Math 2.0

Algebraic geometry Projective and enumerative geometry Problem

Hilbert's fifteenth problem

Builds on Enumerative geometrySchubert calculusIntersection theory

Problem

Give a rigorous foundation for the enumerative calculus of Hermann Schubert. Determine exactly the range of validity of his methods, and check the numbers of enumerative geometry that he and his contemporaries found.

Remarks

Hilbert posed this problem in his list of 23 problems presented in 1900. Schubert had computed many enumerative numbers with symbolic rules, but these rules lacked proofs, and some classical answers were later found to need correction. The modern foundations are intersection theory and the cohomology of Grassmannians and flag varieties, in which Schubert calculus becomes a set of theorems. Much of the problem is regarded as settled in this way, but checking all of Schubert's numbers is a separate task, and the problem is often listed as partially solved.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph
Enumerative geometry, concept, no evidence levelEnumerative geometryconceptIntersection theory, concept, no evidence levelIntersection theoryconceptSchubert calculus, concept, no evidence levelSchubert calculusconceptHilbert's fifteenth problem, problem, no evidence levelHilbert's fifteenth p…problem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Enumerative geometry concept
  2. Intersection theory concept
  3. Schubert calculus concept
  4. Hilbert's fifteenth problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Hilbert%27s_fifteenth_problem
About this page
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AboutSchubert calculus, Enumerative geometry
UsesEnumerative geometry, Schubert calculus, Intersection theory
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Changed2 November 2026

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