144 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…

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.

Graded character conjecture. We have

$n!$ conjecture. The dimension of $D_{\mu}$ is equal to $n!$.

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 $3$…

Let $S$ be the surface in the Hilbert-scheme setup, let $\mathcal{M}_n$ be the corresponding moduli space, and let $I\in\mathcal{M}_n^{\mathbb{T}}$ be a fixed ideal. Let…

Minor flattening conjecture. The displayed equality holds.

Tangency flattening conjecture. The displayed equality holds.

The nesting conjecture. The variety $V_{\underline{\boldsymbol{h}}}$ is a generically reduced elementary component if and only if

Exceptional-curve conjecture. $\operatorname{EC}(a+b+c)$ corresponds to the exceptional curves of $G\text{-}\operatorname{Hilb}(\mathbb{C}^{3})$.

Let $(S,h)$ be a general polarized K3 surface of genus $16$, let $S'$ be its unique nontrivial Fourier–Mukai partner, and let $V_{10}$ and $V_{10}'$ be the rank $10$ vector spaces…

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…

Let $0 \in J \subseteq I$ and set $K \coloneqq I \setminus J$. Let $X_J$ be the corresponding partial resolution of the Kleinian singularity, let $g:X_J\to X_0$ be the natural morp…

Let $(C,0)$ be a planar curve singularity, and let $L(q,t,0)$ be the specialization at $a=0$ of its Hilbert $L$-function. A $t$-zero is a complex number $\xi$ such that…

Let $(C,0)$ be a planar curve singularity. Let $L(q,t,a)$ be its Hilbert $L$-function and let $\mathrm{H}^{\mathrm{mot}}(q,t,a)$ be its normalized motivic superpolynomial. Cheredni…

Component correspondence conjecture. The irreducible components of $\mathfrak{K}_{r,k}(X)$ are in bijection with the irreducible components of $\operatorname{Hilb}_r(X)$ whose gene…

Let $L\subset\mathrm{Hilb}_n(\mathbb C^2)$ be the Lagrangian cut out by $\sum_{i=1}^n x_i^r=0$ for $r>0$, let $P$ be the Procesi bundle, let $\Omega^*$ be the bundle of differentia…