Chow-Lam Recovery

TL;DR

This paper investigates the conditions for recovering subvarieties of Grassmannians from their Chow-Lam forms, extending classical results and identifying cases where recovery is impossible.

Contribution

It provides necessary conditions for recovery and presents examples where recovery fails, advancing understanding beyond the classical projective space case.

Findings

Recovery always possible for projective space (Chow 1937)

Necessary conditions for recovery in general Grassmannians

Examples showing recovery is not always possible

Abstract

We study the conditions under which a subvariety of the Grassmannian may be recovered from certain of its linear projections. In the special case that our Grassmannian is projective space, this is equivalent to asking when a variety can be recovered from its Chow form; the answer is "always" by work of Chow in 1937. In the general Grassmannian setting, the analogous question is when a variety can be recovered from its Chow-Lam form. We give both necessary conditions for recovery and families of examples where, in contrast with the projective case, recovery is not possible.