987 problems

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 $k$ 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…

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

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

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…

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…

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…

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…

For every polarized smooth projective complex manifold $(X,L)$, the polarized manifold $(X,L)$ is $K$-polystable if and only if the Kähler class $c_1(L)$ admits a constant scalar c…

Does every polynomial $\phi\in\mathbb{R}[x_1,x_2,x_3,x_4]$ satisfying $\det\bigl(\operatorname{Hess}\phi(x)\bigr)=-1$ and having Lorentzian Hessian, namely…

For every integer $m\ge 1$, let $X_m^4=\{x_0^m+x_1^m+x_2^m+x_3^m+x_4^m+x_5^m=0\}\subset\mathbb{P}^5$ be the Fermat fourfold. The Hodge conjecture asserts that every rational Hodge…

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 $C$ be a smooth projective curve and let $(L,V)$ be a generated linear series with $\deg L=d$ and $\dim V=r+1$, where $r\ge 2$. Let $M_{L,V}$ be its syzygy bundle, defined by t…

For every integer $n\ge 2$ and every admissible marked configuration of $n$ complete pairwise disjoint timelike affine worldlines in Minkowski space, let $F_1,\ldots,F_n$ be the as…

For every polynomial $f\in\mathbb{C}[x_1,x_2,x_3,x_4]$ satisfying $\det\bigl(\operatorname{Hess}(f)\bigr)\in\mathbb{C}^{\times}$, the gradient map…

Let $V$ be an arbitrary variety over an arbitrary field $k$, let $n$ be an integer, and let $\bar{k}$ be an algebraic closure of $k$. Write $T_{\ell}$ and $T_{DR}$ for the $\ell$-a…

Let $Y$ be a smooth projective variety and $f:Y\dashrightarrow Y$ be a rational dominating map. A smooth birational model is a smooth projective variety $\hat Y$ together with a bi…

Let $X$ be a complex algebraic manifold, and let $k$ be a positive integer for which the degree doubling formula is defined. A braid factorization is considered up to Hurwitz equiv…

Connectedness conjecture. The closures of ${\cal V}$ and ${\cal W}$ should be connected without the assumption that there are irreducible tuples.

Let $X$ be a $d$-dimensional algebraic variety with at worst Gorenstein canonical singularities. The bounded denominator conjecture asserts that the denominator of the rational num…

Let $q_1$ and $q_2$ be two distinct odd primes, and let $V$ be a nonconstant real-valued $q_1\be Z\binom{}{}\oplus q_2\be Z$-periodic potential. For $[?]$

Zero-dimensional intersection conjecture. Given $d,m\geq 1$ and $m\leq r\leq \binom{d+m}{m}$, we have

Let $\beta=\beta^1\beta^2$ be the braid and let $r_1$ and $w$ be as in the construction of the open subset $\mathcal{U}_{r_1,w}(\beta)\subseteq X(\beta)$. Assume that…