14 problems

Let $\uppi\colon \EuScript{X}\to\mathop{\mathrm{Spec}}\EuScript{R}$ be a flopping contraction, and let $\Lambda_{\mathrm{con}}(\uppi)$ denote its contraction algebra, the stable en…

Unobstructedness and smooth-centre conjecture. The point $[A]$ is smooth in $\mathcal{M}_d$, equivalently, $A$ is unobstructed; moreover, the dimension of the centre is constant in…

The paper considers the contraction algebra $A_{8}$, presented as a quotient of the free algebra ${\mathbb C}\langle x,y\rangle$, and a third method for constructing a flat deforma…

Let $f$ be any potential, and let $\textnormal{Jdim}(f)$ denote its Jacobian dimension. Let $\EuScript{J}\mathrm{ac}(f)$ denote the Jacobian algebra of $f$. A crepant resolution is…

The Realisation Conjecture. Every $f\in\mathbb C\langle\kern -2.5pt\langle \mathsf{x}\rangle\kern -2.5pt\rangle$ whose Jacobi algebra satisfies

Realisation Problem. If $f\in\mathbb{C}\langle\!\langle x,y\rangle\!\rangle$ satisfies $\operatorname{GKdim}\EuScript{J}\mathrm{ac}(f)\leq 1$, then $\EuScript{J}\mathrm{ac}(f)$ is…

The Classification Conjecture. The complete local cDV singularities are isomorphic if and only if their contraction algebras have equivalent bounded derived categories:

Donovan–Wemyss conjecture. Any potential algebra of a rose quiver, and more generally any potential algebra arising from a quiver in the relevant wider list, can be realised as a c…

Let $R_W$ be a $2$-generated superpotential algebra assigned to a divisorial contraction to a curve in a $3$-fold, and let $[\mathbf{rep}_n R_W]_{virt}$ denote its virtual represen…

Quasi-homogeneity conjecture. The threefold $X$ is quasi-homogeneous if and only if the potential $\Phi$ is quasi-homogeneous. The source introduces this as a conjectural relation…

Polarized Donovan–Wemyss conjecture. The spaces $X$ and $X'$ are formally isomorphic at their singular points if and only if $\Lambda(Y,\Phi)$ and $\Lambda(Y',\Psi)$ are isomorphic…

Infinity-contraction algebra reconstruction conjecture. The completions of the stalks at $p$ and $q$ are isomorphic if and only if $A^\infty_{con}$ and $B^\infty_{con}$ are derived…

Let $\upvarphi$ be a contraction with an irreducible contractible flopping curve, and let $\mathrm{A}_{\mathrm{con}}$ denote its contraction algebra. The contraction algebra analyt…

Suppose that $Y\to Y_{\mathrm{con}}$ and $Z\to Z_{\mathrm{con}}$ are flopping contractions of a single curve in smooth projective $3$-folds, to points $p$ and $q$ respectively. Ass…