52 problems

Let $X=X^5_9$ or $X=X^3_8$, and let $\mathcal{C}_0$ denote the cone of moving curves of $X$. Let $F$ be the anticanonical curve class, and let $(0)$-Weyl lines and $(1)$-Weyl lines…

Weyl expected dimension conjecture. For every effective divisor $D$ on $\widehat{X^4_8}$, the dimension of its space of global sections equals its Weyl expected dimension:

Laface–Ugaglia conjecture. The dimension of the space of global sections of an effective divisor on $X^3_s$ depends on the multiplicities of containment in its base locus of Weyl l…

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…

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

Weil-conjecture identity. Replacing $s$ by $1$ in the point-counting identity yields

Vafa–Witten's blowup conjecture. There is a universal function $Z_a$, independent of $X$ and $v$, such that

Dimension conjecture. The correct dimension is $\dim X$ in both cases; that is,

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…

Braden–Nekrasov conjecture. The $U(1)$ instanton on noncommutative ${\bf R}^4 \approx {\bf C}^2$ can be transformed, via the specified field redefinition, to a $U(1)$ instanton on…

Braden–Nekrasov conjecture. Instantons constructed on blowups of ${\bf C}^2$ should be related to instantons on noncommutative ${\bf R}^4$.

Let $p_1,\ldots,p_n$ be $n\geq 9$ points in $\mathbb{P}^2$ that do not satisfy any nontrivial algebraic constraint, and let $X$ be the blow-up of $\mathbb{P}^2$ at these points. Co…

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 $Z$…

Let $Z$ be a finite scheme on $\mathbb{P}^2$ and consider the surface obtained by blowing up $\mathbb{P}^2$ along $Z$. Let the residual étale algebra of $Z$ mean the étale algebra…

Let $E$ be the anticanonical divisor of a noncommutative projective plane, and consider a blow-up at points on $E$. A phantom category is a nontrivial admissible full triangulated…

$-1$-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 $X^n_s$ be the blow-up of $\mathbb{P}^n_\mathbb{C}$ at $s$ points in general position. The Weyl chamber decomposition conjecture. For $X^3_8$ and $X^5_9$, the nef chamber decom…

Let $M$ be a Kähler manifold with an extremal metric, and let $S\subset M$ be a submanifold invariant under a maximal torus in the Hamiltonian isometry group. Consider the blowup o…

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…

Strong $\Delta$-Conjecture. If $s\geqslant 11$ and $R$ is a rational de Fernex non-positive ray with $R^2=0$, then $R$ is not effective and therefore is a good ray.

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 $h^0(\mathcal{O}(D))$…