10 problems

Let $X_1$ and $X_2$ be Gushel–Mukai varieties, and let $\mathsf{A}(X1)$ and $\mathsf{A}(X2)$ be their associated Lagrangian subspaces. Assume that neither subspace contains decompo…

Generalized Pfaffian HPD conjecture. For every $0\leq t\leq \lfloor n/2\rfloor$, there exist noncommutative resolutions of singularities of $X$ and $Y$ which are Homologically Proj…

Higher-dimensional Pfaffian HPD conjecture. There exists a noncommutative resolution of singularities $(Y,\CR)$ of $Y$ which is Homologically Projectively Dual to $X$.

Let $\mathbf{G}$ be the Grassmannian in its Plücker embedding, let $\mathbf{Pf}$ be its projective-dual Pfaffian variety, and write $\mathbf{P}^*$ for the dual projective space. A…

Let $B$ be a generalized del Pezzo surface with $b_2(X)=b_2(B)+1$, and let … be a smooth genus-one fibered Calabi–Yau threefold with a 5-section. Assume that $X$ is the degeneracy…

Categorical crepant resolution conjecture. The categorical crepant resolution $\widetilde{D}^b({\mathrm {Pf}}(2k,V^{\vee}))$ has the non-rectangular Lefschetz decomposition

Homological projective duality conjecture. The homological projective dual variety of the Cayley plane is a non-commutative resolution of the Cartan cubic hypersurface in…

Let $W$ be a vector space of dimension $m$, and let $\operatorname{Pf}(2k,W)$ denote the closure of the locus of bivectors in $\Lambda^2W$ of rank $2k$. Let…

Let $W$ be a vector space of dimension $2n$, let $V\subset\Lambda^2W^*$ satisfy $\dim V=2n=\dim W$, and let $V^\perp\subset\Lambda^2W$ be its orthogonal. Define the Pfaffian hypers…

Let $W$ be a vector space of dimension $2n$. For $1\le k\le n-1$, let $\operatorname{Pf}_k(W^*)\subset\mathbb{P}(\Lambda^2W^*)$ be the variety of skew-forms of rank at most $2n-2k$…