352 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'…

Let $S$ be a smooth surface, and let $C$ be a reduced and irreducible curve on $S$. Bounded Negativity Conjecture. There is a constant depending only on $S$ such that … is bounded…

Guo–Yang polynomiality conjecture. For each $k$, the coefficient of $g^{-k}$ in this expansion is a polynomial in $n$ and in the multiplicities of the arguments, and only the multi…

Kontsevich's volume formula. The Masur–Veech volume is

Strictness conjecture. All the inclusions in this sequence are strict.

Let $Q$ be a regular local ring, let $f$ be a non-zero element of its maximal ideal, and set $R=Q/(f)$. Assume that $R$ is an isolated hypersurface singularity, and for finitely ge…

Serre's intersection conjectures. The following inequalities and equalities should hold:

Let $H_g((d),\boldsymbol{\nu})$ be the one-part double Hurwitz number, where $\boldsymbol{\nu}=(a_1,\ldots,a_n)$ is a partition of degree $d$, and let $r$ be the number of simple b…

Let $\operatorname{{\mathsf M}}$ be a matroid of rank $r+1$, with characteristic-polynomial coefficients…

Let $S$ be a strictly henselian trait with geometric generic point $\bar\eta$. Let $p:X\to S$ be regular, flat, proper, and generically smooth. Write $X_s$ and $X_{\bar\eta}$ for t…

Self-intersection conjecture. One has

Let $q$ be a prime power, let $\mathcal H_q^{(3)}\subset\mathbb P^4$ be a Hermitian threefold, and let $S\subset\mathbb P^4$ be a hypersurface of degree $d\le q$, all defined over…

Let $F/F_0$ be the unramified quadratic extension and let $G(\mathbb{W})\times G(\mathbb{V})$ act on the formal scheme $\mathcal{N}_{m,n}$. For a regular element…

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…

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…

Generalized Bloch conductor formula. If $p:X\to S$ is an arithmetic $S$-scheme with $S$-proper singular locus $Z$, then

Let $\Theta_{g,n}$ be Norbury's CohFT class and let $Z_\Theta$ be its associated partition function. Let $Z_{\mathrm{BGW}}$ denote the Brézin--Gross--Witten tau-function of the Kor…

Let $g,m\geq 0$, $n\geq 1$, and let $\textbf{d}=(d_1,\ldots,d_n)\in(\Bbb Z_{\geq 0})^n$, with $|\textbf{d}|:=d_1+\cdots+d_n$. For a nondegenerate balanced genus-$g$ stable rooted t…

Let $\mathbb{V}$ be the relevant hermitian space. For a basis $\{x_1,\dots,x_{2n-m},y_1,\dots,y_m\}$ of $\mathbb{V}$, let $\mathcal{Z}(x_i)$ and $\mathcal{Y}(y_j)$ be the associate…

Let $E$ be an elliptic curve, let $\overline{M}_{g,n}$ be the moduli space of stable curves, and let $\mathcal C_g(\gamma_1,\ldots,\gamma_n)$ denote the class defined in the paper…

Let $d$ be a positive integer, and let $u$ and $mu$ be partitions of $d$. Double Hurwitz numbers count degree-$d$ ramified covers of $mathbb{C}mathbb{P}^1$ with ramification profi…

Let $G$ be a complex linear algebraic group, let $CH^i(BG)_{(p)}$ be the $p$-localized Chow group of its classifying space, and let $BP^*(BG)$ denote its Brown–Peterson cohomology.…

Let $X$ be a smooth elliptic Calabi–Yau $4$-fold, and consider its vertical four-cycles, meaning four-cycles arising from the vertical part of the elliptic fibration. Their interse…