26 problems

Let $X$ be a $K3$ surface over a field of positive characteristic. A $K3$ surface is called supersingular if it has maximal possible Picard rank, namely $22$, and it is unirational…

Shioda's conjecture. The surface $X$ is unirational if and only if it is Shioda supersingular.

Let $X$ be a supersingular K3 surface over an algebraically closed field. Artin's conjecture. The surface $X$ is unirational. The paper proves this conjecture, resolving Artin's qu…

Unirationality conjecture. Unirationality characterizes supersingularity for symplectic varieties.

Artin–Shioda conjecture. Every supersingular $K3$ surface is unirational.

A variety is unirational if it admits a dominant rational map from some projective space, and rationally connected if two general points can be joined by a rational curve. Uniratio…

Iskovskikh–Manin conjecture. For every integer $n\geq 4$ there exists a non-unirational, smooth, degree-$n$ hypersurface in $\mathbb{P}^n$.

A K3 surface is a connected simply-connected projective algebraic surface with trivial canonical class; it is supersingular when its Picard group has rank $22$. Unirationality conj…

Let $X\subset \mathbb{P}^n$ be a very general hypersurface of degree $d$, with $d\leq n$. Unirationality conjecture. If $d$ is large compared to $n$—for example, if $d=n$ for…

Deformation-invariance conjecture. Deformation invariance fails for the class of unirational varieties. The claim proposes that in a connected family of smooth projective varieties…

Flenner's conjecture. The threefold $X$ does not admit a nontrivial $\mathbb C_+$-action.

Let $d,n$ be integers with $d,n\geq 1$, and let $X\subseteq\mathbb{P}^{n}$ be a degree $d$ projective hypersurface containing a projective linear subspace of dimension $d-2$. Conte…

Bogomolov–Karzhemanov–Kuyumzhiyan conjecture. Every unirational algebraic variety $X$ is birationally stably flexible.

Let $\mathcal{V}_{\bf r}$ be the Severi variety of unicuspidal rational curves of type $({\rm S},{\bf r})$, under the hypotheses and notation of the codimension conjecture above. U…

Let $\pi\colon X\to W$ be a conic bundle threefold over a geometrically rational smooth surface $W$, defined over a field $k$ of characteristic different from $2$. Folklore conject…

Let $X$ be a cubic fourfold, let $F(X)$ denote its Fano variety of lines, and let $S$ be an associated K3 surface. Let $\mathcal{C}_d$ be the Hassett divisor of discriminant $d$, a…

Let ${\mathcal E}_{g,\phi}$ be the moduli space of polarized Enriques surfaces $(S,H)$ with $H^2=2g-2$ and … Here $g>1$ and $\phi>0$, and $g$ is the arithmetic genus of curves in…

Artin--Shioda--Rudakov--Safarevic conjecture. A K3 surface is supersingular if and only if it is unirational.

Bastianelli–Cortini–Flamini–Supino conjecture.

Let $X$ be a unirational variety over an algebraically closed field of characteristic zero, and let $H^i_{nr}$ denote unramified cohomology. Unramified-cohomology finiteness conjec…

Let $X$ be a unirational variety over a field $k$. A rational tower is a sequence of dominant rational maps … over $k$, with rational source $X_n$ and geometrically rational irredu…

Let $k$ be a number field and let $K/k$ be a cyclic extension of degree $3$. For a polynomial $f\in k[t]$ of degree $6$, let $\mathcal{S}_{f}$ denote the variety defined by the nor…

Let $X$ be a unirational algebraic variety over a field $\mathbf{k}$. The field $\mathbf{k}(X)(y_1,\ldots,y_m)$ is said to admit an infinitely transitive model if it has a model th…

Let $\mathcal M_h$ be the moduli space of $10$-dimensional polarised O'Grady varieties with a non-split polarisation $h$, and let $2d=h^2$ be its Beauville degree. Unirationality c…

Let $\mathcal{K}_{36}$ denote the moduli space of primitively polarized smooth $\mathrm{K3}$ surfaces of genus $36$. Unirationality conjecture. The moduli space $\mathcal{K}_{36}$…