15 problems

Let $\mathbb{F}_e$ be a Hirzebruch surface and let $H_m$ be an arbitrary polarization. Let $\mathbf{v}\in K(\mathbb{F}_e)$ be a character such that there are $H_{\lceil m\rceil}$-p…

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

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. Let $D$ be a divisor satisfying the paper's condition; assume $D$ is effect…

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 $\mathbb{F}_{e,r}$ 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 $\mathbb{F}_{e,r}$ 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…

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

Let $Q$ be the quiver for the local Hirzebruch surface $K_{\mathbb{F}_0}$, let $e_i$ be its vertex dimension vectors, and let $\mathrm{d}=(1,1,1,1)$ be the dimension vector of a si…

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…

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