17 problems

Let $G$ be a multigraph, and let $\Gamma$ be the metric multigraph obtained from $G$ by assigning length $1$ to every edge. Let the Baker–Norine rank of a divisor be computed eithe…

Brill–Noether conjecture. The gonality of a graph is bounded above by a linear function of its genus $g$.

Let $G$ be a finite, undirected, connected, loop-free multigraph of genus $g$. Define the Brill-Noether number by … A divisor on $G$ has a degree and a rank in the usual graph-theo…

Let $\overline{\mathcal{M}}_g$ be the moduli space of stable curves of genus $g$. For an effective divisor $D$ on $\overline{\mathcal{M}}_g$ written as … its slope is…

Fix a degree $d$ and geometric genus $g$, and let the normalization of the Severi variety parametrize plane curves with these invariants. Let $A$, $B$, and $C$ be the divisor class…

Let $X$ be the variety under consideration, let $D\in \operatorname{Div}(X)\otimes \mathbb{Q}$ be a divisor class on the boundary of the effective cone, and let $\eta\in X$ be a ge…

Gonality-tight graph conjecture. The following hold:

High-order gonality conjecture. Then

Let $\overline{\mathcal{M}}_{0,n}$ be the moduli space of stable $n$-pointed genus-zero curves, and let $E$ be an $F$-nef divisor, meaning that it has nonnegative intersection with…

Minimal-degree conjecture. Divisors of the form $D_{x,y}$ have minimal degree among divisors of positive rank on Ferrers rook graphs.

Let $\overline{\mathcal{M}}_{g,2}$ be the moduli space of stable genus-$g$ curves with two marked points, and let $\psi_1$ and $\psi_2$ be the corresponding cotangent line bundles.…

Let $\overline{\mathcal{M}}_{g,2}$ be the moduli space of stable genus-$g$ curves with two marked points, and let $\psi_1$ and $\psi_2$ denote the corresponding cotangent line bund…

Threefold numerical-effectivity conjecture. Then $L$ is numerically equivalent to an effective divisor.

Let $\Gamma$ be a smooth tropicalization of an algebraic curve $C$. An internal divisor is a divisor lying in a cell $\sigma$ of $|\mathcal{D}_{\Gamma}|$ parametrizing divisors wit…

Let $\overline{\mathcal{M}}_{0,n}$ be the moduli space of stable $n$-pointed genus-zero curves, and let $\overline{\operatorname{Eff}}(\overline{\mathcal{M}}_{0,n})$ denote its eff…

For an effective divisor $D=a\lambda-b\delta$ on $\overline{\mathcal M}_g$ with $a,b>0$, define its slope by … Here $g$ tends to infinity. Asymptotic sharp-slope conjecture. There…

Let $\Gamma$ be a metric graph of genus $g$. A minimal rank-determining set is a rank-determining set of $\Gamma$ that has no proper subset that is rank-determining. Cardinality bo…