1,884 problems

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…

In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety V has a resolution, which is a non-singular variety W with a proper birationa…

In algebraic geometry, the Virasoro conjecture states that a certain generating function encoding Gromov–Witten invariants of a smooth projective variety is fixed by an action of h…

In mathematics, specifically arithmetic geometry, the Tate conjecture is a 1963 conjecture of John Tate that would describe the algebraic cycles on a variety in terms of a more com…

In mathematics, the standard conjectures about algebraic cycles are several conjectures describing the relationship of algebraic cycles and Weil cohomology theories. One of the ori…

In anabelian geometry, a branch of algebraic geometry, the section conjecture gives a conjectural description of the splittings of the group homomorphism…

In mathematics, more specifically in algebraic geometry, Parshin's conjecture states that for any smooth projective variety X defined over a finite field, the higher algebraic K-gr…

In mathematics, the Nakai conjecture is an unproven characterization of smooth algebraic varieties, conjectured by Japanese mathematician Yoshikazu Nakai in 1961. It states that if…

In mathematics, the Nagata–Biran conjecture, named after Masayoshi Nagata and Paul Biran, is a generalisation of Nagata's conjecture on curves to arbitrary polarised surfaces.

In mathematics, the Nagata conjecture on curves, named after Masayoshi Nagata, governs the minimal degree required for a plane algebraic curve to pass through a collection of very…

In mathematics, specifically algebraic geometry, Donaldson–Thomas theory is the theory of Donaldson–Thomas invariants. Given a compact moduli space of sheaves on a Calabi–Yau three…

In mathematics, the Jacobian conjecture is a conjecture concerning polynomials in several variables that states that if a polynomial function from an -dimensional space to itself h…

In algebraic geometry, general elephant is an idiosyncratic name for a general element of the anticanonical system of a variety, introduced by Miles Reid. For 3-folds the general e…

In mathematics, Fujita's conjecture is a problem in the theories of algebraic geometry and complex manifolds. It is named after Takao Fujita, who formulated it in 1985.

In algebraic geometry, the Fröberg conjecture is a conjecture about the possible Hilbert functions of a set of forms. It is named after Ralf Fröberg, who introduced it in Fröberg.…

In algebra, the Dixmier conjecture, stated by Jacques Dixmier in 1968, originally asked whether any endomorphism of the first Weyl algebra $A_{1}$ over a field of characteristic ze…

In deformation theory, a branch of mathematics, Deligne's conjecture is about the operadic structure on Hochschild cochain complex. Various proofs have been suggested by Dmitry Tam…

Pierre René, Viscount Deligne is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 1978 Field…

In mathematics, the Bass–Quillen conjecture relates vector bundles over a regular Noetherian ring A and over the polynomial ring $A[t_{1},\dots,t_{n}]$. The conjecture is named for…

In mathematics, especially algebraic geometry, the Bass conjecture says that certain algebraic K-groups are supposed to be finitely generated. The conjecture was proposed by Hyman…

In algebraic geometry, the abundance conjecture is a conjecture in birational geometry, more precisely in the minimal model program, stating that for every projective variety $X$ w…

Let $X$ be a projective manifold, meaning a smooth projective variety, whose tangent bundle $T_X$ is nef. Campana–Peternell conjecture. The variety $X$ should be homogeneous. This…