17 problems

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 $f:\mathcal{X}\rightarrow S$ 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 $\X=\G/\P$ 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…

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…

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

Let $J_1$ and $J_2$ be the ideals under consideration, with $J_1$ monomial-free. Quadratic-generation conjecture. If $J_1$ is monomial-free, then $J_2$ is quadratically generated.…

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…

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…

Let $n\geq 3$ be an arbitrary integer, and let $\underline w_0$ be a reduced expression. For each Plücker variable, let ${\bf m}_J$ be the corresponding minimal monomial and let…

Extremal transition conjecture. The following hold: (1) the structure constants of the small quantum product of $X_{\rm res}$ are polynomials in $q_1,\dots,q_l$ with coefficients r…

Let $\mathcal{X}\rightarrow D$ be a polarized toric degeneration with intersection complex $(\check B,\check{\mathcal P})$. Let $\omega_t$ be a Kähler form on $\mathcal{X}_t$ repre…

Consider the Gross–Siebert toric degeneration of Hitchin's moduli spaces and the large complex degeneration of the same moduli space. The central fiber of the first is the special…

Millson's conjecture. Give the ring $$ the associated term-order filtration from HMM and HMSV, with associated graded ring isomorphic to

Toric degeneration conjecture. The degeneration $f$ is birationally equivalent to a toric degeneration