62 problems

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:

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 $\overline{\mathcal{M}}_{g,2}$ 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 $\mathcal M_g^{\leq 1}$ be the moduli space of stable curves of genus $g$ with degenerations of torus rank at most $1$, and let…

Top-codimension conjecture. For all $g\geq 1$,

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 ${\mathbf M}_g$ 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…

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…

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…

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

Invariance criterion. If

Let $G$ be a semisimple algebraic group, let $P$ be a parabolic subgroup, and let $X=G/P$ be a general flag variety. For nonnegative integers $n$ and curve class $\beta$, write…

Gorenstein conjecture. The tautological rings of the filtration of $\overline{M}_{g,n}$ are finite-dimensional Gorenstein algebras.

Let $g$ be a nonnegative integer, let $P$ be a finite set of markings, and let $k$ be a nonnegative integer. Consider the essential tautological classes in…

Faber's strong conjecture. These pairings are perfect for $k=0,1,2,\dots$, and $\mathcal R_c^\bullet({\mathcal M}_g)$ is a free $\mathcal R^\bullet({\mathcal M}_g)$-module of rank…

Low-codimension pairing conjecture. For $i\leq 3$, these pairings are perfect. If $i=4$, the pairing has rank $\dim \mathsf{R}^4(\mathcal{M}_{g,n}^{\operatorname{ct}})$. The conjec…

Let $\overline{\mathcal{M}}_{g,n}$ be the moduli space of stable curves, let $H^k(\overline{\mathcal{M}}_{g,n})$ denote its cohomology, and let $RH^k(\overline{\mathcal{M}}_{g,n})$…

Chow tautological-generation conjecture. The Chow group

Taut 10 conjecture. All cohomology groups

Taut 20 conjecture. The cohomology groups

Let $g\geq 2$, let $\ell$ be a positive integer, and let $\mu=(m_1,\ldots,m_k)$ be a partition of $\ell(2g-2)$ with $m_i\neq-\ell$ for every $i$. Let $\mathcal{P}^\ell(\mu)$ be the…

Generation conjecture for Gorenstein kernels. The following hold: (P1) $d6a5_{g,\mathsf A}\in\mathsf{K}_{g,n}$ for all $g$ and all $s\times n$ matrices $d538$; and (P2) the system…