68 problems

Let $A$ be the algebra of regular functions on a quasihomogeneous isolated surface singularity $X$ defined by $Q(x,y,z)=0$ over a field of characteristic $p$, with the notation…

Seiberg–Witten invariant conjecture. The constants are the Seiberg–Witten invariants of the link, namely

Fix a rational homology sphere link and consider the general' $\mathbb{Q}$-Gorenstein singularity with that link. Suppose that its fundamental cycle has arithmetic genus at least…

Let $(X,0)$ be a complete intersection surface singularity whose link is an integral homology sphere. Neumann–Wahl's splice-type equations conjecture. The equations defining…

Let $(Y,0)$ be a normal surface singularity with rational homology sphere link, and let $tilde{Y}$ be its minimal good resolution. An end-curve function for an end-curve is a funct…

Nash's conjecture. The number of exceptional curves in the minimal resolution of $(S,s)$ equals the number of irreducible components of $\mathcal{H}$.

Complete-intersection semigroup-condition conjecture. If $X$ is a complete intersection, then $\Delta$ satisfies the semigroup condition.

Let $(X,o)$ be a Gorenstein surface singularity with integral homology sphere link, and let its splice diagram encode the topology of the link. Gorenstein splice-type conjecture.…

Let $(X,o)$ be a Gorenstein normal surface singularity whose link $M$ is an integral homology sphere, meaning $H_1(M;{\mathbb Z})=\{0\}$. Let $\Delta$ be its splice diagram. Integr…

Let $(X,o)$ be a ${\mathbb Q}$-Gorenstein normal surface singularity with ${\mathbb Q}$-homology sphere link, and let $\Delta$ be the splice diagram associated to its resolution gr…

Let $(X,o)$ be a ${\mathbb Q}$-Gorenstein surface singularity with ${\mathbb Q}$-homology sphere link, and let its resolution graph be equipped with the associated discriminant-gro…

The $A_n$-base conjecture. If $f\colon (X,C)\to (S,0)$ is a Mori conic bundle, then $(S,0)$ is a Du Val singularity of type $A_n$.

Let $M_N$ be the KSBA moduli stack of index one covers, let $M_N^{\operatorname{Bub}}$ be the moduli stack of bubble-tree index one cover DM stacks, and let…

A rational homology disk singularity is a normal surface singularity whose link bounds a rational homology disk; a Wahl singularity is a cyclic quotient surface singularity of the…

Let $f\in\mathbb{C}[[x,y,z]]$ be an isolated hypersurface singularity. Let $\mu$ be its Milnor number and $p_g$ its geometric genus. Durfee's conjecture. … with equality only when…

Let $f\in\mathbb{C}[[x,y,z]]$ be an isolated hypersurface singularity, with Milnor number $\mu$ and Tjurina number $\tau$. Almirón's conjecture. … This is described as a weak versi…

Let $C_0$ and $C_1$ be the curves and let $g_0$ and $g_1$ be the polynomial function germs associated with them. Write $V(g_j)$ for the zero locus of $g_j$ and let the monodromy of…

Let $r\in\mathbf{Z}_{\geq 2}$, let $\mathcal{S}$ be the ring in which the ideal $\mathfrak{a}$ is defined, and let $J_r$ be the ideal generated by the monomials described above: th…

Consider the two-dimensional semi-log canonical hypersurface singularities in the families labelled (1)--(3) in the paper. Semistability conjecture. All singularities in these thre…

Frobenius uniqueness conjecture. A Wahl chain $\texttt{W}$ in $\mathbb{W}_9$ is determined by its index.

Let $(V_i,0)\subset(\mathbb{C}^3,0)$, for $i=1,2$, be germs of isolated surface singularities. Let $K_1,K_2$ be their links, and let $\Delta_1,\Delta_2$ be the characteristic polyn…

Birbrair–Gabrielov conjecture. If $(X_0,0)$ and $(X_1,0)$ are ambient topologically equivalent and outer bi-Lipschitz equivalent, then $X_0$ and $X_1$ are ambient bi-Lipschitz equi…

Let $G\subset\operatorname{GL}(2,\mathbb{C})$ be a finite subgroup, let $X=\mathbb{C}^2/G$ with boundary divisor $B$, and let $\mathcal{M}_{\theta}$ denote the moduli space of…

Frobenius liftability conjecture. Every cusp singularity is $F$-liftable.

DCC conjecture. The set of volumes of these orbifold pairs satisfies the descending chain condition, and its minimum nonzero value is