Mathematics Problem Archive

On a projective non-singular algebraic variety over $\mathbb{C}$, any Hodge class is a rational linear combination of classes of algebraic cycles....

A collection of conjectures about algebraic cycles on smooth projective varieties, including Lefschetz standard conjecture and Künneth standard conjec...

For a minimal model $X$ of non-negative Kodaira dimension, the canonical divisor $K_X$ is semi-ample....

Find effective uniform bounds for the heights of rational points on algebraic curves....

Rigorously justify Schubert's enumerative geometry....

On a projective algebraic variety, is every Hodge class a rational linear combination of classes of algebraic cycles?...

For an elliptic curve $E$ over the rationals, does the rank of its group of rational points equal the order of vanishing of its $L$-function at $s=1$?...

For varieties over finite fields, are the $\ell$-adic representations arising from étale cohomology related to algebraic cycles in the expected way?...

If the canonical bundle of a variety is nef, must it be semiample?...

Bass conjecture: for every finitely generated $\mathbb Z$-algebra $A$ and every $n\geq0$, is the G-theory group $K'_n(A)$ finitely generated? Equivale...

Fröberg conjecture on the Hilbert functions of a set of forms....

Fujita conjecture regarding the line bundle $K_{M} \otimes L^{\otimes m}$ constructed from a positive holomorphic line bundle $L$ on a compact complex...

General elephant problem: do general elephants have at most Du Val singularities?...

Nagata's conjecture on curves, specifically the minimal degree required for a plane algebraic curve to pass through a collection of very general point...

Nagata–Biran conjecture that if $X$ is a smooth algebraic surface and $L$ is an ample line bundle on $X$ of degree $d$, then for sufficiently large $r...

Nakai conjecture: if a complex algebraic variety has a ring of differential operators generated by its contained derivations, then it must be smooth....

Section conjecture on splittings of group homomorphisms from fundamental groups of complete smooth curves over finitely-generated fields $k$ to the Ga...

Zariski multiplicity conjecture on the topological equisingularity and equimultiplicity of varieties at singular points...

Are infinite sequences of flips possible in dimensions greater than 3?...