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PreprintarXiv licencearXiv

Higher-order tangent lines to smooth hypersurfaces

  • Roya Beheshti
  • Shibashis Mukhopadhyay
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Abstract

We prove that for every smooth hypersurface X⊂ℙn of degree d≥ n ≥ 5 over an algebraically closed field of characteristic zero, the space of pointed lines (ℓ,p) having intersection multiplicity at least six with X at p has the expected dimension 2n−7. The proof uses deformations of ruled surfaces preserving prescribed contact with X. As an application, we show that a smooth hypersurface of degree d≥ 6 in ℙ5 contains at most 225d3−1370d2+1800d planes. This bound has optimal order of growth in d.

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Cite

BibTeXRIS
BibTeX
@misc{beheshti2026higherorder,
  title = {{Higher-order tangent lines to smooth hypersurfaces}},
  author = {Roya Beheshti and Shibashis Mukhopadhyay},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.28558},
  eprint = {2610.28558},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.28558},
}
RIS
TY  - UNPB
TI  - Higher-order tangent lines to smooth hypersurfaces
AU  - Roya Beheshti
AU  - Shibashis Mukhopadhyay
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.28558
UR  - https://arxiv.org/abs/2610.28558
AB  - We prove that for every smooth hypersurface X⊂ℙ^n of degree d≥ n ≥ 5 over an algebraically closed field of characteristic zero, the space of pointed lines (ℓ,p) having intersection multiplicity at least six with X at p has the expected dimension 2n−7. The proof uses deformations of ruled surfaces preserving prescribed contact with X. As an application, we show that a smooth hypersurface of degree d≥ 6 in ℙ^5 contains at most 225d^3−1370d^2+1800d planes. This bound has optimal order of growth in d.
ER  -