43 problems

Batyrev–Popov conjecture. Cox rings of del Pezzo surfaces are presented by ideals of quadrics.

Let $X_w^v$ be a Richardson variety in the Grassmannian, let $G_{k,n,\ell}|_w^v$ be the corresponding restricted set of relations, and let…

Let $F=F(n_1,\ldots,n_l,n)$ be a partial flag manifold, and let $X\subset F$ be a generic complete-intersection Calabi–Yau variety. Let $P$ be the toric degeneration of $F$, let…

Let $G$ be of type $A_n$, let $\underline{w}_0$ be any reduced decomposition of the longest Weyl-group element, and let $\lambda$ be a weight. Write $Q_{\underline{w}_0}(\lambda)$…

Let $G$ be a group of type $A_n$, and consider toric limits arising from $(G/B,{\mathcal O}(-K))$. Gorenstein toric limits conjecture. The limits of $(G/B,\mathcal O(-K))$ are Gore…

Let $f:\mathcal{X}\rightarrow S$ be a toric degeneration, and write $X=\mathcal{X}_0$. Let $Z$ be the singular set of the degeneration, and let $\mathcal{M}_X$ denote the induced l…

Let a Calabi–Yau hypersurface or complete intersection admit a toric degeneration. A Lagrangian torus fibration is a fibration whose regular fibers are Lagrangian tori and whose si…

Let $\X=\G/\P$ be a homogeneous space with a dominant weight $\varpi$ yielding a projective embedding $\X\hookrightarrow\bP(V_\varpi)$, and let $\bs$ be a reduced expression for…

Let $\X=\G/\P$ be a homogeneous space with a projective embedding $\X\hookrightarrow\bP(V_\varpi)$, and let $\bs$ be a reduced expression for $\wP$. The Chevalley polytope Newton–O…

Fix the monomial orders on $\mathbb{C}[\xi,\psi]$ and $\mathbb{C}[s,t,\mathbf{x}]$ described in the source. Let ${\rm Gr}(d,n)$ be the Grassmannian, let ${\rm CFFLV}(d,n)$ be the C…

The Hibi–Li ideal gives the PBW, or Feigin–Fourier–Littelmann–Vinberg, degeneration of the Grassmannian, with associated polytope ${\rm FFLV}(d,n)$. Let ${\rm CFFLV}(d,n)$ be the C…

Let $B$ be the base of an integral affine torus fibration, let $B(\mathbb{Z})$ be its set of integer points, and let a toric degeneration of Calabi–Yau manifolds carry a relatively…

Let $X$ be a generic Fano variety with klt singularities. A polarized cluster variety is a cluster variety equipped with a polarization; write it as $U=Y\setminus D$ with polarizat…

A $\mathbb{Q}$-Fano variety is a Fano variety with $\mathbb{Q}$-factorial terminal locally toric singularities. Let deformation-equivalence classes be taken among $\mathbb{Q}$-Fano…

Infinite-dimensional extension conjecture. The results in Sections 4 and 5 should also hold when the ambient manifold is infinite-dimensional.

Let $B_c^\sigma$ be the matching fields considered in the paper, with $\sigma \in S_n$ and $c \in [p-1]$. Suppose that the matching field polytope associated to $B_c^\sigma$ gives…

Let $f$ be a mirror partner to a smooth Fano variety $X$, and let $T_f$ be the singular toric variety defined by the spanning fan of the Newton polytope of $f$. A qG toric degenera…

Let $\Delta$ be the relevant polytope, let $\mathbb{P}_{\Delta}$ be the associated toric variety, let $\mathbb{D}_{\Delta}$ be its toric boundary data, and let $Z$ be the specified…

Let $Y$ be a smooth complete variety with toric degeneration $Y_0$, let $Y_0^\vee$ be its $f$-mirror partner, and suppose there is a geometric transition … Let $k_Z$ and $m_Z$ deno…

Let $Y$ be a smooth complete algebraic variety that is the generic fiber of a flat family with special fiber $Y_0$, where $Y_0$ is a complete intersection in a complete toric varie…

Sturmfels–Xu conjecture. The optimal number of linear inequalities is $21$.

Let $Y_{sing}$ be a toric Fano threefold with smoothing $Y_{sm}$, and let … be the Laurent polynomial whose sum runs over the vertices $v$ of the fan polytope of $Y_{sing}$. Write…

String valuation tropicalization conjecture. For every reduced expression $\underline w_0$, the weight vector $\mathbf w_{\underline w_0}$ lies in the relative interior of a maxima…

Coates–Corti–Galkin–Golyshev–Kasprzyk conjecture. The toric variety defined by the Newton polytope of $f$ is the central fibre of a degeneration of the corresponding Fano manifold.

Let $X$ be a Mori dream space, and let $Y_\bullet$ be a flag for which the global semigroup $\Gamma_{Y_\bullet}(X)$ is defined. The Cox ring of $X$ is denoted by…