24 problems

Nef-cone conjecture. For any $g\geq2$, the nef cone on ${\cal A}_g^*$ is given by the divisors $aL-bD$ satisfying

$-1$-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 $Y$ 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…

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 $C$ 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…

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…

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

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…

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 $\overline{M}_{0,n}$ be the moduli space of stable $n$-pointed rational curves, and let $\delta_S$ denote the class of the boundary divisor indexed by $S$, with…

Oguiso's conjecture. If $X$ has a Cartier divisor $D$ with nef-boundary class, then $X$ contains a rational curve.

F-conjecture. A divisor on $\overline{\mathrm{M}}_{0,n}$ is nef if and only if it is F-nef.

Let $S$ be the indexing set used in the source, and let $\mathcal{X}_{n+2,(n-2)}$ be the corresponding space. For a divisor $D$ of the form given in the source, define the intersec…

Let $m\geq 5$ be prime, let $m\mid n$, and let $d_1=\cdots=d_n$. For the associated $\mathbb{Z}_m$-tuple, let $\mathfrak{D}(\mathbb{Z}_m,E_m;\vec d)$ be the divisor defined in equa…

Let $\overline{M}_{0,n}$ be the Deligne–Mumford moduli space of stable $n$-pointed rational curves. A line bundle on $\overline{M}_{0,n}$ is F-nef if it has nonnegative intersectio…

Let $\overline{\operatorname{M}}_{0,\operatorname{n}}$ be the moduli space of stable $n$-pointed genus-zero curves. An F-conjecture. A divisor $D$ on…

Let $\operatorname{Nef}(\overline{M}_{0,6})$ be the nef cone. A Fakhruddin ray is an extremal ray spanned either by $D(\mathfrak{sl}_2,1,(1,1,1,1,1,1))$ or by one of the 127 pullba…

Let $\mathfrak{g}$ be a simple Lie algebra. Let $D:=\operatorname{Sym} D(\mathfrak{g},\ell,\vec{\lambda})$ be a symmetrized conformal block divisor on $\overline{M}_{0,9}$ associat…

Let $\overline{\mathbf{M}}_{0,n}$ be the moduli space of stable $n$-pointed genus-zero curves, let $A_{\alpha}$ be the divisor considered in the paper, and let $p_i$ denote the sta…

Let $\overline{\mathbf{M}}_{0,n}$ be the moduli space of stable $n$-pointed genus-zero curves, and let $\overline{NE}_1$ denote the closure of the real cone of effective curves on…