Partial λ-geometries of small nexus

In a partial λ-geometry, each two points are joined by 0 or λ blocks; each two blocks have 0 or λ points in common; the block size k is constant; and for each nonflag (p, G), there are precisely e blocks X with p in X such that X∩G is not empty. Generalized quadrangles are partial 1–geometries with nexus e = 1. If λ = 2, then e ⩾ 3; and the first author has determined that partial 2–geometries with nexus 3 exist precisely for the values k = 3, 4, 8, 24. We prove (1) that if λ > 2 and k > e + 1, then e > 2λ (2) that λ = 3, e = 7 implies k is one of 7, 15, 21, 24 or 36. We call a partial λ-geometry extremal if |G∩H∩K| > 1 implies |G∩H∩K| ⩾ λ. There are no extremal partial λ-geometries with e = λ²- λ. Such a geometry with e = λ²- λ + 1 and k > e is called a λ-quadrangle. We determine all λ-quadrangles with λ > 2. They are constructed from quadratic forms of Witt index 4 on finite 8–dimensional vector spaces.