32 problems

The compatibility conjecture. The isomorphism $\eta$ induces an isomorphism of perfect obstruction theories

Let $k\geq 1$, and consider the Hilbert scheme $(\mathbb P^2)^{[k]}$ together with the appropriate theta bundle arising from the strange-duality construction. Let the expected size…

Generation conjecture. The elements $t_1,t_2,u_1,s_i^j$ for $1\leq j\leq 2g$ and $i=1,2$ generate $H^*(R_{C,d};\mathbb{Q})$.

Hessian residue formula conjecture. The intersection product is an integer and satisfies

Let $\lambda=(\lambda_1,\lambda_2,\dots,\lambda_r)$ be a partition, and let $\operatorname{\mathsf{Quot}}_d(\mathbb{P}^1,N,r)$ denote the Quot scheme of degree-$d$ quotients of the…

Let r,d\begin{in}\mathbb{N}\end{in} be two positive integers and let $X$ be an irreducible smooth threefold. For a point $[\mathcal F]$ of the Quot scheme…

Let $\bar{Y}_f$ be the enhanced $y$-ified Khovanov--Rozansky module of the link $L_f$. Let $\widetilde{Q}_{S}^{\vec{x},\vec{y}}$ be the bigraded $\mathbb{C}[\vec{x},\vec{y}]$-modul…

Let $L_f$ be the link of the plane curve germ $f=0$. Let $\widetilde{\operatorname{Quot}}^{\vec{x},\vec{y}}$ be the bigraded $\mathbb{C}[\vec{x},\vec{y}]$-module assembled from the…

Let $L_f$ be the link of the plane curve germ $f=0$, with $b$ components, and let $\mathbb{C}[\vec{x},\vec{y}]$ act on its $y$-ified Khovanov--Rozansky homology. Let…

Let $f(x,y)=0$ be a plane curve germ as above, let $\bar{X}_f(a,q,t)$ be the normalized Khovanov--Rozansky homology polynomial, and let $\Psi$ be the map from the relevant symmetri…

Assume $R=\mathbb{C}[[x]][y]/(f)$ defines a generically separable degree-$n$ cover of the $x$-axis, fully ramified at the origin. For an integer composition $\nu$ of $n$, let…

Hilb-vs-Quot conjecture. For any plane curve germ,

Let $X$ be the affine four-space occurring in the definition of the Nekrasov genus, let $E=\bigoplus_{i=1}^{N}\mathcal{O}_X\langle y_i\rangle$ and…

Let $Y=\mathbb{C}^d$, let $N>1$, set $E=\bigoplus_{i=1}^{N}\mathcal{O}_Y\langle y_i\rangle$, and, for $r=0,1,\ldots,N-2$, set $V=\bigoplus_{i=1}^{r}\mathcal{O}_Y\langle v_i\rangle$…

Let $Y=\mathbb{C}^d$, let $E=\bigoplus_{i=1}^{r+1}\mathcal{O}_Y\langle y_i\rangle$ and $V=\bigoplus_{i=1}^{r}\mathcal{O}_Y\langle v_i\rangle$. Nekrasov's conjecture for Quot scheme…

Let $X=\mathbb{C}^4$, let $E=\bigoplus_{i=1}^{N}\mathcal{O}_X\langle y_i\rangle$ and $V=\bigoplus_{i=1}^{r}\mathcal{O}_X\langle v_i\rangle$, and let $\alpha\in K_{\mathsf{T}}(X)$.…

Let $A_0(q),A_1(q),B_0(q),B_1(q)$ be the series introduced from the degree-zero reduced Segre and Verlinde series, with $N=1$. Degree-zero coefficient formulas. For…

Let $r,N>0$, let $S=\mathbb{C}^2$, and set $E=\bigoplus_{i=1}^N\mathcal{O}_S\langle y_i\rangle$ and $V=\bigoplus_{i=1}^r\mathcal{O}_S\langle v_i\rangle$. Strong Segre symmetry. The…

Let $X$ be the nonsingular quasiprojective variety considered in the construction, let $R$ be the corresponding relation algebra, and let … be the derived affine scheme equipped wi…

Dualized exterior-power vanishing conjecture.

Symmetric-power conjecture. For every $k\leq n+r(a+1)$,

Exterior-power conjecture. For every $k\leq n+r(a+1)$,

Let $S$ be a base scheme, let $\boldsymbol{\operatorname{Quot}}_{S}(\mathcal{E},d)$ and $\boldsymbol{\operatorname{Quot}}_{S}(\mathcal{H},d)$ be the derived Quot schemes associated…

The Quot formula conjecture. For every $i\in[0,\min\{d,\delta\}]$ and every $\alpha\in B_{i,\delta-i}$, there is a strong relative Fourier–Mukai transform

Let $\mathrm{S}$ be a smooth projective surface, and let $\mathcal{E}$ be a locally free sheaf on $\mathrm{S}$. For a positive integer $l$, let…