22 problems

A Mori contraction is a contraction associated with a $K$-negative extremal ray in the Mori program. A log variety is nonexceptional when it is not exceptional in the sense of comp…

Let $X$ be a smooth, complete toric variety. Write $N_{1}(X)$ for the real vector space of numerical equivalence classes of $1$-cycles on $X$, and let $NE(X)$ be the cone generated…

Let $f:X\to Y$ be a Mori fiber space with $X$ of dimension three, and call $Y$ its target space. A variety has canonical singularities when all discrepancies are nonnegative. Canon…

Let $F$ be an isolated fiber of a Fano–Mori contraction, assume that $F$ is a local complete intersection, and suppose its codimension is “small”. Isolated-fiber normal-bundle conj…

Andreatta–Wiśniewski's equality conjecture. If

Let $\varphi:X\rightarrow Z$ be a Fano–Mori contraction of a smooth fourfold, and let $E$ be an exceptional divisor. Fujita's conjecture. There are no such contractions with $E$ no…

Let $X$ be a Fano manifold. Denote its Picard number by $\rho_X$, its pseudo-index by $\iota_X$, and its dimension by $\dim(X)$. Generalized Mukai conjecture. One has … This genera…

Let $\overline{M}_{g,n}$ be the moduli space of stable $n$-pointed curves of genus $g$, and let a divisor mean a divisor on this space. A 1-dimensional stratum is a component of th…

Beltrametti–Schneider–Sommese conjecture. There is an integer $d_0$ such that every threefold on ${\cal Q}^{5}$ of degree $d\geq d_0$ is a minimal model.

Let $X$ be a Fano manifold, let $n=\dim X$, and define the $(-K_X)$-slope of a subsheaf $\mathcal{E}\subseteq T_X$ by … The tangent bundle has $(-K_X)$-slope $\mu(T_X)=(-K_X)^n/n$.…

F-conjecture. The cone $\overline{\text{NE}}_1(\overline{\rm{M}}_{g,n})$ is generated by the cycle classes of F-curves. Equivalently, a line bundle on $\overline{\rm{M}}_{g,n}$ is…

F-conjecture. The cone $\overline{\text{NE}}_1(\overline{\rm{M}}_{g,n})$ is generated by the cycle classes of F-curves. Equivalently, a line bundle on $\overline{\rm{M}}_{g,n}$ is…

A Kähler space is a compact complex space admitting a Kähler form; a contraction is a morphism that contracts the curves in an extremal ray of the cone of curves. Mori's cone theor…

Let $\pi:\mathcal{X}\to B$ be a Fano fibration. Let $A$ be the affine space of curve classes having intersection $1$ with a general fiber, and let $\mathcal{P}$ be the convex hull…

Let $n>0$ be a fixed integer and let $\varepsilon>0$ be a real number. An extremal contraction is a morphism $\pi:X\to S$ as in the claim, with $X$ a $\mathbb{Q}$-factorial $n$-dim…

Let $\overline{\mathcal{M}}_{0,n}$ be the moduli space of stable $n$-pointed genus-zero curves, and let $\operatorname{NE}(\overline{\mathcal{M}}_{0,n})$ be its Mori cone. A one-di…

Let $(X,\Delta)$ be a Kawamata log-terminal pair, and let $e^{\rm str}_{\rm alg}(X,\Delta)$ denote its algebraic stringy Euler number. Monotonicity conjecture. The algebraic string…

Let $g\colon X\dashrightarrow X^+$ be a birational morphism between $[200~\mathbb{Q}$-Gorenstein varieties, together with birational morphisms $f\colon X\to Z$ and…

Let $\sigma\colon Y\to X$ be a Mori extraction or a Mori flipping contraction. A general member $S_Y\in|{-K_Y}|$ determines a surface $S_X:=\sigma(S_Y)\in|{-K_X}|$. General elephan…

Let $X$ be a manifold of $K3^{[k]}$-type, and let $R$ be the class of a primitive $1$-cycle in $X$. The Beauville–Bogomolov square of $R$ is denoted by $q(R)$. Hassett–Tschinkel's…

Width-decreasing conjecture. Let $X$ be a CP-manifold which is not a product of positive-dimensional varieties. If $\tau(X)>0$, then there exists a surjective morphism $f:X'\to X$…

Let $\overline{M}_{0,n}$ be the moduli space of pointed stable rational curves. A vital curve is a one-dimensional boundary stratum of $\overline{M}_{0,n}$. F-conjecture. Any effec…