of Shokurov's b-semi-ampleness conjecture. For all $m\gg 0$ and any $\epsilon\in\mathbb Q_{>0}$, the divisor

Shokurov's toric-model conjecture. Every maximal log Calabi–Yau pair $(X,\Delta_X)$ whose underlying variety $X$ is a rational $3$-fold has a toric model.

Optimal index conjecture. The index $m$ of $X$ is the largest possible index among all terminal Calabi-Yau varieties of dimension $n$, and also among all canonical Calabi-Yau varie…

Boundedness conjecture for log Fano varieties. Then $(X,\operatorname{Supp}(D))$ belongs to a finite number of algebraic families. This conjecture is known to be true when…

Let ${\EuScript{N}}_d({\Phi})$ denote the set of positive integers occurring as complement indices in the setting of the assumptions referred to in the source. Boundedness conjectu…

Let $f\colon X\to Z\ni o$ be a contraction from a threefold with only terminal singularities such that $-K_X$ is $f$-nef and $f$-big. An $n$-complement of $K_X$ is a complement wit…

Let $\mathcal{Y}\to B$ be the one-parameter family of smooth, multiply connected Calabi–Yau threefolds over … where $\xi$ is a primitive fifth root of unity, and write $Y_t$ for th…

Let $X$ be a nonsingular toric weak Fano $d$-fold. Two such varieties are weakly-F-equivalent if they are connected by a sequence of equivariant blow-ups, blow-downs, and flops thr…

Let $X$ and $Y$ be nonsingular toric Fano $d$-folds, and let $\varphi:Y\longrightarrow X$ be an equivariant morphism. An equivariant blow-up is the blow-up along a $T_N$-invariant…

Pseudo-symmetry or F-equivalence conjecture. Any nonsingular toric Fano $d$-fold is either pseudo-symmetric or F-equivalent to the $d$-dimensional projective space $\mathbf P^{d}$.

Boundedness of $B$-canonical representations. There exist positive integers $B_l$ and $B'_l$ such that

Shokurov's codimension bound conjecture. The following inequality holds:

Strong factorization conjecture. There exists a diagram

Weak factorization conjecture. Every such birational map admits this factorization, with the stated preservation of the open set and normal-crossings boundary.

Let $(X,B)$ be a log variety, let $x\in X\setminus LCS(X,B)$ be a closed point, and let $H\in\operatorname{Div}(X)\otimes\mathbb{Q}$ be ample and normalized at $x$. Effective-build…

The Base Change Conjecture. Then $K_Y+B_Y$ is $\mathbb{Q}$-Cartier and, if $f':(X',B^{X'})\to Y'$ is induced by a birational base change $\sigma:Y'\to Y$, then

Let $(X,B)$ be a log variety, and let $a(x)=a(x;X,B)$ denote the minimal log discrepancy at a closed point $x\in X$. Lower semicontinuity conjecture. For every closed point…

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

Flip conjecture. There exists a unique birational map $\phi:X\dashrightarrow X^+$ to a $\mathbb{Q}$-factorial projective variety $X^+$ with only terminal singularities, together wi…

Katsylo's conjecture. The following properties are equivalent: $\dim V=\dim W$; and there exists a $G$-equivariant birational map