Cluster type smooth Fano threefolds

Abstract

We prove that most rational smooth Fano threefolds are compactifications of 𝔾_m³. More precisely, we study which smooth Fano threefolds are cluster type, i.e., for which smooth Fano threefolds X one can find a boundary B∈ |−K_X| and an embedding 𝔾_m³↪ X∖ B. We prove that many important classes of smooth Fano threefolds are cluster type. For instance, a smooth Fano threefold which is the blow-up of either ℙ³ or ℚ³ and is general in its deformation space is of cluster type. As a consequence, we show that among the $105$ families parametrizing all smooth Fano threefolds, the general element of precisely $87$ families is of cluster type. We prove that a general prime smooth Fano threefold of genus $7$ is not of cluster type. This is the only family whose general element is rational but not of cluster type. Finally, we prove that cluster type projective klt varieties are cylindrical and we give a criterion for such a cylinder to be anticanonically polar. Using this, we reduce the cylindricity problem for smooth Fano threefolds to the families 𝒩^o 1.3, 𝒩^o 1.5, and 𝒩^o 1.6.

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BibTeX RIS

BibTeX

@misc{silva2026cluster,
  title = {{Cluster type smooth Fano threefolds}},
  author = {Eduardo Alves da Silva and Fernando Figueroa and José Ignacio Yáñez and Joaquín Moraga},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.36656},
  eprint = {2610.36656},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.36656},
}

RIS

TY  - UNPB
TI  - Cluster type smooth Fano threefolds
AU  - Eduardo Alves da Silva
AU  - Fernando Figueroa
AU  - José Ignacio Yáñez
AU  - Joaquín Moraga
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.36656
UR  - https://arxiv.org/abs/2610.36656
AB  - We prove that most rational smooth Fano threefolds are compactifications of 𝔾_m^3. More precisely, we study which smooth Fano threefolds are cluster type, i.e., for which smooth Fano threefolds X one can find a boundary B∈ |−K_X| and an embedding 𝔾_m^3↪ X∖ B. We prove that many important classes of smooth Fano threefolds are cluster type. For instance, a smooth Fano threefold which is the blow-up of either ℙ^3 or ℚ^3 and is general in its deformation space is of cluster type. As a consequence, we show that among the $105$ families parametrizing all smooth Fano threefolds, the general element of precisely $87$ families is of cluster type. We prove that a general prime smooth Fano threefold of genus $7$ is not of cluster type. This is the only family whose general element is rational but not of cluster type. Finally, we prove that cluster type projective klt varieties are cylindrical and we give a criterion for such a cylinder to be anticanonically polar. Using this, we reduce the cylindricity problem for smooth Fano threefolds to the families 𝒩^o 1.3, 𝒩^o 1.5, and 𝒩^o 1.6.
ER  -

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