Free rational curves on low degree hypersurfaces and the circle method

Tim Browning, Will Sawin

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


We use a function field version of the Hardy–Littlewood circle method to study the locus of free rational curves on an arbitrary smooth projective hypersurface of sufficiently low degree. On the one hand this allows us to bound the dimension of the singular locus of the moduli space of rational curves on such hypersurfaces and, on the other hand, it sheds light on Peyre’s reformulation of the Batyrev–Manin conjecture in terms of slopes with respect to the tangent bundle.

Original languageEnglish (US)
Pages (from-to)719-748
Number of pages30
JournalAlgebra and Number Theory
Issue number3
StatePublished - 2023
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory


  • circle method
  • free rational curve
  • function field
  • hypersurface


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