102 problems

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'…

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

Let $\mu=(m_1,\ldots,m_n)$ be a partition of $2g-2$, and let $\mathbb P\mathcal H(\mu)$ be a varying stratum of holomorphic differentials up to scaling. Let … where…

Let $L_{K3}$ denote the K3 lattice, and let $N\hookrightarrow L_{K3}$ be a lattice of signature $(1,r)$. Let ${\mathcal F}_N$ be the moduli space of $N$-quasi-polarised K3 surfaces…

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 $\overline{\mathcal M}_{g,n}$ 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 $\overline{\mathcal M}_{g,n}$ be the moduli space of stable $n$-pointed curves of genus $g$, and let $R^k(\overline{\mathcal M}_{g,n})$ denote the degree-$k$ part of its tautol…

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 $\mathcal{C}_g$ be the universal curve of genus-$g$ curves and let $\mathcal{J}_g$ be the universal Jacobian. A relation in the tautological ring is an algebraic relation among…

Let $\pi\colon {\mathcal M}^{\rm rt}_{g,N}\to {\mathcal M}_g$ be the forgetful map from the moduli space of stable curves with rational tails to the moduli space of smooth genus-…

Let $L$ be a linear combination of dual graphs, let $\dim L$ denote its dimension, and let $\tau_k(L)$ be the recursively defined operation on such combinations. Condition (ii) is…

A tautological relation is a linear combination of dual graphs representing the zero class in the cohomology of the moduli space $\overline{\mathcal M}_{g,n}$ of stable curves. A u…

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 $\mathfrak{r}_l$ be the linear invariance operators between tautological rings of moduli spaces of possibly disconnected stable curves, … Here $\overline{M}^{\bullet}_{g,n}$ de…

Let $E=\sum_i c_i\Gamma_i$ be a linear combination of codimension $k$ tautological classes in $\overline{M}_{g,n}$, with $k<3g-3+n$. Assume that $\mathfrak{r}_l(E)=0$ for all $l$,…

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…

Hain–Looijenga–Faber–Pandharipande perfect pairing conjecture. For $0\leq i\leq 3g-3+n$, the natural product

Hain–Looijenga–Faber–Pandharipande socle conjecture.

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…

Let $\overline{\mathcal M}_{g,n}$ be the moduli space of stable $n$-pointed curves of genus $g$, and let $R^*(\overline{\mathcal M}_{g,n})$ be its tautological ring, generated by M…

Let $M_{g,n}$ be the moduli space of smooth $n$-pointed curves of genus $g$, with tautological ring $R^*(M_{g,n})$. Restricted generation conjecture. The ring $R^*(M_{g,n})$ is add…

Exactness conjecture. The restriction sequence is exact in all degrees. More generally, the analogous tautological sequences for the compact-type and rational-tail loci are conject…