99 problems

Let $C$ be a Petri curve, let $L$ be a line bundle on $C$, and let $(L,V)$ be a linear system with $V\subset H^0(L)$; write $D_{L,V}$ for its dual span bundle. Butler's stability c…

Let $C$ be an integral curve of genus $g$, and for each positive integer $n$ define … Here ${\rm Cliff}_1(C)$ is the rank-one Clifford index. Mercat's conjecture. … The conjecture…

Expected-dimension conjecture. If $\operatorname{codim}_{\mathcal{M}_g}\mu(\mathcal{H})\le g-5$ and $\mathcal{X}(d,g,r)\ge 0$, then $\mathcal{H}$ has the minimal possible dimension…

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

Set $\rho_{g,k}:=3g-3-\binom{k+1}{2}$. Let $C$ be a general curve of genus $g\geq 2$, and consider the moduli space of stable rank-two vector bundles on $C$ with canonical determin…

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…

Strong Maximal Rank Conjecture. The locus $Q^r_d(C)\subset G^r_d(C)$ consisting of linear series for which $\nu(L,V)$ fails to have maximal rank is a determinantal locus of the exp…

Donagi–Morrison conjecture. There exists a line bundle $M\in\operatorname{Pic}(S)$ adapted to $|H|$ such that $|A|$ is contained in the restriction of $|M|$ to $C$ and

Let $(S,H)$ be a polarized K3 surface, and suppose that the complete linear system $|H|$ is base point free. For smooth curves $C$ in $|H|$, the gonality is the least integer $k$ f…

Auel–Haburcak's conjecture. If $g\geq3$ and $g\neq7,8,9$, every expected maximal Brill–Noether locus is maximal with respect to containment.

Brill–Noether conjecture. For general $C$, the locus $W^r_dC$ is nonempty if and only if

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…

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…

Pflueger's Brill–Noether conjecture. Analogues of the Brill–Noether theorem should hold for the transmission locus $W^{\tau}(C,p,q)$. Pflueger's conjecture proposes that the expect…

Let ${\mathcal M}_g$ be the moduli space of smooth curves of genus $g$, and let … be the locus of curves possessing an $r$-dimensional theta-characteristic. Farkas's conjecture. Fo…

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

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

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…

Aprodu–Farkas strong maximal rank conjecture. The determinantal variety

Maximal Rank Conjecture. These restriction maps have maximal rank, meaning that each is either injective or surjective. Equivalently, the dimension of the space of degree-$k$ polyn…

Pure codimension conjecture. If $\rho(g,r,n)\geq 0$, then $P^r_{g,n}$ has pure codimension one whenever it is non-empty. The Petri locus records failure of injectivity of the Petri…

Petri-type injectivity conjecture. The Petri map $\mu_N$ is injective.

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…

Generation conjecture. The elements $t_1,t_2,u_1,s_i^j$ for $1\leq j\leq 2g$ and $i=1,2$ generate $H^*(R_{C,d};\mathbb{Q})$.

Brill–Noether conjecture. A generic curve $C$ in the moduli space of curves of genus $g$ is Brill–Noether general.