Projective Variety Recovery from Unknown Linear Projections

We study how a smooth irreducible algebraic variety $X$ of dimension $n$ embedded in $\mathbb{C} \mathbb{P}^{m}$ (with $m \geq n+2$), which degree is $d$, can be recovered using two projections from unknown points onto unknown hyperplanes. The centers and the hyperplanes of projection are unknown: the only input is the defining equations of each projected varieties. We show how both the projection operators and the variety in $\mathbb{C} \mathbb{P}^{m}$ can be recovered modulo some action of the group of projective transformations of $\mathbb{C} \mathbb{P}^{m}$. This configuration generalizes results obtained in the context of curves embedded in $\mathbb{C} \mathbb{P}^3$ and results concerning surfaces embedded in $\mathbb{C} \mathbb{P}^4$. We show how in a generic situation, a characteristic matrix of the pair of projections can be recovered. In the process we address dimensional issues and as a result establish a necessary condition, as well as a sufficient condition to compute this characteristic matrix up to a finite-fold ambiguity. These conditions are expressed as minimal values of the degree of the dual variety. Then we use this matrix to recover the class of the couple of projections and as a consequence to recover the variety. For a generic situation, two projections define a variety with two irreducible components. One component has degree $d(d-1)$ and the other has degree $d$, being the original variety.