17 problems

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 $f(x,y)=0$ 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 $E=\bigoplus_{i=1}^{r+1}\mathcal{O}_Y\langle y_i\rangle$ 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…

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

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

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

Compatibility conjecture. The restriction $\iota_{n,\alpha}^*{\mathbb E}$ agrees, as a symmetric perfect obstruction theory, with ${\mathbb E}_{\operatorname{crit}}$.

Let $X$ be a smooth projective surface with $\chi(\mathcal{O}_X)=1$, let $f$ be the invariants of an ideal sheaf of $k$ points, let $V$ be a vector bundle such that…

Assume a general sheaf in $M(e)$ is locally free, let $V$ have Chern character $e^*+f$, and assume $\chi(e\cdot f)=0$. Let $\operatorname{Quot}(V,f)$ parametrize quotients…