237 problems

For every genus $g\ge 2$, let $\pi:\mathcal{X}_g\to\mathcal{M}_g$ be the universal family of polarized K3 surfaces of genus $g$, and let $o_S\in\mathrm{CH}^2(S)$ denote the Beauvil…

Let $X_4\subset\mathbb{P}^3_{\mathbb{C}}$ be the Fermat quartic surface $X_4=\{[x_0:x_1:x_2:x_3]\mid x_0^4+x_1^4+x_2^4+x_3^4=0\}$. Find an explicit finite collection of automorphis…

Let $X$ be a projective K3 surface with a generic polarization $H$, and let $F$ be an $H$-polystable coherent sheaf on $X$. Kaledin–Lehn's formality conjecture. The differential gr…

Strange duality conjecture. For any $g\geq6$, this map is an isomorphism.

Skorobogatov's conjecture. The Brauer–Manin obstruction is the only obstruction to the Hasse principle for algebraic K3 surfaces over number fields.

For each $g\ge 3$, let $\mathcal F_g'$ be the smooth dense open moduli space of primitively polarized K3 surfaces of degree $2g-2$ with trivial automorphism groups, and let … be th…

Projective-normality and quadratic-generation conjecture. If $g\geq8$ (equivalently, $q(L_2-2\delta)\geq6$), then this embedding is projectively normal; and if $g\geq10$ (equivalen…

Multiple cover formula.

Let $X$ be as above, let $i$ be as above, and let $S$ be the finite set of bad primes. For each good prime $p\in\mathbb P\setminus S$, let…

Let $r_{g,h}$ denote the BPS count associated to a class $\beta$ on a K3 surface whose norm square satisfies … Let $q$ and $y$ be formal variables. Katz–Klemm–Vafa product formula.…

Strong uniform boundedness conjecture. If $X$ is a K3 surface over a number field of bounded degree $d$, then

Let $\mathcal{M}^{r}_{g,d}$ denote the Brill–Noether locus of smooth genus-$g$ curves possessing a linear series $g^r_d$, and let $\rho(g,r,d)=g-(r+1)(g-d+r)$ be the Brill–Noether…

Joyce's conjecture. For every $\sigma\in\operatorname{Stab}^{\ast}(X)$ and $\alpha\in N(X)$, there is an invariant $J^{\alpha}(\sigma)\in\Lambda$ such that $J^{\alpha}(\sigma)$ doe…

Let $X$ be a $K3$ surface, and let ${\operatorname{Aut}}_s(X)$ be the group of symplectic automorphisms of $X$, meaning automorphisms that act trivially on a nonzero holomorphic…

Kuga--Satake Hodge conjecture. The embedding $\iota_V$ is induced by a correspondence in $A^\ast(X\times KS(V)\times KS(V))$. This is a special case of the Hodge conjecture; the pa…

Várilly-Alvarado's conjecture. The order

Let $(S,H)$ be a polarized K3 surface and let $C\in|H|$ be a smooth irreducible curve of genus $g\ge 2$. Suppose $A$ is a complete basepoint-free $g^r_d$ on $C$ with $d\le g-1$ and…

Universal Severi irreducibility conjecture. For every $0\leq \delta\leq g$, the universal Severi variety $\mathcal{V}_{g,\delta}$ is irreducible.

Positive Lyapunov exponent conjecture. One has $\Lambda>0$.

Artin–Rudakov–Shafarevich–Shioda conjecture. A K3 surface over $k$ is supersingular if and only if it is unirational.

Let $\mathcal{F}_{2d}$ be the moduli space of polarized K3 surfaces of degree $2d$, and let $\overline{\mathcal{F}_{2d}}^{{\rm Sat},\tau_{\rm ad}}$ be its Satake compactification f…

Let $(X,L)$ be a very general polarized $K3$ surface. A rational curve in $|L|$ is a curve of geometric genus $0$. Nodal rational curves conjecture. All rational curves in $|L|$ ar…

Shioda's conjecture. A K3 surface is unirational if and only if it is Shioda-supersingular.

Let $X$ be a K3 quartic surface defined over a field of characteristic $3$. The 58-line bound conjecture. The surface $X$ contains $112$ lines or at most $58$ lines. The theorem pr…

Let $(F,g)$ be a polarized irreducible holomorphic symplectic fourfold deformation equivalent to the Hilbert scheme of length-two subschemes of a K3 surface. Let…