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Motivic classes of Fano schemes of lines on intersections of two quadrics

  • Lena Ji
  • Fumiaki Suzuki
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Abstract

Let X⊂ℙN be a smooth complete intersection of two quadrics. We find formulas in the Grothendieck ring of varieties for the class of X and for the class of its Fano scheme of lines, thereby proving the first two cases of a conjecture of Belmans et al. We also find a formula for the class of the relative Fano schemes of linear subspaces (of any dimension) in the fibers of quadric fibrations over curves, under a simple degeneration assumption. As consequences, we compute the rational Chow motive of the relative Fano scheme, and we provide evidence for a conjecture of Shah on a residual category in a semiorthogonal decomposition of the relative Fano scheme of lines.

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Cite

BibTeXRIS
BibTeX
@misc{ji2026motivic,
  title = {{Motivic classes of Fano schemes of lines on intersections of two quadrics}},
  author = {Lena Ji and Fumiaki Suzuki},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.36233},
  eprint = {2609.36233},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.36233},
}
RIS
TY  - UNPB
TI  - Motivic classes of Fano schemes of lines on intersections of two quadrics
AU  - Lena Ji
AU  - Fumiaki Suzuki
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.36233
UR  - https://arxiv.org/abs/2609.36233
AB  - Let X⊂ℙ^N be a smooth complete intersection of two quadrics. We find formulas in the Grothendieck ring of varieties for the class of X and for the class of its Fano scheme of lines, thereby proving the first two cases of a conjecture of Belmans et al. We also find a formula for the class of the relative Fano schemes of linear subspaces (of any dimension) in the fibers of quadric fibrations over curves, under a simple degeneration assumption. As consequences, we compute the rational Chow motive of the relative Fano scheme, and we provide evidence for a conjecture of Shah on a residual category in a semiorthogonal decomposition of the relative Fano scheme of lines.
ER  -