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Singular loci in varieties of tensors

Christopher Chiu, Alessandro Danelon, Jan Draisma

TL;DR

This work develops a unified, functorial lens on varieties of tensors by introducing Vec-varieties, i.e., varieties parameterised functorially by a vector space $V$. The authors prove a global description of singularities: there exists a unique closed Vec-subvariety $X^{\text{sing}}$ with $X^{\text{sing}}(V)=\operatorname{Sing}(X(V))$ for large $\dim V$, and establish local equations away from $X^{\text{sing}}$. They further connect to infinite-dimensional, $\mathrm{GL}$-equivariant geometry via the embedding and shift theorems, and define Vec-varieties of linear type to build a semi-functorial weak resolution: a system of smooth spaces $\Omega(V)$ with a proper, birational map to $X(V)$ for large $V$. This framework offers a principled path toward uniform, functorial control of singularities in tensor-geometry problems, with concrete constructions using torsors and incidence varieties. The results synthesize a broad class of tensor-geometry phenomena and illuminate how uniform singularity descriptions and weak resolutions can be achieved in a GL-equivariant, finite-type setting.

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.

Singular loci in varieties of tensors

TL;DR

This work develops a unified, functorial lens on varieties of tensors by introducing Vec-varieties, i.e., varieties parameterised functorially by a vector space . The authors prove a global description of singularities: there exists a unique closed Vec-subvariety with for large , and establish local equations away from . They further connect to infinite-dimensional, -equivariant geometry via the embedding and shift theorems, and define Vec-varieties of linear type to build a semi-functorial weak resolution: a system of smooth spaces with a proper, birational map to for large . This framework offers a principled path toward uniform, functorial control of singularities in tensor-geometry problems, with concrete constructions using torsors and incidence varieties. The results synthesize a broad class of tensor-geometry phenomena and illuminate how uniform singularity descriptions and weak resolutions can be achieved in a GL-equivariant, finite-type setting.

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.
Paper Structure (28 sections, 27 theorems, 69 equations)

This paper contains 28 sections, 27 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Local equations for $X$ away from $X^\text{sing}$
  4. Semi-functorial resolution of singularities
  5. Relation to the existing literature
  6. Notation
  7. Background
  8. Polynomial functors and $\mathbf{Vec}$-varieties
  9. A well-founded order
  10. Connection to Schur functors
  11. Shifting
  12. $\mathbf{Vec}$-varieties
  13. The embedding Theorem and its consequences
  14. Topological Noetherianity of polynomial functors.
  15. The shift theorem
  16. ...and 13 more sections

Key Result

Theorem 1.2

Let $X$ be a $\mathbf{Vec}$-variety. Then there exists a unique closed $\mathbf{Vec}$-subvariety $Y$ of $X$ such that $Y(V)=\operatorname{Sing}(X(V))$ for all $V \in \mathbf{Vec}$ with $\dim(V) \gg 0$. This $Y$ satisfies $Y(V) \supset \operatorname{Sing}(X(V))$ for all $V \in \mathbf{Vec}$.

Theorems & Definitions (93)

  • Example 1.1
  • Theorem 1.2: Main Theorem
  • Theorem 1.3
  • Example 1.4
  • Example 1.5
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Remark 2.4
  • Remark 2.5
  • ...and 83 more