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We consider the standard contact structure on the supercircle, S^{1|1}, andthe supergroups E(1|1), Aff(1|1) and SpO(2|1) of contactomorphisms, definingthe Euclidean, affine and projective geometry respectively. Using the newnotion of (p|q)-transitivity, we construct in synthetic fashion even and oddinvariants characterizing each geometry, and obtain an even and an odd supercross-ratios. Starting from the even invariants, we derive, using a superizedCartan formula, one-cocycles of the group of contactomorphisms, K(1), withvalues in tensor densities F_\lambda(S^{1|1}). The even cross-ratio yields aK(1) one-cocycle with values in quadratic differentials, Q(S^{1|1}), whoseprojection on F_{3/2}(S^{1|1}) corresponds to the super Schwarzian derivativearising in superconformal field theory. This leads to the classification of thecohomology spaces H^1(K(1),F_\lambda(S^{1|1})). The construction is extended tothe case of S^{1|N}. All previous invariants admit a prolongation for N>1, aswell as the associated Euclidean and affine cocycles. The super Schwarzianderivative is obtained from the even cross-ratio, for N=2, as a projection toF_1(S^{1|2}) of a K(2) one-cocycle with values in Q(S^{1|2}). The obstructionto obtain, for N\geq 3, a projective cocycle is pointed out.

On the projective geometry of the supercircle: a unified construction of the super cross-ratio and Schwarzian derivative