51 problems

Shafarevich's conjecture. For every positive integer $e$, only finitely many lattices occur as $\operatorname{Pic}(\overline{X})$ for K3 surfaces $X$ over number fields $K$ with…

Let $K_{2l}$ be the moduli space of quasi-polarized K3 surfaces of degree $2l$, and let the Noether–Lefschetz divisors be the divisors parametrizing K3 surfaces acquiring additiona…

Let $\mathscr{H}_{k,g}$ be the Hurwitz stack parametrizing degree $k$ covers by genus $g$ curves of $\mathbb{P}^1$, up to automorphisms of the target $\mathbb{P}^1$, and let…

Let $(R,\mathfrak{m})$ be a local complete intersection ring of dimension $3$. Let … be the punctured spectrum of $R$. Gabber's conjecture. The group $\operatorname{Pic}(U_R)$ is t…

Let $\overline{\mathcal H}$ be the compactified Hurwitz space, and let its irreducible boundary components be the irreducible components of the complement of the locus parametrizin…

Griffiths–Harris conjecture. For $3 \le r \le d$, every irreducible component of $\operatorname{NL}_{r,d}$ has codimension at least

Tautological Picard group conjecture. The rational Picard group of $\widetilde{\mathcal H}^{\dagger}_{d,g}$ is

Let $k$ be a field and let $\mathcal{M}_\sim(k)$ be the category of motives over $k$ for an adequate equivalence relation $\sim$. Let $\operatorname{Pic}(\mathcal{M}_\sim(k))$ deno…

Let $r$ be a positive integer, and consider the moduli stack of smooth $r$-spin $1$-pointed curves of genus $1$. Jarvis–Picard finiteness conjecture. The Picard group of this modul…

Let $n\geq 3$, let $X$ be an $(n-1)$-rational normal scroll, and let $\operatorname{Hilb}^X_{n-can}$ be the Hilbert scheme of canonical $n$-gonal curves inside $X$. Denote by…

Let $f:\mathcal{X}\rightarrow S$ be a toric degeneration, and write $X=\mathcal{X}_0$. Let $Z$ be the singular set of the degeneration, and let $\mathcal{M}_X$ denote the induced l…

Let $X$ be an $n$-dimensional toric Fano variety, and let $\rho$ denote its Picard number. The preceding discussion suggests that varieties with maximal Picard number are $S_3$-bun…

Let $X$ be a smooth Fano variety of Picard number $\rho(X)=1$ over an algebraically closed field of characteristic zero. An endomorphism of degree greater than $1$ is a self-map of…

Let $C$ be a smooth projective curve, let $S$ be a finite set of points containing all bad points of the group scheme $\mathcal{G}$, and let $q^*$ be the pullback map from the Pica…

Let $X$ be a smooth toric Fano variety of dimension $d$, and let $\rho_X$ denote its Picard rank. Batyrev's conjecture. The Picard rank satisfies … This conjecture concerns the max…

Let $Z$ be the compact surface serving as the base of an elliptic Calabi–Yau 3-fold, and let $\rho(Z)$ denote its Picard number. Morrison's maximal Picard rank conjecture. The maxi…

Let $X$ be a smooth projective curve, let $\mathcal{G}$ be a parahoric Bruhat–Tits group scheme, and let $S\subseteq X$ be a nonempty subset containing the bad points $\mathcal{R}$…

Let $n\geq 1$ be a height and let $p$ be a prime. Write $\operatorname{Pic}_{n,p}$ for the Picard group of the $K(n)$-local category,…

Picard-rank finiteness conjecture. The Picard number of $\mathcal{Y}_s$ is greater than or equal to $2$ for at most finitely many values of $s\in\mathbb{P}^1(\mathbb{C})$.

Fausk–Lewis–May conjecture. The canonical map

Consider the four families of quartic surfaces in $\mathbb P^3$ defined by homogeneous polynomials that are, respectively, symmetric in all four variables; symmetric in three varia…

Reduced Homological Vanishing Conjecture. For the parameters under consideration, this map is an isomorphism

Let $C$ be a smooth projective curve over $\mathbb F_q$, let $S\subset C$ be a reduced effective divisor, and consider smooth projective double covers $D\to C$ unramified over $S$.…

Let $X=D53F^1\timesD53F^1$ and let $M_{H_m}({\bf v})$ be the moduli space associated with a character ${\bf v}=(r,\nu,\Delta)\in K(X)$, where $r\geq 2$ and $\Delta\geq\frac{1}{2}$.…

Strengthened independence conjecture. For “most” sections, the following maps are injections: