Families and presentations
- Family and presentation
- A family groups fourfolds related by smooth deformation; a presentation constructs a general zero locus in a specified ambient variety. Known matches are grouped together; distinct catalogue entries can still describe the same family.
- Family label
- The entries are Picard rank, anticanonical volume, and anticanonical section count, followed by a distinguishing letter. Thus 5-120-31-a has $\rho=5$, $\mathrm{c}_1^4=120$, and $\mathrm{h}^0(X,\omega_X^\vee)=31$.
- Preferred model and presentation list
- The preferred model is the catalogue's chosen representative presentation, displayed by default when opening a family page. Selecting another presentation changes its defining data and polyvector/Hochschild data; the family invariants stay fixed.
- Connected components
- When a zero locus has several isomorphic connected components, the displayed invariants describe one component.
- FanoK3 reference
- A recorded case label links to the corresponding example in Bernardara–Fatighenti–Manivel–Tanturri, Fano fourfolds of K3 type.