6 problems

Let $X$ and $\widetilde{X}$ be nonsingular projective toric $4$-folds, and let $\psi:\widetilde{X}\rightarrow X$ be a 2-blow-up, meaning an equivariant blow-up along a $T_N$-invari…

Nonrationality conjecture. These fourfolds are not rational if their semiorthogonal decompositions do not contain the derived category of a commutative K3 surface.

Moduli-deformation conjecture. There is an infinite series of associated moduli spaces that can be deformed one to another.

Derived-category conjecture for Kuechle fourfolds.

Birationality conjecture. There is an infinite series of such moduli spaces that can be deformed one to another and therefore are birational.

Derived-category conjecture for $X_{10}$.