16 problems

Ciliberto–Di Gennaro conjecture. The hypersurface $X$ is either factorial, or contains a plane and has at least $(d-1)^2$ nodes, or contains a quadric surface and has exactly…

Let $(R,\mathfrak m)$ be a Noetherian local complete intersection ring. Say that $R$ is a UFD in codimension at most three if every localization $R_{\mathfrak p}$ at a prime ideal…

Let $X:=V(f_{1},\ldots,f_{k})\subset\mathbb P_{k+3}$ be a nodal complete intersection threefold such that the sequence $(\deg(f_{1}),\dots,\deg(f_{k}))$ is non-decreasing, and supp…

Let $X\subset \mathbb{P}^5$ be a complex projective complete intersection threefold with isolated singularities, obtained as a complete intersection on a smooth fourfold…

Let $X_v$ be a Schubert variety and let $e_u$ be a torus-fixed point. Write $\mathcal{I}_{\mathrm{not\ factorial}}$ for the order ideal of intervals describing pairs $(u,v)$ for wh…

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

Nodal hypersurface factoriality conjecture. Under any one of these three conditions, $V$ is $\mathbb{Q}$-factorial.

Birational-rigidity conjecture. The quartic threefold $V$ is birationally rigid.

Let $X_w$ be a Schubert variety, and let $({\mathfrak S},\prec_I)$ be the poset of permutation intervals. The order ideal ${\mathcal I}_{\mathrm{not\ factorial}}$ records intervals…

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

Let $R$ be a Noetherian, strictly henselian, local unique factorization domain, and let $R[[X]]$ denote its formal power series ring in one variable. Danilov's conjecture. The ring…

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 of four quadrics in $\mathbb P_{7}$, and let a node mean an ordinary double point. Factoriality conjecture. (a) If $X$ has at most nine nod…

Let $M_{H}(r,c_{1},c_{2})$ be the moduli space of $H$-semistable sheaves on a rational, ruled surface $S$, where $H$ is a polarization. Drézet's local factoriality conjecture. If t…