7 problems

Let $X\subset\operatorname{\mathbb{P}}^4$ be a nodal hypersurface, meaning a hypersurface of degree $d$ whose singularities are all ordinary double points. Assume that $X$ has at m…

Let $V\subset\mathbb{P}^{4}$ be a hypersurface of degree $d$ with at most isolated ordinary double points, and let $\mathrm{Sing}(V)$ denote its singular locus. Factoriality conjec…

Node-count conjecture. The inequalities

Factoriality conjecture for sextic double solids. If $X$ has at most four singular points, then $X$ is factorial.

Let $F_g$ be the free group of rank $g$, let $G$ be a reductive group, and let $DG$ denote its derived subgroup. Write $\mathcal{X}(F_g,G)$ for the character variety of representat…

Let $X \subset \mathbb{P}^4$ be a hypersurface of degree $d$ whose singular locus consists of $k$ ordinary multiple points $p_1,\ldots,p_k$ of multiplicities $m_1,\ldots,m_k$. Fact…

Let $X$ be a nodal complete intersection threefold in $\mathbb{P}^5$ of hypersurfaces $F_n$ and $F_k$ of degrees $n$ and $k$, respectively, with $n\geq k$, and suppose that $F_k$ i…