20 problems

Generic dimension and non-smoothability conjecture. For generic $L_r^n$, equality holds in the stated bounds for $\dim T^1_l$ when $l>0$ and for $\dim T^1_{-1}$, with some exceptio…

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…

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…

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 $p$ 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…

The $X_2$ multiple-line conjecture. For every $i$, $(X_2)_i$ is curvilinear and $\operatorname{length}(X_2)_i=\delta_{P_i}$. If $P_i$ is an $A_{2m_i-1}$ singularity, then $L_i$ cut…

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

Oblomkov–Shende conjecture.