24 problems

Moyano et al.'s conjecture. The Picard number of a surface should be determined by its Newton–Okounkov bodies if one allows for infinitesimal flags.

Let $G$ be a plabic graph, and let $rG$ denote any rotation of $G$. Its associated Newton–Okounkov body is denoted by $\Delta_G$. Rotation invariance conjecture. All rotations of a…

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…

SHGH conjecture. Every negative curve on $Y$ is a rational $(-1)$-curve.

Valuative Nagata conjecture. If

Let $X$ be a smooth projective variety of dimension $d$ over an algebraically closed field $\kappa$, and let $L$ be a big line bundle on $X$. For each valuation $\nu$ in the higher…

Let $X$ be a projective variety of dimension $d$, let $L$ be a big line bundle on $X$, and let $D$ be a simple normal crossing divisor whose dual cone complex $\Sigma(D)$ is pure o…

Higher rank valuations take values in lexicographically ordered groups, and their associated higher rank skeleta arise as tangent cones of dual cone complexes. For a big line bundl…

A conservative reaction network is a chemical reaction network in which each species occurs in at least one nonnegative conservation law. Let the mixed-volume bound be the upper bo…

Sharpness conjecture. If $X$ is $\mathbb P^2$, $\mathbb P^1\times\mathbb P^1$, or a Hirzebruch surface, then

Let $S$ be a smooth projective algebraic surface, let $D$ be a big divisor on $S$, and write $\rho=\rho(S)$. A smooth projective surface $\tilde S$ with a proper projective morphis…

Let $S$ be a smooth projective surface and let $D$ be an ample divisor on $S$. Consider the set of Newton–Okounkov bodies of $D$ with respect to all rank-$2$ valuations of $K(S)$.…

Good-ordering finite-generation conjecture. Good orderings whose associated value semigroups are finitely generated exist for all Lie types.

Let $X$ be a smooth projective toric surface, $D$ an ample torus-invariant divisor, $R=(1,1)$ a general point, and $v\in\mathbb{Z}^2$ a primitive direction. Any admissible torus-in…

Semi-algebraicity conjecture. Every line bundle possesses a Newton–Okounkov body that is a semi-algebraic set.

The proposed bijection. The resulting point is the corresponding point in $NO_G^r$.

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…

Extended vertex-bound conjecture. The number of vertices of this Newton–Okounkov body is at most $2\varrho+2$.

Let $X$ be a smooth projective surface, $L$ a big ${\mathbb R}$-divisor, and let $f\colon \tilde{X}\to X$ be a proper birational morphism. Let ${Y_{\bullet}}$ be an admissible flag…

Let $\Lambda_{\mathbb R}$ be the real weight space, let $\Lambda^+_{\mathbb R}$ be its positive cone, and let $\mathcal C_{(>,S)}\subset\Lambda_{\mathbb R}\times\mathbb R^N$ be the…

Let $G$ be a simply connected simple complex algebraic group. For a suitable good ordering of the positive roots, let $\Gamma_{\lambda}={\operatorname*{Es}}_P(\lambda)$ denote the…

Let $G$ be a complex reductive group, let $B$ be a Borel subgroup, let $w_0$ be the longest Weyl-group element, and let $\mathbf{i}$ be a reduced expression of $w_0$. For a dominan…

Minimal valuation conjecture. If $C$ is sufficiently general and $s\geqslant 8+\frac{1}{36}$, then