Vihaan Dheer

Open problem. Does every separated Noetherian scheme have the resolution property? Equivalently, can one construct such an $X$ and a coherent sheaf $\mathcal F$ that is not a quoti…

Kuznetsov's conjecture. The fourfold $X$ is rational if and only if … for an ordinary, untwisted projective K3 surface $S$.

For a smooth Fano variety $X$ of Hodge-Tate type, the modern form of Dubrovin's conjecture predicts … The refined Gamma conjecture II also identifies categorical and quantum data.…

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…

DK hypothesis. K-equivalence should imply an equivalence … More generally, if $p^*K_X\geq q^*K_Y$, one expects a fully faithful embedding from $D^b\operatorname{Coh}(Y)$ into…

Conjecture. Every standard graded Artinian Gorenstein algebra of embedding dimension three over a field of characteristic zero has the weak Lefschetz property.

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…

It remains open whether a standard graded Artinian Gorenstein algebra of embedding dimension 4 can have a non-unimodal Hilbert function; the analogous phenomenon is impossible in d…