24 problems

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

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…

Motivic jet-space integral conjecture. The identity

Let $J^N$ be the finite arc algebra and let $\mathfrak{i}_k$ and $\mathfrak{j}_k$ be the support ideals generated respectively by the initial and terminal arc coefficients as defin…

Let $J[0^N,(1)^{N'}]$ and $A[0^N,(1)^{N'}]$ be the finite arc algebras, let $\mathfrak{a}$ be the local-cohomology support ideal, and let $\mathsf{th}$ denote the homomorphisms ind…

Let $J[\delta,(1)^N]$ and $A[\delta,(1)^N]$ be the finite arc algebras introduced in the source. Finite arc-algebra singularity conjecture. The algebra $J[\delta,(1)^N]$ is Cohen–M…

Let $\mathcal{C}$ be the cone in the source, let $\mathpzc{Arc}_q^N$ be the mapping space associated with the deformation $\mathbb{C}[z,q]/(z^{N+1}-q^{N+1})$, and set…

Threshold conjecture. One has

Intersection conjecture. For some $q$, the intersection

Let $C$ be a connected curve over $k$ with only one singular point $p$, let $f:\bar C\to C$ be its normalization, and let $\bar\zeta_{C,p}(t)$ be the reduced auto-Igusa zeta series…

Let $C$ be a connected reduced curve over $k$ with exactly one singular point $p$, and let $\bar C\to C$ be its normalization of degree $e$. Let $\mathfrak{l}_{P_i(n)}$ denote the…

Let $X$ be a variety over $k$, let $p$ be a point of $X$, and let $\mathcal{A}_n(X,p)$ be its auto-arc space at level $n$. Let $\kappa(p)$ be the residue field at $p$, and write…

Let $(0\in X)$ be an isolated singularity, and let $0\in U\subset X$ be a contractible, Stein neighborhood. Neighborhood-independence conjecture. The homeomorphism types of…

Let $X$ be a complex space, let $W$ be an irreducible component of $\operatorname{Arc}^{\circ}(X)$ or $\operatorname{ShArc}(X)$, and let $[\phi_1],\dots,[\phi_m]$ be arcs in $W$. C…

Let $X$ be a complex space, and let $\operatorname{Arc}^*(X)\subset \operatorname{Arc}(X)$ be the space of arcs that are not contained in $\operatorname{Sing} X$. A finite type fam…

Let $\phi:\mathcal{X}=[M/G]\to X=M/G$ be the proper birational morphism associated with a $G$-variety $M$. Let $\gamma:\mathcal{E}\to\mathcal{X}$ be a twisted arc whose generic poi…

Let $\mathcal{X}$ be a Deligne–Mumford stack of finite type over $D$, and let $\mathcal{J}_{\infty}\mathcal{X}$ be its moduli space of twisted arcs. For $n\in\mathbb{Z}_{\ge0}$, a…

Let $\mathcal{X}$ be a Deligne–Mumford stack of finite type over $D$. A stacky formal disk is a connected normal Deligne–Mumford stack birational and finite over $D$, and a twisted…

Let $D=\operatorname{Spec}A$ be as above, let $M$ be a $D$-variety endowed with an action of a finite group $G$, and let a G-arc be a $G$-equivariant $D$-morphism $E\to M$ from a…

Let $X\in\mathbbmss{Sch}_{\kappa}$ be affine. Suppose that $X\to\mathfrak{n}$ and $X\to\mathfrak{m}$ are smooth morphisms, where $\mathfrak{n}\hookrightarrow\mathfrak{m}$ are eleme…

Strictness conjecture. If $\mathfrak{x}$ has positive dimension, each inclusion in this chain is strict. The smooth schemes form the basic class of $\mathfrak{x}$-stable schemes, a…

Let $(X,O)$ be a normal surface singularity, and let … be its minimal resolution, with exceptional divisor $E=\bigcup_{k=1}^r E_k$. For each exceptional component $E_k$, let … and…

Let $G=GL_n$, let $\mathbf{r}=(r_1,\ldots,r_k)$ determine the partial flag variety and the corresponding flag representation $\widetilde{V}_\bullet$, and let…

Let $G=GL_n$ and let $V_\bullet$ be the universal flag representation used to define the contact loci $\operatorname{Cont}^{\geq\lambda}(V_\bullet)$ and…