Classification of (0,2)-geometries embedded in AG (3,q)

Abstract

In De Clerck and Delanote (Des. Codes Cryptogr, 32:103-110, 2004) it is shown that if a (0, alpha) -geometry with alpha >= 3 is fully embedded in AG (n, q) then it is a linear representation. In De Feyter (J. Combin Theory Ser A, 109(t): 1-23, 2005; Discrete math, 292:45-54,2005) the (0, 2)-geometries fully embedded in AG(3, q) are classified apart from two open cases. In this paper, we solve these two open cases. This classification for AG (3, q) is used in De Feyter (Adv Geom, 5: 219-292, 2005) to classify the (0, 2)-geometries fully embedded in AG(n, q)

The embedding in AG(3,q) of (0,2)-geometries with

“…In fact in [6] we prove, using other methods than in this paper, that cases (2) and (3) do not occur. So the results in [7], in this paper and in [6] add up to the complete classification of (0, 2)-geometries embedded in AG(3, q), q = 2 h , h > 1, which have at least one plane of type IV (those without planes of type IV were already classified in Theorem 1.3). This classification for AG (3, q) is then used in [8] to obtain the complete classification of (0, 2)-geometries embedded in AG(n, q), q = 2 h , h > 1, which have at least one plane of type IV.…”

“…There are no examples of geometries in case (2) or (3). In fact in [6] we prove, using other methods than in this paper, that cases (2) and (3) do not occur. So the results in [7], in this paper and in [6] add up to the complete classification of (0, 2)-geometries embedded in AG(3, q), q = 2 h , h > 1, which have at least one plane of type IV (those without planes of type IV were already classified in Theorem 1.3).…”