8 problems

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…

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

Dao–Kurano's conjecture. If $\dim(M)\leqslant \frac{d}{2}$, then

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…

Theta-pairing sign conjecture. If $n$ is odd, then the pairing

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…