53 problems

Pagani–Ricolfi–van Zelm's conjecture. The classes $\mathsf{w}_{g,A,d}^r(\theta)$ lie in the tautological ring $R^*(\overline{\mathcal{M}}_{g,n})$ for any choice of $g,d,A$ and gene…

Let $G$ be a graph, and let $\Gamma$ be the associated $\mathbb Q$-graph. For an integer $r\geq 1$, let $D(G,r)$ and $D(\Gamma,r)$ be the minimal degrees of a $g^r_d$ on $G$ and…

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…

Fix positive integers $g,r,d$ with $r\geq 3$ and $\rho(g,r,d)\geq 0$, and let $C$ be a general curve of genus $g$. For a linear series $\ell=(L,V)\in G^r_d(C)$, let…

A BN-triple is a triple $(g,r,d)$ specifying a Brill–Noether curve of genus $g$, degree $d$, and dimension $r$. For a general BN-curve $X\subset\mathbb{P}^r$ of type $(g,r,d)$, con…

The Strong Maximal Rank Conjecture. The locus of line bundles $L\in W^r_dC$ for which $\phi^k_L$ is not of maximal rank has dimension

Let $(E,V)$ be a generated coherent system of type $(n,d,v)$ on a general curve $X$ of genus $g\ge3$, where $n,d,v$ are positive integers and $v>n$. Write $D_{E,V}$ for its dual sp…

Let $\Gamma$ be a metric graph of genus $g$, and let $d,r$ be integers satisfying $0<2r\leq d<g$. The Brill–Noether rank $w_d^r(\Gamma)$ is the tropical Brill–Noether rank. Jensen–…

Large-genus containment threshold conjecture. If $e<T$, then

General secant non-containment conjecture. If

Secant non-containment conjecture. If

Tropical Brill–Noether cycle conjecture. Assume $\rho\geq 0$. Then there exists a canonical tropical subvariety $Z^r_d\subseteq W^r_d$ of pure dimension $\rho$ such that

Let $\mathcal{M}^{r}_{g,d}$ denote the Brill–Noether locus of smooth genus-$g$ curves possessing a linear series $g^r_d$, and let $\rho(g,r,d)=g-(r+1)(g-d+r)$ be the Brill–Noether…

Bounded strong Donagi–Morrison conjecture. There is a bound $\beta$ depending on $C$ and $S$ such that, if $d<\beta$, then there is a line bundle $N\hookrightarrow E_{C,A}$ with…

Strong Donagi–Morrison conjecture. There is a nontrivial line bundle $N\hookrightarrow E_{C,A}$ with $h^0(S,N)\geq 2$ such that $E_{C,A}/N$ is stable.

Nonemptiness conjecture. $Z_n(L)$ is not empty for any $g\leq 2n-2$.

Prym-Petri expectation. If $-r\leq \rho(2g-1,r,2g-2)<r$, then

Let ${\bf r}=(r_1,\ldots,r_n)\in \mathbb{N}_{>0}^n$, and let $\mathcal{V}_{\bf r}\subset M^n_{d,g;\mathrm{S}}$ be the subvariety of maps $f:\mathbb{P}^1\rightarrow\mathbb{P}^n$ wit…

Brill–Noether existence conjecture for once-marked graphs. For any once-marked graph $(G,u)$ of genus $g$, every partition $\lambda$ with $|\lambda|\leq g$ belongs to the divisor c…

Let $(S,H)$ be a polarized K3 surface and let $C\in|H|$ be a smooth irreducible curve of genus $g\ge 2$. Suppose $A$ is a complete basepoint-free $g^r_d$ on $C$ with $d\le g-1$ and…

Let $d$, $g$, $r$, and $n$ be nonnegative integers with … A BN-curve is a stable map from a general curve of genus $g$ to $\mathbb{P}^r$ of degree $d$ corresponding to the unique c…

Aprodu–Farkas strong maximal rank conjecture. The determinantal variety

Let $G$ be a connected loopless multigraph, and let $g:= |E(G)| - |V(G)| + 1$ denote its cyclomatic number. Brill–Noether gonality conjecture. Then … This is the $r=1$ case of the…

Let $C$ be a curve of genus $g$, and let $E$ be a semistable vector bundle on $C$ of rank $n$ and degree $d$. Write $\mu(E)=d/n$, and let $\gamma_1$ denote the Clifford index of…

Tropicalization surjectivity conjecture. The map $\operatorname{Trop}$ is surjective. Moreover, if $g=\lvert\bm{\mu}\rvert$, then it is a bijection.