20 problems

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 $-1$.

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

Let $C$ 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…

Extremal-pair conjecture. Every $\xi$ above the Rudakov $\delta$-surface has an extremal pair, and every controlling exceptional bundle of an extremal pair is contained in two extr…

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 $\overline{M}_{0,n}$ be the moduli space of stable rational curves with $n$ marked points. A boundary divisor $\delta_I$ is associated to a partition of the markings into $I$ a…

Let $\overline{\mathcal{M}}_{0,n}$ be the moduli space of stable $n$-pointed genus-zero curves, and let $D_\Gamma$ be the divisor associated with an irreducible hypertree $\Gamma$…

Let $X$ be an irreducible holomorphic symplectic variety with polarization $H$, and let $N_1(X,\mathbb{Z})$ be its group of curve classes modulo homological equivalence. Write…

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 $g\beq 4$, let $\overline{\mathcal{M}}_g$ be the moduli space of stable curves of genus $g$, and let $s(\overline{\mathcal{M}}_g)$ denote its slope, defined as the infimum of t…

Let $G$ be the Grassmannian and let $\mathcal{E}$ and $\mathcal{K}_d$ be the bundles appearing in the construction of the line varieties, with $H=c_1(\mathcal{O}(1))$ on…

Let $V$ be smooth and projective, and let $W\subset V$ be a very moving subvariety: through a generic point $y\in V$, and for a generic vector subspace $W'\subset T_{V,y}$ of rank…

Let $(F,g)$ be a polarized irreducible holomorphic symplectic fourfold deformation equivalent to $S^{[2]}$, where $S$ is a K3 surface. Let $\mathrm{N}^1_+(F,g)$ be the positive hal…