23 problems

Let ${\mathcal{L}}=|ah+bF-\sum_{i=1}^{h}m_iP_i|$ be a system on the Hirzebruch surface ${\mathbb{F}}_e$, and let ${\mathcal{M}}$ be its residual system after repeatedly subtracting…

Let $X$ be a smooth projective surface, let $C\subset X$ be the framing divisor, and let $\pi:\tilde X\to X$ be the blowup of a point $x_0\notin C$. Let…

Let $X$ be a smooth projective variety with a $G$-action, let $Z\subset X$ be a $G$-invariant submanifold of codimension $c\geq 2$, and set $\widetilde X=\operatorname{Bl}_Z(X)$ an…

Let $X$ be a smooth projective variety with an action of a group $G$, and let $Z\subset X$ be a $G$-invariant submanifold of codimension $c\geq 2$. Write … with $c-1$ copies of $Z$…

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

Let $X=X^5_9$ or $X=X^3_8$. Write $\overline{\operatorname{Eff}}(X)$ for the pseudoeffective cone of divisors, $\operatorname{WCD}(\overline{\operatorname{Eff}}(X))$ for its Weyl c…

Let $\Gamma \simeq \mathbb P^k \subset X$ be a $k$-dimensional linear subspace in a smooth Fano hypersurface $X \subset \mathbb P^{n+1}$ of degree $d$. Let…

Let $X$ be the blowup of $\mathbb{P}^2$ at $n$ very general points, and let $D=dH-\sum_i m_iE_i$ be a divisor class with $d\geq 0$. Call $D$ special when both $h^0(\mathcal{O}(D))$…

Nagata's conjecture. If $n\geq 10$ and $t>\sqrt{n}$, then $A_t$ is ample. In particular,

Let $\Phi$ be the multiplicative class specified in Examples 3 and 3, and let $Z_a$ be the power series defined by the virtual $\chi_y$-genus blowup relation. For the generating se…

Let $X^n_{n+3}$ be the blow-up of $\mathbb{P}^n$ at $n+3$ points. Let $L_I$ denote the linear space spanned by the relevant subset of base points, and let $\sigma_t$ denote the $t$…

Vanishing and global-generation conjecture. For every effective divisor $D$ on $\widehat{X^4_8}$:

Weyl-line intersection conjecture. If

Let $X$ be the blowup of $\mathbb{P}^2$ at $n$ sufficiently general points, and let $\mathcal{Q}_n$ and $\Delta_n^{\preccurlyeq}$ be as defined in the source. Strong Delta-conjectu…

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 $X_\mathcal K$ be the blowup associated with the Klein line configuration, with $H$ the pullback of a line and $E_4,E_3$ the exceptional divisors over its quadruple and triple…

Let $X_\mathcal K$ be the blowup associated with the Klein line configuration, let $H$ be the pullback of a line, and let $E_4$ and $E_3$ denote the exceptional divisors over the q…

Exceptional-line projective-dimension conjecture. Then

Effective bounded negativity conjecture. One has

cscK blowup conjecture with moment-map perturbation. If $m>2$, there exist $\delta_0,\varepsilon_0>0$ such that, whenever $\mu(p)+\delta\Delta\mu(p)=0$ for some…

Harbourne-Gimigliano-Hirschowitz conjecture. The linear system $|D|$ has expected dimension

SHGH negative-curve formulation. Every reduced irreducible curve $C\subset X$ with $C^2<0$ is an exceptional curve, and for every nef divisor $F$ either

Splitting-type minimization conjecture. The pair $(a_C,b_C)$ is the solution $(a,b)$ satisfying