281 problems

Open problems. Determine the smallest $d$ for which $H_{d,0}$ is reducible, and the smallest $d$ for which $H_d$ is reducible. The presently known bounds are … Indeed, both spaces…

Let $S$ be a smooth projective surface over the field of complex numbers ${\mathbb{C}}$. Consider its Hilbert scheme of points $\mathsf{Hilb}_n(S)$ and orbifold symmetric product…

Expected-dimension conjecture. If $\operatorname{codim}_{\mathcal{M}_g}\mu(\mathcal{H})\le g-5$ and $\mathcal{X}(d,g,r)\ge 0$, then $\mathcal{H}$ has the minimal possible dimension…

Let $C\subset\mathbb{C}^2$ be a singular curve with a planar singularity at the origin, and let $L$ be the algebraic link obtained by intersecting $C$ with a sufficiently small $3$…

Hilbert-scheme genus conjecture. The variety $\Sigma(\mathcal{X})$ is a smooth curve of genus $3$ if and only if $\operatorname{Aut}(\mathcal{X})$ is finite.

Let $(C,p)$ be a reduced irreducible plane curve singularity, let $C^{[l]}_p$ be its punctual Hilbert scheme of length-$l$ subschemes, let $\boldsymbol{\mu}$ be its Milnor number,…

Projective-normality and quadratic-generation conjecture. If $g\geq8$ (equivalently, $q(L_2-2\delta)\geq6$), then this embedding is projectively normal; and if $g\geq10$ (equivalen…

Multiple cover formula.

Let the Hilbert scheme of surfaces in $\mathbb{P}^4$ parametrize surfaces in projective four-space, and consider its irreducible components whose general members correspond to smoo…

Let $N\geq 3$ and $k$ be a positive integer. Set $l={N-1+k \choose N}$, and let $\mathfrak{m}$ be the maximal ideal in $\mathbb{C}[x_1,x_2,\dots,x_N]$. Consider the Hilbert scheme…

Holomorphic anomaly equations. Assuming the quasi-Jacobi form property,

Let $\beta$ be a braid on $n$ strands. Let $\operatorname{Hilb}^n(\mathbb{C}^2,\mathbb{C})$ be the indicated Hilbert scheme, let $\mathcal{T}$ be its tautological rank-$n$ bundle,…

Let $X$) be a projective Calabi–Yau $4$-fold and let $L$ be a line bundle on $X$. Define … Here $M(q)=\prod_{i=1}^{\infty}(1-q^i)^{-i}$ is the MacMahon function. Cao–Kool's conject…

Dijkgraaf–Moore–Verlinde–Verlinde conjecture.

Let $\mathcal{P}$ be the Procesi bundle on $\operatorname{Hilb}^n(\mathbb{C}^2)$, let $\mathcal{P}^*$ denote its dual, and let $\mathcal{O}(d)$ be the indicated determinant-line-bu…

Lehn's conjecture. The generating series satisfies

Let $\Gamma$ be a finite subgroup acting on $\mathbb C^2$, let $\Gamma_n$ be its wreath product acting on $\mathbb C^{2n}$, and let … be the natural morphism over…

Let $M$ be a two-dimensional moduli space of meromorphic Higgs bundles, and let $n$ be a positive integer. Boalch's conjecture. The Hilbert scheme of $n$ points on $M$ is itself a…

Let $X$ be a smooth projective surface, let $X^{[n]}$ be the Hilbert scheme of $n$ points on $X$, and let $L$ be a line bundle on $X$. For the reduced generating series…

Let $X$ be a smooth projective complex surface, and let $X^{[n]}$ be the Hilbert scheme of $n$ points on $X$. For line bundles $L_1,\ldots,L_N$ on $X$, let ${\rm ch}_k(L_i^{[n]})$…

Conjectural formula for $G_4$. One has

Flag Hilbert scheme conjecture. The following hold:

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…

Severi's conjecture. A general embedding of a general curve of genus $g$ has maximal rank.

Kleppe–Ellia conjecture. The family $W$ is an irreducible component of $(H(d,g)^{sc})_{\operatorname{red}}$, and $H(d,g)^{sc}$ is generically non-reduced along $W$.