12 problems

Let a geometric characterization theorem be the theorem characterizing toric triples by the properties (i), (ii), and (iii) stated in the introduction. In positive characteristic,…

Let $\mathcal{G}_{r}=\mathbb{P}(r,\ldots,r)$ be the canonical nontrivial $\mu_{r}$-gerbe over $\mathbb{P}^{d}$, let $H$ denote the hyperplane class of $\mathbb{P}^{d}$, and let $q$…

The generalized Manin conjecture. If $\mathcal{X}(F)$ is Zariski dense in $\mathcal{X}$, then there exists a thin subset $Z\subset\mathcal{X}(F)$ and a positive constant $C$ such t…

Torus and rationality conjecture. There exists a subcomplex $Y \subseteq X_A$ homotopic to $\mathbb{T}^n$. Furthermore, $\mathcal{X}$ is rational, with local systems on $Y$ corresp…

Torus conjecture. The complex $X_{A_{\mathcal{X}}}$ is homotopic to a real torus $\mathbb{T}$.

Maximal-length fullness conjecture. Every strong exceptional collection of line bundles of maximal length on a Fano toric Deligne–Mumford stack is a full strong exceptional collect…

Let $\mathfrak{X}_{\mathbf{\Sigma}}$ be a weak-Fano compact toric stack, and let $V_i$ be the $K$-theory classes whose $hat{\Gamma}$-integral flat sections give an asymptotic basis…

The dimension-three H-triviality conjecture. There are infinitely many $\mathrm{H}$-trivial line bundles on $\mathbb{P}_{\mathbf{\Sigma}}$ if and only if there exists…

Let $\mathcal{X}=[U/\mathbb G_m^2]$ be the toric stack associated to the projectivized quadric cone, with good moduli space $X$, and let $s,t$ denote the first Chern classes of the…

Let $\mathcal X$ be an Artin stack of finite type over an algebraically closed field $k$ of characteristic $0$, with an action of a torus $T$ and a dense open substack $T$-equivari…

Let $\Delta$ be a convex lattice polytope in $\bR^n$ containing the origin in its interior. Let $X$ be the toric Fano stack associated with $\Delta$, let $W:(\bC^*)^n\to\bC$ be a L…

Let $\Delta$ be a convex lattice polytope in $\bR^n$ containing the origin in its interior. Let $X$ be the associated toric Fano stack, let $\bT$ be the $n$-dimensional torus actin…