1,585 problems

Let $\Sigma$ be the $d$-Weddle scheme associated with a general set of $\binom{d+n}{n}$ points in $\mathbb{P}^{n+1}$, as in the preceding theorem. Reducedness and irreducibility co…

For every integer $d\geq 2$, any two nonsingular projective toric $d$-dimensional weak Fano varieties are connected by a finite zigzag of projective toric birational morphisms, eac…

Let $Q$ be a stable partition of length $\ell$, let $J_Q$ be a Jordan matrix of type $Q$, and let $\mathcal N_{J_Q}$ be the variety of nilpotent matrices commuting with $J_Q$. For…

For $1\leq i<j\leq n$, let $\mathcal{S}_{ij}$ be the Schubert variety in ${\rm Gr}(2,n)$, let $\mathcal{S}_{ij}^2$ be its square image, let $P$ be the associated symmetric matrix,…

Spinor-modification hyperbolic-equivalence conjecture. If $X'/S$ is a spinor modification of $X/S$, then $X'/S$ is hyperbolic equivalent to $X/S$.

Let $k$ be a number field, let $M\in \mathrm{M}_n(\mathcal{O}_k)$, and suppose that the triangularization variety $X_M$ has a rational point, $X_M(k)\neq\varnothing$. An irreducibl…

The big quantum cohomology of G/P is generically semisimple.

Four-variable Hessian conjecture ($\mathrm{HC}_4$). Must $\varphi^L$ be a polynomial? Equivalently, must every four-variable gradient polynomial map with constant nonzero Jacobian…

If $S$ is a surface with ADE singularities, it is known that $\operatorname{Hilb}^n(S)$ is reduced for $n\leq 7$, while reducedness for arbitrary n remains open. Its description th…

Conjecture. For every smooth quasi-projective variety $X$, … The lower bound $\operatorname{Rdim}D^b(X)\geq\dim X$ is known quite generally. The difficult direction is to construct…

Let $X$ and $Y$ be smooth projective varieties. Orlov's motivic conjecture predicts … where $h(X)_{\mathbb Q}$ denotes the rational Chow motive. A Fourier-Mukai equivalence produce…

Open problems. Determine the smallest $d$ for which $H_{d,0}$ is reducible, and the smallest $d$ for which $H_d$ is reducible. The presently known bounds are … Indeed, both spaces…

Let $X$ be a smooth projective variety with an algebraic action of a complex torus $T_{\mathbb C}$, and let $Y=X/\!/T_{\mathbb C}$ be a smooth GIT quotient with no orbifold singula…

Let $X=\operatorname{Tot}_{\mathbb{P}^2}\!\left(\mathcal{O}_{\mathbb{P}^2}(-1)\oplus\mathcal{O}_{\mathbb{P}^2}(-2)\right)$. For every effective curve class…

For the Simpson stratification of the Hodge moduli space of Higgs/de Rham objects over a smooth complex projective curve of genus $g\geq 2$, Simpson's nestedness conjecture asserts…

For every integer $N\ge 2$ and every integer $s\ge 1$, let $I\subseteq k[x_0,\ldots,x_N]$ be the homogeneous defining ideal of a general set of $s$ points in $\mathbb{P}^N$ over an…

For every integer $n\ge 10$ and every homogeneous cubic form $C\in\mathbb{Z}[x_1,\ldots,x_n]$, there exists a nonzero vector $\mathbf{x}\in\mathbb{Z}^n$ such that $C(\mathbf{x})=0$…

For every smooth projective surface $S$ with numerically trivial canonical class, and for all tautological insertions (in particular, line bundles $L_1,\ldots,L_N$ and indices…

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For a general smooth curve $C$ of genus $g\ge 2$ and a nontrivial $2$-torsion line bundle $\eta\in \operatorname{Pic}^0(C)[2]$, the Prym-canonical line bundle $K_C\otimes\eta$ sati…

Let $\pi\colon X\to S$ be a smooth projective family, let $s_0\in S$ and write $X_{s_0}=\pi^{-1}(s_0)$. Given an effective line bundle $L$ on $X$ and a prescribed singular Hermitia…

Let $D\colon\mathrm{Span}(C,E)\to\mathrm{Cat}$ be a six-functor formalism, and let $P$ and $I$ denote respectively the classes of $D$-proper and $D$-acute{e}tale morphisms. The con…

For every smooth complex projective variety $X$ and every slope-stable ample holomorphic vector bundle $E\to X$, does there exist a smooth strongly pseudoconvex Finsler metric on…

For every linear space $L\subseteq\mathbb{P}^n$, determine the factorization of its principal matroid determinant $E_L$ into irreducible polynomial factors, including the multiplic…

For each $n\in\{4,5\}$, let $S$ be a nondegenerate Euclidean $n$-simplex, and let $M(S)$ be the simplex formed by trisecting the dihedral angles of $S$. If $M(S)$ is regular, then…