27 problems

Let $\overline{\mathcal{M}}_{0,n}$ be the moduli space of stable $n$-pointed genus-zero curves, and let its boundary divisors be the irreducible components of…

Let $g$ be a genus and let $D$ be an effective divisor on $\overline{\mathcal{M}}_g$. Its slope $s(D)$ is the smallest rational number $a/b\geq 0$ such that … is an effective combi…

Let $X$ be a smooth Calabi–Yau threefold arising from the Kreuzer–Skarke database, and let a non-trivial Hilbert basis element be a divisor class in the Hilbert basis under conside…

Let $(F,g)$ be a $g$-polarized irreducible holomorphic symplectic fourfold deformation equivalent to the Hilbert scheme of length-two subschemes of a K3 surface. Let $E^*$ be the s…

Cone-of-curves formulation. The extremal rays of $\mathcal{B}_1$ that do not lie on $\operatorname{Nul}(Y)$ are spanned by classes of $(-1)$-curves, and

Negative-curve conjecture. The curve $C$ is a $(-1)$-curve of $Y$, meaning that it is a smooth rational curve with self-intersection $-1$.

Boundary generation conjecture. The cone $(M{0,n})$ should be generated by the irreducible components of the locus of singular curves:

Let $X$ be the variety under consideration, let $D\in \operatorname{Div}(X)\otimes \mathbb{Q}$ be a divisor class on the boundary of the effective cone, and let $\eta\in X$ be a ge…

Polyhedrality conjecture. The cone $\Lambda_{\rm eff}(V,{\cal L})$ is a rational finitely generated polyhedral cone.

Let $X$ be a smooth Calabi–Yau threefold, and let its effective cone be the cone generated by effective divisor classes and its movable cone be the cone generated by divisor classe…

Let $C\subset\mathbb{P}^n$ be a rational normal curve, and let $X_n=\operatorname{Bl}_C\mathbb{P}^n$. For an integer $d$, write $\operatorname{Sec}_d(C)$ for the $d$-secant variety…

Let $\overline{LM}_7$ be the Losev–Manin space and let $\operatorname{Bl}_e \overline{LM}_7$ denote its blow-up at the identity point $e$ of the embedded torus. Let Table 27 be the…

Let $\phi: \overline{M}_{0,7}\rightarrow \operatorname{Bl}_e \overline{LM}_7$ be the natural reduction map. Let the divisors in Tables 1, 2, and 3 denote the specified $S_7$-equiva…

Let $a,b,c$ be positive pairwise coprime integers, let $\mathbb P(a,b,c)$ be the corresponding weighted projective plane, and let $\pi\colon X(a,b,c)\to\mathbb P(a,b,c)$ be the blo…

For any integer $g\ge 3$, consider the perfect-cone compactification ${{\mathcal A}_{g}^{\operatorname{Perf}}}$ and the five surfaces obtained by embedding the surfaces $S_A$,…

Let $\overline{{\mathcal A}_{3}}$ be the toroidal compactification of the moduli space of complex principally polarized abelian threefolds. Let $S_A$, $S_F$, and $S_D$ be the three…

Slope convergence conjecture. The sequence $s_g$ is decreasing, or at least convergent, and its limit is conjectured to be either $0$ or $6$. The slope is a central invariant in th…

Let $C$ be a prime divisor on ${\rm Bl}_{s}(\mathbb{P}^{2})$. Negative-curve conjecture. If … then $C$ is a $(-1)$-curve. This conjecture describes the expected negative curves on…

Standard-form negativity conjecture. If $s\geq1$ and $D^{2}<0$, then $D$ is not effective. This is presented as a consequence expected from the SHGH conjecture and concerns the eff…

Effective-divisor spanning conjecture. The method laid out in this paper produces a set of effective divisors spanning the effective cone of $M(\xi)$ for every $\xi$ above Rudakov'…

Let $X=F(Y)$ be the variety of lines on a very general cubic fourfold, and write … for some $1\leq\lambda\leq\frac{8}{5}$, where $g$ and $c$ are the cycle classes used in this desc…

Let $\overline{M}_{0,n}$ be the moduli space of stable $n$-pointed rational curves, and let $\operatorname{Eff}_1(\overline{M}_{0,n})$ denote its cone of effective curves. The one-…

Let $\overline{\mathcal{M}}_{0,n}$ be the moduli space of stable $n$-pointed genus-zero curves, and let $\overline{\operatorname{Eff}}(\overline{\mathcal{M}}_{0,n})$ denote its eff…

For an effective divisor $D=a\lambda-b\delta$ on $\overline{\mathcal M}_g$ with $a,b>0$, define its slope by … Here $g$ tends to infinity. Asymptotic sharp-slope conjecture. There…

Let the Naruki space be the moduli space of marked cubic surfaces, and let an Eckhart divisor denote one of the degeneracy divisors on this space. The effective cone is the cone ge…