345 problems

Let the moduli space of curves carry its intersection theory, and let the KdV hierarchy be the integrable hierarchy associated with the Korteweg–de Vries equation. Witten's conject…

Let $cmathcal{M}_g$ be the moduli space of smooth genus-$g$ curves, and let $cmathsf{R}^*(cmathcal{M}_g)$ denote its tautological ring, with tautological classes $ckappa_i$. Faber'…

Let $\mathcal{M}_g$ be the moduli space of smooth projective curves of genus $g$ over $\mathbb{C}$, let $\mathcal{A}_g$ be the moduli space of $g$-dimensional principally polarized…

Pixton's conjecture. The $3$-spin relations are complete in Chow and cohomology:

Let $\operatorname{\overline{M}}_{g,n}$ be the moduli space of stable $n$-pointed curves of genus $g$. A divisor $L$ is F-nef if it has non-negative intersection with every F-curve…

Let $[C,\tau]\in\mathcal{R}_{g,\ell}$ be a general level $\ell$ curve of genus $g\geq 5$, where $C$ is a smooth curve and $\tau$ is a non-trivial torsion line bundle. Its paracanon…

It is a long-standing question to determine the lower bound for the slopes of effective divisors in $\overline{\mathcal M}_g$. Harris–Morrison's slope conjecture. The Brill–Noether…

Let $\overline{\mathcal{M}}_{0,n}$ be the moduli space of stable $n$-pointed genus-zero curves, and let its boundary divisors be the irreducible components of…

Let a cohomological field theory be given, and let its double ramification hierarchy be the Hamiltonian integrable system constructed from double ramification cycles. Let its Dubro…

Let $\mathcal{C}\to U$ be a universal curve over a Zariski open subset $U$ of the moduli space of curves of genus $g\geq 2$. For a point $b\in U$, write $\mathcal{C}_b$ for the fib…

Mirzakhani's conjecture. All totally geodesic subvarieties of dimension at least $2$ are covering constructions.

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

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…

Let $g,n,k$ be as above, let $\overline{\operatorname{DRC}}$ denote the pluricanonical double ramification cycle on $\overline{\mathcal{M}}_{g,n}$, and let…

Hain–Looijenga conjecture. These pairings are perfect for every $k\geq0$; in addition, $R_c^*(M_g)$ is a free $R^*(M_g)$-module of rank one. This conjecture proposes a form of dual…

Let $R_{g,A}\in CH^g(\overline{\mathcal{M}}_{g,n})$ be the double ramification cycle, and let $\Omega_{g,A}\in CH^*(\overline{\mathcal{M}}_{g,n})$ be Pixton's mixed-degree graph-su…

Let $C$ be a general canonical curve of genus $g\geq 7$, embedded by its canonical linear system in ${\mathbf P}^{g-1}$, and let $N_{C/{\mathbf P}^{g-1}}$ denote its normal bundle.…

Let $\overline{\mathcal M}_{g,n}$ be the moduli space of stable curves of genus $g$ with $n$ marked points, and let $R^*(\overline{\mathcal M}_{g,n})$ denote its tautological ring.…

Let $g$ be a genus and let $D$ be an effective divisor on $\overline{\mathcal{M}}_g$. Its slope $s(D)$ is the smallest rational number $a/b\geq 0$ such that … is an effective combi…

Faber's conjecture. These intersection pairings are perfect for all $i$. This is one of the proposed descriptions of the tautological ring; the source states that the proportionali…

Let an F-curve be an irreducible one-dimensional boundary stratum in the moduli space $\overline{\operatorname{M}}_{g,n}$, and let a type 6 F-curve denote an F-curve in the sixth c…

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…

The classical Grassmannian carries an action of the Virasoro algebra, and the moduli space of curves is considered with respect to this action. In the super setting, one analogousl…

Let the double ramification (DR) and Dubrovin–Zhang (DZ) constructions associate integrable hierarchies of partial differential equations to a cohomological field theory. A Miura t…