147 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…

For every smooth projective surface $S$ with numerically trivial canonical class, and for all tautological insertions (in particular, line bundles $L_1,\ldots,L_N$ and indices…

For a smooth threefold $X$ and an integer $\ell\geq 1$, characterize the points $[Z]\in \operatorname{Hilb}^{\ell}(X)$ having maximal singularity, namely those satisfying…

Let $p\colon C\to B$ be a smooth proper family of curves, let $L$ be a line bundle on $C$, and let $\pi_n\colon C^{[n]}\to B$ be the relative Hilbert scheme of length-$n$ subscheme…

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 ${\rm ch}_k(L_i^{[n]})$…

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 $3$…

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 $X$ be a smooth 3-fold and let $\mathscr{L}_1,\mathscr{L}_2$ be line bundles on $X$ satisfying … Let…

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 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 $N\geq 1$ and let $\mathfrak{m}=(x_1,\ldots,x_N)\subset\mathbb{C}[x_1,\ldots,x_N]$. Briançon–Iarrobino conjecture. The ideal $\mathfrak{m}^k$ has the maximum dimension tangent…

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…

Flag Hilbert scheme conjecture. The following hold:

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 $W$.

Let $\mathcal H_\lambda(z,w)$ be the partition function appearing in the source, let $\phi(0):=0$, and for every nonzero partition $\lambda$ define … where the sum is over the boxe…

Fixed-point formula conjecture. The motivic generating series satisfies

Let $S$ be a smooth projective surface and let $L$ be a line bundle on it. Define … Here $H$ is the divisor corresponding to $L$, $K$ is the canonical divisor, and … Let…

Let $\Gamma$ be a finite subgroup of $SL_2(\mathbb{C})$, let $\Gamma_n=\Gamma^n\rtimes S_n$, and let $X_{\Gamma,n}$ be the union of connected components of the $\Gamma$-fixed locus…

Let $V$ be a surface with $p_g(V)>0$, let $K$ be an effective canonical divisor, and fix $m\in H^2(V,\mathbb{Z})$. Let … be the zero-cycle on…

Let $(V,g)$ be a surface with a Kähler metric $g$, let $m\in H^2(V,\mathbb{Z})$, and let $\beta$ be a real closed $2$-form of type $(1,1)$. Let $\tau$ be a $Spin^c(4)$-structure re…

The $(q,t)$ Catalan character conjecture. In particular, $\chi_{\mathbf e\cdot L(\mathsf{triv})}(q,t)=C_n^{(k)}(q,t)$, the $(q,t)$ Catalan number. This conjecture refines the corre…

Let $X$ be a quasi-projective surface. For each $n$, let $H^*(X^{[n]})$ be expressed in the Heisenberg monomial basis $(\mathfrak h_\mu(n))$. Constant Conjecture. The structure con…

Higher-dimensional polygraph conjecture. The coordinate ring of $Z(n,l)$ over $E=\mathbb{A}^d$ is a free $k[\mathbf{z}]$-module.

Normality and Cohen–Macaulayness conjecture. For all $d$ and $n$, $X_n(\mathbb{C}^d)$ is normal and Cohen–Macaulay.