173 problems

For a general smooth curve $C$ of genus $g\ge 2$ and a nontrivial $2$-torsion line bundle $\eta\in \operatorname{Pic}^0(C)[2]$, the Prym-canonical line bundle $K_C\otimes\eta$ sati…

Let $(a_1,\ldots,a_n)$ be a partition of $2g-2$ with non-negative parts. For sufficiently large $r$, let $\mathcal{W}_{g; a_1, \dots, a_n}(r)=r^{g-1}W^r_{g,n}(a_1, \dots, a_n)$ be…

For $g\geq2$, $n\geq1$, and $d_1,\ldots,d_n\geq1$ satisfying $|\mathbf d|=3g-3+n$, let $C(\mathbf d)$ denote the primitive normalized Witten intersection number, and write…

Faber's Gorenstein conjectures. (D) The intersection pairings

Let $C_g^{2g-1}$ be the $(2g-1)$-fold fiber product of the universal curve over $M_g$, let $\mathbb{E}$ be the Hodge bundle, and let $K_i$ and $D_{ij}$ denote the tautological clas…

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

Zograf's conjecture. For any fixed $n\geq 0$, as $g\to\infty$,

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…

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 $\overline{M}_{0,n}$ be the moduli space of stable $n$-pointed rational curves, and let $\delta_S$ denote the class of the boundary divisor indexed by $S$, with…

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 $g\beq 4$, let $\overline{\mathcal{M}}_g$ be the moduli space of stable curves of genus $g$, and let $s(\overline{\mathcal{M}}_g)$ denote its slope, defined as the infimum of t…

Let $g$ be the genus of a smooth proper curve and let $k$ be a non-negative integer. Write $H^{2g+k}_c(\mathcal{M}_g)$ for the compactly supported cohomology of the moduli space…

Let $\overline{\mathcal{M}}_{g,2}$ be the moduli space of stable curves with two marked points, let $\lambda_g$ denote the top Chern class of the Hodge bundle, and let…

Let $g,n,k$ be as above, let $A=(a_1,\ldots,a_n)$ satisfy $\sum_{i=1}^n a_i=k(2g-2)$, and let $\widetilde A=(a_1+k,\ldots,a_n+k)$. For $k\geq 1$, assume that $A$ is not of the form…

Let $\mathcal M_g^{\leq 1}$ be the moduli space of stable curves of genus $g$ with degenerations of torus rank at most $1$, and let…

Logarithmic double ramification tautologicality conjecture. The logarithmic double ramification cycle belongs to

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 $g,k,u\geq 0$ and let $A=(a_1,\ldots,a_n)\in\mathbb{Z}^n$ satisfy … Let $\overline{{\mathcal M}}_{g,A}^{k,r}$ be the moduli space of twisted $r$-spin structures, let…

Let ${\mathbf M}_g$ be the moduli space of smooth genus-$g$ curves, and let ${\cal R}^*({\mathbf M}_g)$ be its tautological algebra, with tautological classes $\kappa_i$. Faber's c…

For $g\geq 3$, define the slope of the moduli space of stable curves by … where $\delta=\delta_0+\delta_1+\cdots+\delta_{\lfloor g/2\rfloor}$ and…

Let $\overline{\mathcal{M}}_g$ be the moduli space of stable curves of genus $g$, and let $\kappa(\overline{\mathcal{M}}_g)$ denote its Kodaira dimension. Low-genus Kodaira-dimensi…

Let $\overline{\mathcal{M}}_{g,n}$ be the moduli space of stable $n$-pointed curves of genus $g$, let $NE_1(\overline{\mathcal{M}}_{g,n})$ denote its Mori cone of curves, and let a…

Let ${\mathcal M}_{g,n}^{rt}$ be the moduli space of stable $n$-pointed genus-$g$ curves with rational tails, and let … be the forgetful morphism. For positive integers…

Multinomial monotonicity conjecture.