26 problems

Let $QH=\mathbb{C}P^1\times\mathbb{C}P^1$ and let $\Sigma_2$ be the second Hirzebruch surface. For integers $a,b\geq 0$, let $\square_{a,b}$ and $\triangle_{a,b}$ denote the homolo…

Let $\mathbb F_{e,r}$ be the blow-up of the Hirzebruch surface $\mathbb F_e$ at $r$ very general points. Let $D$ be a divisor satisfying the paper's condition; assume $D$ is effect…

Let $\mathbb{F}_{e,r}$ denote the blow-up of the Hirzebruch surface $\mathbb{F}_e$ at $r$ very general points $p_1,\dots,p_r$. A curve is irreducible and reduced if it is an irredu…

Let $F_n=\mathbb P(\mathcal O\oplus\mathcal O(-n))$ be the Hirzebruch surface over $\mathbb P^1$, in the range $n\geq 3$ considered by the source. Parity conjecture. There are two…

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…

Toric Residue Mirror Conjecture for the first Hirzebruch surface. The series expansion at the vertex $v_{\mathcal T}\in\operatorname{Sec}(A)$ of the toric residue

Given integers $m,a\geq 1$, let $X_{2m,a}$ be the double cover of the Hirzebruch surface $\mathbb{F}_{2m}$ branched along the union of the exceptional section $\Delta_\infty$ and a…

Let $\mathbb F_{e,r}$ be the blow-up of the Hirzebruch surface $\mathbb F_e$ at $r$ very general points, and let $\widetilde{C}_e$ be the strict transform of the negative section…

Let $\mathbb F_{e,r}$ be the blow-up of the Hirzebruch surface $\mathbb F_e$ at $r$ very general points, and let $C_e$ be the negative section of $\mathbb F_e$ with strict transfor…

Let $\mathbb{F}_{e,r}$ be the blow-up of a Hirzebruch surface $\mathbb{F}_e$ at very general points, and let $\widetilde{C}_e$ be the strict transform of its negative section. A no…

Let $X$ be a rational member of the class of manifolds studied by Cristofaro-Gardiner, Holm, Mandini, and Pires, which generalizes finite toric blowups of $\mathbb{CP}^2$ and…

Let $H_b:=\mathbb{C}P^2_1\#\overline{\mathbb{C}P}\!\,^2_b$ be the one-fold blow-up of $\mathbb{C}P^2$, with line area $1$ and exceptional divisor area $0\leq b<1$. For $z\geq 1$, d…

Special-rational-point conjecture. No special rational point has a staircase. This question is related to whether $b=1/3$ has a descending staircase.

Scrollar-invariant tableau conjecture. The multiset of scrollar invariants of the partition $\lambda$ with respect to $\varphi$ is

Moe's four-cusps conjecture. The curve $C$ has at most four cusps.

Stair–Block invariance conjecture. The action of every element $T\in\mathcal{G}$ preserves the sets $\mathrm{Stair}$ and $\mathrm{Block}$.

Let $b\in(1/3,1)$, let $\mathcal S^U_{\bullet,n}$ be the staircases from Theorem with centers in $(6,\infty)$, and let $\Phi$ and $Sh$ be the transformations described in the surro…

Let ${\bf B}$ be a center-blocking class with corresponding blocked $b$-interval … Write $H_b$ for the Hirzebruch surface problem at parameter $b$, and let $\mathcal S_b$ denote a…

Let $e\geq 2$ and let $\mathcal{V}$ be an exceptional bundle on the Hirzebruch surface $\mathbb{F}_e$. Small-polarization stability conjecture. The bundle $\mathcal{V}$ is…

The Hirzebruch surfaces $\mathbb F_0$ and $\mathbb F_2$ have corresponding $q$-refined Block-Göttsche tropical invariants, and the classical Abramovich–Bertram–Vakil formula relate…

Autoequivalence conjecture. For any exceptional object $\mathcal{E}\in D(\mathbb{F}_2)$, there exists an autoequivalence $\Phi\in\operatorname{Auteq}(D(\mathbb{F}_2))$ such that…

Let $C$ be a rational cuspidal curve with four or more cusps on a Hirzebruch surface $\mathbb{F}_e$. Let $(V,D)$ be the minimal embedded resolution of $C$, and let…

Let $C$ be a rational cuspidal curve on a Hirzebruch surface. A cusp is a singularity whose local germ is irreducible. Four-cusp conjecture. $C$ has at most four cusps. The paper's…

Let $(S,\theta)$ be an isometry class of integral involutions of the K3 lattice, and let Node (1) and Isolated point denote real isotopy types of real anti-bicanonical curves with…

Let $\mathcal{L}_n(a,b;m_1,\dots,m_r)$ be a linear system on the Hirzebruch surface $\mathbb{F}_n$, with the notation and $-1$-systems defined by the intersection pairing and reduc…