Obstructions to deforming space curves lying on a complete intersection of quadrics

Abstract

Let S=S_2,…,2⊂ℙⁿ⁺² be a smooth complete intersection of n quadrics containing a line E, where n≥ 3, and set D=(n−1)h+E, where h=[𝒪_S(1)]. We study the deformations of smooth curves C∈ |mD| and their obstructions. The linear system |D| defines a birational morphism f:S→S̅ that contracts E to a point and realizes an unprojection in the sense of Papadakis--Reid. When n=3, the surface S=S_2,2,2⊂ℙ⁵ is a K3 surface and E is a (−2)-curve. We prove that the complementary open surface S∖ E admits first-order infinitesimal deformations in ℙ⁵∖ E that are obstructed at the second order. Using these obstructions, we show that, for every sufficiently large m with char knmid m, there exists a generically non-reduced irreducible component W_m of the Hilbert scheme Hilb^scℙ⁵ of smooth connected curves in ℙ⁵. Its general member C lies on a complete intersection S_2,2,2 of three quadrics, satisfies C∼ mD, and has h⁰(N_C/ℙ⁵)−dim_[C]Hilb^scℙ⁵=3. In contrast, when n>3, we prove that, for every sufficiently large m with char knmid m, the corresponding family of curves in |mD| gives rise to a generically smooth irreducible component of Hilb^scℙⁿ⁺².

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BibTeX RIS

BibTeX

@misc{nasu2026obstructions,
  title = {{Obstructions to deforming space curves lying on a complete intersection of quadrics}},
  author = {Hirokazu Nasu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.30962},
  eprint = {2610.30962},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.30962},
}

RIS

TY  - UNPB
TI  - Obstructions to deforming space curves lying on a complete intersection of quadrics
AU  - Hirokazu Nasu
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.30962
UR  - https://arxiv.org/abs/2610.30962
AB  - Let S=S_2,…,2⊂ℙ^n+2 be a smooth complete intersection of n quadrics containing a line E, where n≥ 3, and set D=(n−1)h+E, where h=[𝒪_S(1)]. We study the deformations of smooth curves C∈ |mD| and their obstructions. The linear system |D| defines a birational morphism f:S→S̅ that contracts E to a point and realizes an unprojection in the sense of Papadakis--Reid. When n=3, the surface S=S_2,2,2⊂ℙ^5 is a K3 surface and E is a (−2)-curve. We prove that the complementary open surface S∖ E admits first-order infinitesimal deformations in ℙ^5∖ E that are obstructed at the second order. Using these obstructions, we show that, for every sufficiently large m with char knmid m, there exists a generically non-reduced irreducible component W_m of the Hilbert scheme Hilb^scℙ^5 of smooth connected curves in ℙ^5. Its general member C lies on a complete intersection S_2,2,2 of three quadrics, satisfies C∼ mD, and has h^0(N_C/ℙ^5)−dim_[C]Hilb^scℙ^5=3. In contrast, when n>3, we prove that, for every sufficiently large m with char knmid m, the corresponding family of curves in |mD| gives rise to a generically smooth irreducible component of Hilb^scℙ^n+2.
ER  -

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