Singularities of improper affine maps and their Hessian equation

We study improper affine spheres with some admissible singularities, called improper affine maps, and associated to the unimodular Hessian equation. In particular, we characterize when a curve of R3R3 is the singular curve of some improper affine map with prescribed cuspidal edges and swallowtails. Also, we consider improper affine maps with isolated singularities and show some similarities and differences between the Hessian +1 equation and the Hessian −1 equation. As a consequence, we construct global examples with the desired singularities.