Singular loci in varieties of tensors

TL;DR

This paper studies the singular loci of tensor varieties viewed as Vec-varieties, proving they are proper closed subvarieties and can be described by finitely many equations, with initial steps toward resolving their singularities.

Contribution

It establishes that singular loci of Vec-varieties are proper closed subvarieties and can be characterized by finitely many polynomial equations, advancing understanding of their geometric structure.

Findings

Singular locus of Vec-varieties is a proper closed subvariety.

Singular loci can be described by finitely many polynomial equations.

Initial results on functorial resolution of singularities for Vec-varieties.

Abstract

A Vec-variety is a suitable functor from finite-dimensional vector spaces to finite-dimensional varieties. Most varieties in the geometry of tensors, e.g. the variety of d-way tensors of slice rank at most r, are of this form. We prove that the singular locus of a Vec-variety is a proper closed Vec-subvariety, analogously to the situation for ordinary finite-dimensional varieties. Via earlier work of the third author, this implies that these singular loci admit a description by finitely many polynomial equations. A natural follow-up question to our main result is whether a Vec-variety also admits a suitably functorial resolution of singularities. We establish some preliminary results in this direction in the regime where the dimension of evaluations of a Vec-variety grows linearly with that of the input vector space.