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PreprintCC BY 4.0arXiv

A Sarkisov Involution from a Rational Cubic Curve on a Quartic Double Solid

  • Joseph Cutrone
  • Nicholas Marshburn
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Abstract

Building on an earlier classification of smooth weak Fano threefolds of Picard number two with small anticanonical morphism, we prove that Case~28 of the table of E1-E1 links from that classification is geometrically realizable. We construct the link by blowing up a smooth rational cubic curve on a degree-two del Pezzo threefold V2, realized as a quartic double solid. A lattice calculation on an anticanonical K3 surface shows that the anticanonical morphism of the blow-up is small, and the covering involution of the anticanonical double cover realizes the flop. The resulting Sarkisov link is a birational involution with the numerical invariants of E1-E1 Case~28.

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Cite

BibTeXRIS
BibTeX
@misc{cutrone2026sarkisov,
  title = {{A Sarkisov Involution from a Rational Cubic Curve on a Quartic Double Solid}},
  author = {Joseph Cutrone and Nicholas Marshburn},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.26364},
  eprint = {2610.26364},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.26364},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A Sarkisov Involution from a Rational Cubic Curve on a Quartic Double Solid
AU  - Joseph Cutrone
AU  - Nicholas Marshburn
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.26364
UR  - https://arxiv.org/abs/2610.26364
AB  - Building on an earlier classification of smooth weak Fano threefolds of Picard number two with small anticanonical morphism, we prove that Case~28 of the table of E1-E1 links from that classification is geometrically realizable. We construct the link by blowing up a smooth rational cubic curve on a degree-two del Pezzo threefold V_2, realized as a quartic double solid. A lattice calculation on an anticanonical K3 surface shows that the anticanonical morphism of the blow-up is small, and the covering involution of the anticanonical double cover realizes the flop. The resulting Sarkisov link is a birational involution with the numerical invariants of E1-E1 Case~28.
ER  -