Birational type

The birational type over $\mathbb Q$ of the characteristic-zero realization space, with one entry per $\mathbb Q$-irreducible component, joined by | . Examples: rational, empty (not realizable in characteristic 0), K3 surface, genus 1 curve (11a3) (with Cremona label), point over Q(a)/(a^2 - a + 1), K3 surface x P^1, Calabi-Yau threefold (...).

Free affine factors are split off and written x P^k. The type refers to the space after the projective quotient, so where the literature writes "genus 1 curve × PGL₃" this reads genus 1 curve. Computed for rank 3 on at most 11 elements and rank 4 on at most 9. See also birational_type_components and birational_type_method.

Full details: docs/birational_type.md.