41 problems

Let $\operatorname{\overline{M}}_{g,n}$ be the moduli space of stable $n$-pointed curves of genus $g$. A divisor $L$ is F-nef if it has non-negative intersection with every F-curve…

Let $Y$ be a Calabi–Yau manifold, meaning that $Y$ has trivial canonical bundle and $h^1(\mathcal{O}_Y)=0$. Let $\mathrm{Nef}^+(Y)$ denote the convex hull of the rational points of…

Oguiso's conjecture. $X$ contains a rational curve.

Let $g\geq 2$. The partial compactification ${\cal A}_g^{\ast}$ has Picard group generated modulo torsion by the ${\mathbb{Q}}$-line bundle $L$ of modular forms of weight $1$ and t…

Let $C$ be a smooth projective curve of genus $g$ over $\mathbb C$. Denote by $f_1$ and $f_2$ the classes of the fibers of the two projections, and by $\delta$ the class of the dia…

Let $X$ be the blowup of $\mathbb{P}^2$ at $n$ sufficiently general points. Let $\mathcal{Q}_n$ be the nonnegative cone and let $\Delta_n^{\preccurlyeq}$ be the cone defined in the…

Let $X$ be a Calabi–Yau $3$-fold, let $H$ be an ample divisor on $X$, and let $\epsilon>0$. Define … A contraction $\varphi:X\to Y$ is considered when…

Let $Z$ be a Calabi–Yau variety obtained as an anticanonical hypersurface in a toric Fano variety. Let $W$ be the real vector space of toric divisors on $Z$, and let…

Conjecture for arbitrary $\mathbb{Q}$-factorial models. Analogues of Theorems 0.1 and 0.1' should hold for arbitrary $\mathbb{Q}$-factorial models, after replacing $40$ by another…

Let $\overline M_{0,n}$ be the moduli space of stable genus-zero curves with $n$ marked points. A divisor on $\overline M_{0,n}$ is F-nef if it intersects non-negatively with all o…

Let $\overline M_{g,n}$ be the Deligne–Mumford moduli space of stable curves. A divisor on $\overline M_{g,n}$ is F-nef if it intersects non-negatively with all one-dimensional bou…

$-1$-curves conjecture. The Mori cone of $S$ is generated by the light cone and the $-1$-curves. Dually,

Let $(X,\Delta)$ be a klt log Calabi–Yau pair, meaning that $X$ is smooth projective and $\Delta \in |-K_X|$ is linearly equivalent to $\sum_i a_iD_i$ for effective divisors $D_i$…

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 $(X,\Delta)$ be a klt Calabi–Yau pair. Let … where $N^1(X)_{\mathbb{Q}}$ is the rational Néron–Severi space of $X$, and let $\operatorname{Nef}^e(X)$ denote the nef effective c…

Kawamata–Morrison–Totaro Cone Conjecture. There exists a rational polyhedral cone $\Pi$ in $\operatorname{Nef}^e(X)$ that is a fundamental domain for the action of…

SYZ semiampleness conjecture. $X$ admits a birational contraction or a holomorphic Lagrangian fibration induced by $D$.

Kawamata--Morrison cone conjecture. There exists such a rational polyhedral fundamental domain $\Pi$, and the number of $\operatorname{Aut}(X)$-equivalence classes of the relevant…

Let $\overline{M}_{g,n}$ be the moduli space of stable $n$-pointed genus $g$ curves. A divisor on $\overline{M}_{g,n}$ is F-nef if it has nonnegative intersection with every one-di…

Nef-cone conjecture. The cone of nef codimension-two classes is

The square-root genus conjecture. One has

Let $(X,\Delta)$ be a projective KLT pair. Write $\overline{\operatorname{NE}}(X)_{K_X+\Delta>0}$ for the part of the closed cone of effective curves on which $K_X+\Delta$ is posit…

Nefness criterion. The divisor $D$ is nef if and only if these F-nef inequalities hold.

Mirror-family conjecture. For each such face $F_i$, there exists a codimension-one subfamily $\mathcal Y_i$ of $\mathcal Y$ whose members are $\langle\alpha_i\rangle$-polarized rat…

Let $X$ be a projective variety and let $\alpha\in\operatorname{Nef}^k(X)$. Suppose that there exists $\varepsilon>0$ such that … for all $x\in X$. Seshadri criterion. Then…