Generalized Born-Infeld actions and projective cubic curves

We investigate inline image supersymmetric Born\u2013Infeld Lagrangians with a second non\u2013linearly realized supersymmetry. The resulting non\u2013linear structure is more complex than the square root present in the standard Born\u2013Infeld action, and nonetheless the quadratic constraints determining these models can be solved exactly in all cases containing three vector multiplets. The corresponding models are classified by cubic holomorphic prepotentials. Their symmetry structures are associated to projective cubic varieties