Projective Variety Recovery from Unknown Linear Projections
Yirmeyahy Kaminski
TL;DR
This paper addresses recovering a smooth irreducible variety from two unknown projections by extending Kruppa-type equations to higher dimensions via a hierarchy of fundamental matrices. It introduces ordered projections and a generalized algebraic framework (Double Grassmann-Cayley algebra) to encode projection geometry and dual-variety data, establishing conditions under which the projection geometry is uniquely determined up to finite ambiguity. The authors derive dimension bounds tied to the class (degree of the dual variety) and show that for generic centers, the generalized Kruppa equations yield a discrete solution set, enabling reconstruction of the projection operators and thus the ambient variety. A key result is that the intersection of the cones defined by the projected varieties through the centers of projection has exactly two components in general: the original variety of degree $d$ and an extraneous component of degree $d(d-1)$; this provides a principled way to identify the correct reconstruction in practice.
Abstract
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.