7 problems

Kuznetsov's fullness conjecture. If there exists at least one full maximal exceptional sequence, not necessarily consisting of line bundles, then every other maximal exceptional se…

Let $X$ be a rational surface and let $\mathcal{A}$ be a toric system on $X$. Denote by $\mathbb{TV}(\mathcal{A})$ the smooth toric variety associated with $\mathcal{A}$. Bondal's…

Buan–Marsh conjecture. If two complete $\tau$-exceptional sequences over $\Lambda$ differ in only one place, then they do not exist; equivalently, the result proved for $\tau$-tilt…

Uniqueness conjecture. The statement of Theorem $$ holds for all finite-dimensional algebras.

Root-type orbit conjecture. Let $E_1,E_2\in\mathcal{E}$. Assume that there are $g_1,g_2\in B_{n-1}$ such that $g_1E_1$ and $g_2E_2$ have tame type but different root types. Then…

Toric-surface exceptional-sequence conjecture. Either $\mathcal{E}$ is constructible, or it differs from a constructible exceptional sequence by a finite sequence of spherical twis…

Let $X$ be a rational surface of Picard number $\rho\geq 2$. An exceptional toric system is a toric system arising from an exceptional sequence, and it is tame if, for some sequenc…