Algebraic geometry
Polynomial equations define spaces whose shapes, singularities and algebraic cycles pose different open problems.
8 problemsAlgebraic geometry
Shapes defined by polynomial equations, their singularities and algebraic cycles.
Collections
Browse collectionsA few problems to explore
8 totalHodge conjecture
Some topological features of a complex algebraic shape look as though they should come from smaller algebraic shapes inside it. Must they?
Zariski cancellation for affine three-space
If adding one affine coordinate makes a complex variety look like four-dimensional affine space, was it already affine three-space?
Hartshorne’s complete intersection conjecture
Must a smooth complex projective variety of sufficiently small codimension be defined by exactly that many polynomial equations?
Nagata conjecture
How low can a curve’s degree be when it must pass repeatedly through many very general points?
Zariski multiplicity conjecture
Does a complex hypersurface singularity’s local topological shape determine its multiplicity?
Abundance conjecture
Does numerical nonnegativity of a minimal model’s canonical divisor guarantee a globally defined geometric map?