A Classification of the flag-transitive 2- (v, k, 2) designs

In this paper, we provide a complete classification of 2- (v, k, 2) design admitting a flag-transitive automorphism group of affine type with the only exception of the semilinear 1-dimensional group. Alongside this analysis we provide a construction of seven new families of such flag-transitive 2-designs, two of them infinite, and some of them involve remarkable objects such as t-spreads, translation planes, quadrics and Segre varieties. Our result together with those Alavi et al. 1, 2, Praeger et al. 15, Zhou and the first author 37, 38 provides a complete classification of 2- (v, k, 2) design admitting a flag-transitive automorphism group with the only exception of the semilinear 1-dimensional case.