Shadows of Varieties Embedded in Projective Spaces

When two varieties X, X′ embedded in a projective space have the same image, i.e., the same shadow, are they projected from the same points? We prove that two general points of projections are sufficient to identify X. For one point of projection, there are many very different shadows with very different degrees. We give the geometric properties of some of them. These shadows are birational to the variety in which they are a shadow. We compute the minimum degree of all such shadows. For most smooth varieties X⊂Pr, r≥3, it is the integer deg(X)−1.