28 problems

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,…

Stevens's conjecture. The simple reduced curve singularities are exactly those with simple parametrisation.

Let $\{(C_t,o)\}_{t\in(\mathbb{C},0)}$ be a flat deformation of reduced curve singularities. For every $n\in\mathbb{Z}$, let $S_{n,t}$ denote the corresponding space, and let … be…

Functoriality Conjecture. The maps $\{F_{0,n}\}_n$ can be extended to the $S_n$-spaces up to homotopy, in such a way that they induce a degree-zero graded $\mathbb{Z}[U]$-module mo…

Let $C$ be the curve and let … be an embedded resolution of the pair $(\mathbb A^2,C)$. Write $\operatorname{UHilb}^n_{0,1}(C)$ for the open component and…

Graded-root determination conjecture. The graded root $\big(R(C,o),\chi\big)$ is determined by the lattice cohomology module $\mathbb{H}^0(C,o)$.

Let $C$ be a generic curve singularity, in the sense proposed in the surrounding discussion: it is parametrised by a generic point of a component of a base space of a versal deform…

Let $C$ be a Gorenstein monomial curve of Stöhr–Torres type. Stöhr–Torres non-smoothability conjecture. The curve $C$ is not smoothable. The conjecture is motivated by the example…

Let $L_r^6$ be a general homogeneous curve singularity consisting of $r$ lines through the origin in $\mathbb{A}^6$, in the interval $56<r\leq126$. Negative-weight deformation conj…

Let $L_r^n$ be a curve singularity consisting of $r$ lines through the origin in $\mathbb{A}^n$, and consider the minimal free resolution of its coordinate ring. Minimal Betti-numb…

Assume $R=\mathbb{C}[[x]][y]/(f)$ defines a generically separable degree-$n$ cover of the $x$-axis, fully ramified at the origin. For an integer composition $\nu$ of $n$, let…

Let $R$ be the complete local ring of a complex algebraic plane curve germ, with normalization $S$. Let $\mathcal{C}\operatorname{ptPic}$ be its compactified Picard scheme, let…

Hilb-vs-Quot conjecture. For any plane curve germ,

Let $A$ be a commutative ring with $1$, let $q\in A$, and let $\mathit{NH}_d(t)\in A[t]$. Let ${d\brack r}_q$ be the $q$-binomial coefficient, $(t;q)_m$ the finite $q$-Pochhammer s…

For each $d\geq 0$, let $\mathit{NH}_d(t)$ be the normalized cusp Hilbert series, let ${d\brack r}_q$ be the $q$-binomial coefficient, and let $(t;q)_m$ be the finite $q$-Pochhamme…

Let $\alpha=(K(a_1),\dots,K(a_d))$ be a pure-$K$ leading term datum, and let $k$ be any field. Consider the affine variety … defined by … and by the vanishing conditions $X_{ij}=0$…

Let $(X,p)$ be the cusp singularity $x^2=y^3$, and let $\operatorname{Quot}_{d,n}(X,p)$ be its punctual quotient scheme. The motivic polynomiality conjecture. The motive of…

The exact motivic formula conjecture.

Let $X$ be a reduced curve over $\mathbb{F}_q$, and let $\widetilde X$ be its normalization. Let $\#_q$ denote the point-count homomorphism from the completed Grothendieck ring to…

Let $\{(C_t,o)\}_{t\in(\mathbb C,0)}$ be a flat deformation of isolated curve singularities. For each level $n$, let $\ell^*$ denote a vertex of the level set of the generic fibre…

Let $(C_t,o)$ be a $delta$-constant flat deformation of Gorenstein curves. The associated graded graph map … is the map constructed from the deformation, and the graded roots carry…

Let $B_1,\ldots,B_k\subset\mathbb Z^1_{\geqslant 0}$ be support sets, let $d_i:=\min B_i>0$, and let … Assume that $j_\infty=1$, equivalently that the associated map germ is inject…

Let $\mathbf{w}$ be an invertible polynomial in two variables with admissible group of symmetries $\Gamma$, and let $\widecheck{\mathbf{w}}$ be its Berglund–Hübsch transpose with d…

Let $p$ be an $\mathbb F_q$-point of a reduced curve $X$. Write $\widehat Z_{X,p}(x)$ for the associated Cohen–Lenstra series, and let $(xq^{-1};q^{-1})_\infty$ denote the $q$-Poch…

Let $k_{\max}(d)$ be the maximal number of ordinary cusps of a plane curve of degree $d$. Langer's conjecture. … This conjecture asserts that the coefficient of $d^2$ in the known…