53 problems

Fix $r\geq 2$ and $k\geq 2$, and let $v_r^{(k)}(n,t)$ be the sum of the $r$th powers of the coefficients of … with … Denominator-shape conjecture. There is an integer $m\geq 1$, de…

Generalized Fibonacci analogue conjecture. The generating function

Castelnuovo's conjecture. The linear system $\mathcal{L}_{\bullet}$ has dimension $r-10$, and one of the following conditions occurs: (A) a general element…

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…

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…

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…

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

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…

Let $X$ be a curve with compactification $\ol{X}$, let $\sG$ be a reductive group, and let $G$ be its Langlands dual. Let … where $\Bun_G^{\on{level}}$ is the moduli space of $G$-b…

Let $X$ be a smooth projective variety, let $Z$ be a complete toric variety, and let … be a surjective map. Occhetta–Wiśniewski's conjecture. Then $X$ is a toric variety. The paper…

Let $G$ be a reductive group over $\mathbf{Q}$, let $\pi_f$ be a finite part of a cohomological cuspidal automorphic representation, and let $E(\pi_f)$ be a number field over which…

Let $\mathfrak{g}$ be a simple Lie algebra of type $d_4$, $e_6$, $e_7$, or $e_8$, and let $m\geq -1$ be an integer. Set … where $h^{\vee}$ is the dual Coxeter number of…

Let $K$ be a field, let $\bar{K}$ be an algebraic closure of $K$, and let $X_K\subset K^n$ be a nonsingular $K$-rational algebraic variety. Assume that…

Rationality conjecture. A smooth cubic fourfold $X$ is rational if and only if $X\in\mathcal{C}_d$ for some $d$ satisfying ().

Let $X\subset\mathbb{P}^5$ be a smooth cubic fourfold, let $\mathcal{A}_X$ be its K3 category, and let $(S,\alpha)$ be a twisted K3 surface, with $\alpha\in\operatorname{Br}(S)$. T…

Let $X$ be a smooth cubic fourfold, let $F(X)$ be its Fano variety of lines, and let $S^{[2]}$ be the Hilbert scheme of length-two subschemes of a K3 surface $S$. Galkin–Shinder's…

Rationality conjecture for Peskine sixfolds. If the twisting class $\beta$ is trivial, then $X_1^\sigma$ is rational.

Cheltsov's conjecture. In each of these three cases, $X$ is not rational.

Farber–Kaufman–Schütz rationality conjecture. The function $\mathcal{F}_X(x)$ is rational with a single pole of order $2$ at $x=1$; equivalently, there exists $P\in\mathbb{Z}[X]$ s…

Let $n\geq 1$ and set … Assume that $p,q\in\mathbb N$ are coprime, that $p\geq 2n-1$ when $q$ is odd, and that $p\geq 2n$ when $q$ is even. Principal orthosymplectic W-superalgebra…

Let $m\geq 1$ and set … Assume that $p,q\in\mathbb{N}$ are coprime, that $p\geq 2m-1$ when $q$ is odd, and that $p\geq 2m$ when $q$ is even. Orthosymplectic W-algebra rationality c…

Unirationality conjecture. Then $X$ is $k$-unirational.

Rationality conjecture. A fourfold $[X]\in\mathcal M_4$ is rational if and only if it has an associated K3 surface, equivalently if and only if $[X]$ belongs to

Let $X$ be a GM fourfold. An associated K3 surface of $X$ is a K3 surface arising from a period point of $X$ lying on a hypersurface $\mathcal{D}_d$, with a nef algebraic class $h$…