172 problems

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…

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…

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…

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

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.

Faber–Pandharipande conjecture.

Let $\sum_i c_i\Gamma_i=0$ be a tautological equation in codimension $k$ on $\overline{M}_{g,n}$, and let $\mathfrak{r}_l$ be the operators sending decorated graphs on…

Roth–Vakil's conjecture. The coarse space of $\mathcal{M}^{\leq k}_{g,n}$ has an affine stratification with $g+k$ strata; the coarse spaces of $\mathcal{M}^{rt}_{g,n}$ and…

Looijenga's conjecture. There is such a stratification in which every locally closed stratum $S_i$ is affine.

Let $\mathcal{M}_g^0$ be the open moduli space under consideration, and let $\mathcal{Z}_{g,i}$ be the locus of curves $C$ for which there exists a pencil $A\in W^1_k(C)$ such that…

Inductive generation conjecture. Conjecture 2 will produce all tautological equations inductively.

Invariance criterion. If

Let $X$ be a hyperelliptic Riemann surface of genus $g\geq 2$, let $\tau$ be a period matrix associated to $X$, and let … be a fundamental system, where $\mathcal{F}$ is the set of…

Let $\mathcal{S}_g$ denote the moduli space of smooth genus-$g$ spin curves, and let $\mathcal{S}_g^r$ denote the locus of spin curves whose theta-characteristic has at least $r+1$…

Let ${\overline{\mathnormal{M}}}_{0,N}$ be the moduli space of stable $N$-pointed genus-zero curves. An $F$-divisor is a divisor that nonnegatively intersects every $F$-curve; let…

Let $s_g$ be the slope of $\overline{\mathcal{M}}_g$, and let $s(D)$ denote the slope of an effective divisor $D\equiv a\lambda-\sum_{i=0}^{[g/2]}b_i\delta_i$. Slope quotient princ…

Let $\mathcal{M}_{g,P}$ be the moduli space of smooth genus-$g$ curves with marked points $P$, and let $W_{m_*,P}$ denote the combinatorial class associated with ribbon graphs havi…

Let $W_{n,k}^\ast$ be the Kontsevich cycle indexed by nonnegative integers $n$ and $k$, and let $\widetilde{\kappa}_j$ denote the corresponding modified kappa class. Write…