126 problems

Stable-pair rationality and symmetry conjecture. The series $Z_\beta^\mathsf{PT}(q\mid D)$ is the Laurent expansion of a rational function, more precisely of $q^{d_\beta/2}f(q)$ fo…

Orlov's conjecture. If $X$ possesses a full exceptional collection, then $X$ is rational.

Let $X$ be a smooth cubic fourfold, and let $\mathcal{C}_d$ denote the Hassett divisor parametrizing special cubic fourfolds with a labelling of discriminant $d$. The numerical con…

C2-cofiniteness–rationality conjecture. If $V$ is $C_2$-cofinite, then $V$ is rational.

Let $V$ be a vertex operator algebra, let $G\leq\operatorname{Aut}(V)$ be a finite group, and let … be the fixed-point subalgebra. Assume that $V$ is $C_2$-cofinite and rational. O…

Let $Y$ be a cubic fourfold in the Hassett divisor $\mathcal C_{74}$. The Fano variety of lines of a general such $Y$ has no extremal ray. Rationality conjecture. Cubic fourfolds…

Let $X$ be a smooth cubic fourfold, let $F(X)$ be its Fano variety of lines, and let $S$ be a K3 surface. Galkin–Shinder's conjecture. The cubic fourfold $X$ is rational if and onl…

Let $(\Gamma,L)$ be an $n$-dimensional periodic graph, where $\Gamma$ is a directed graph, $L$ is a free abelian group of rank $n$ acting freely on $\Gamma$, and the quotient graph…

Liendo's FML rationality conjecture. If

Let $A_{5+1/2}$ denote the intermediate Lie algebra between $A_5$ and $E_6$, and let $L_k(A_{5+1/2})$ be its affine VOA at positive integer level $k$. Write $r(k)$ for the number o…

Hassett's rationality conjecture. The condition $X\in\mathcal{C}_d$ is equivalent to $X$ being rational.

Carrell's conjecture. A smooth projective variety that admits a holomorphic vector field with exactly one zero is rational.

Let $C_{k,n}$ be the simple graded quotient of $C^{k_1,k_2}$ at $(k_1,k_2)=(k,n)$, where $n$ is a positive integer and $k$ is a positive integer or an admissible level for…

Let $S$ be a rational surface, let $\Delta\subset S$ be a discriminant curve, and let $\pi:X\to S$ be a standard conic bundle. Write $K_S$ for the canonical divisor of $S$,…

Let $G$ be a finite solvable group. Central-extension rationality conjecture. There exists a central extension … with a faithful representation $V$ of $G^c$ such that $V/G^c$ is ra…

Let $f: \mathcal{S} \to B$ be a smooth projective family of complex surfaces over an irreducible base. For each fiber $S_b$, let $\mathcal{T}_{S_b}$ denote its integral polarized t…

Orbifold rationality conjecture. The fixed-point subalgebra $V^g$ is rational if $V$ is rational.

Let $S$ be a rational homology $\mathbb C\mathbb P^2$ with quotient singularities, and write … Assume that $S^0$ is simply connected. Simply connected smooth-locus conjecture. Then…

Let $S$ be a projective surface with quotient singularities such that … and let $S^0$ denote its smooth locus. Assume that $S^0$ is simply connected. Rationality conjecture. Then…

Let $G=SU(N_1)\times SU(N_2)\times\cdots\times SU(N_k)$, and let ${\cal A}_G$ denote the associated net. Complete rationality conjecture. The net ${\cal A}_G$ is completely rationa…

Let $\mathfrak{g}$ be a rigid Lie algebra. A Lie algebra is rational if it can be defined over the rational numbers $\mathbb{Q}$. The source also observes that the eigenvalues for…

Rationality conjecture. The vertex operator superalgebra $V_{c_{p,q}}$ is rational. Moreover, the minimal series modules $L_{c,h_{r,s}}$, with $0<r<p$, $0<s<q$, and…

Butler's conjecture. Under these hypotheses, $SU(r,r(g-1)+d)$ is rational whenever $d$ divides $r+delta$, provided $0<delta<r$.

Width-one conjecture. Under the stated hypothesis, $P$ has width one.

Let $G$ be an adjoint algebraic group over an arbitrary infinite field. Platonov's conjecture. The variety underlying $G$ should be rational, and consequently $G$ should satisfy we…