18 problems

Let $Q$ be a regular local ring, let $f$ be a non-zero element of its maximal ideal, and set $R=Q/(f)$. Assume that $R$ is an isolated hypersurface singularity, and for finitely ge…

Dao–Kurano's vanishing conjecture. The map

Bounded-discrepancy hypersurface singularity conjecture. There exists a hyperplane section $H$ such that $(X,H)$ is plt (lc). Hence $(X\ni P)$ is not weakly exceptional (not except…

Extension conjecture. The sharp bound under the first hypothesis, including Proposition 1 and Theorem 2, should extend to the case where $X$ has only $A_2$-singularities instead of…

Let $R=k[[x_1,\ldots,x_n]]$ define an isolated hypersurface singularity, with $f\in R$ and $k$ a field of characteristic zero. Let $J_f$ be the Jacobian ideal and let $I_f$ denote…

Let $f$ be the hypersurface singularity under consideration at the origin $\boldsymbol{0}in\mathbf{A}^n$, and let $Z_{\mathrm{mot},\boldsymbol{0}}(f;s)$ denote its naïve motivic ze…

Let $Z\subseteq\mathbb{P}^n$ be a hypersurface of degree $d$ with only isolated singular points. Theorem 7.7 gives a bound on the number of these singularities, while Theorem 3.1 g…

Let $X$subseteq$\mathbb{P}^n$ be a hypersurface of degree $d$ with only isolated singular points, and let the bound in Theorem 7.7 and Varchenko's bound be the bounds described in…

Let $f$ be a holomorphic function germ with isolated singularity, let $sp(f)$ denote its spectrum, and let $a\in\mathbb{R}$. An invariant $I$ is semi-continuous if, for every adjac…

Let $f,g:(\mathbb{C}^d,0)\rightarrow(\mathbb{C},0)$ be two germs of holomorphic functions. Their initial forms are their lowest-degree nonzero homogeneous parts, and their Milnor f…

Bruce–Roberts number formula. It was conjectured that

Let $M_f$ be the Milnor algebra of a form $f$, let $X_n^d$ be the variety of degree-$d$ forms in $\mathcal Q_n^d$ with nonzero discriminant, and let … be the associated form morphi…

Associated-form extraction conjecture. For every regular $\operatorname{GL}(V)$-invariant function $I$ on $\mathcal{O}(V)_{d,\Delta}$ there exists a rational…

Associated-form conjecture. For any ${\tt I}\in{\mathcal I}_n^m$ there exists an absolute invariant $\mathbf I$ of forms of degree $n(m-2)$ on $\mathbb C^n$ such that, for all…

Let $X_0$ be an isolated hypersurface singularity with link $L=X_0\cap S$, and suppose $n=2m+2$. The non-vanishing cohomology groups of $L$ are … For codimension $m$ cycles $A$ and…

Let $D$ be the exceptional divisor of a resolution of an isolated rational hypersurface singularity of dimension at least $3$, and let $\Delta=\Delta(D)$ be its dual complex. Ratio…

Let $H$ be a unisingular projective hypersurface, and let $H'$ vary along its equianalytic family. Write $Z^{ea}(H')$ for the equianalytic singularity scheme and let…

A family of hypersurface singularities is minimally obstructed when its equianalytic stratum has the minimal obstruction described in the preceding results. A hypersurface singular…