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On degenerate geometric rank

This paper investigates tensors of degenerate geometric rank by establishing a criterion for their non-liftability, characterizing cases where degeneracy arises from the rank-1 locus, and presenting new examples derived from nonlinear loci.

Original authors: Matěj Doležálek, Paweł Pielasa, Derek Wu

Published 2026-07-22

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Matěj Doležálek, Paweł Pielasa, Derek Wu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery hidden inside a giant, multi-layered box of numbers. In the world of mathematics, these boxes are called tensors. You might know them as the 3D cousins of the flat, 2D grids we call matrices. While a matrix is like a spreadsheet, a tensor is like a stack of spreadsheets glued together. These objects are the secret sauce behind everything from how computers learn to recognize your face to how physicists model the universe.

To understand what's inside these boxes, mathematicians use a tool called rank. Think of rank as a measure of "messiness" or complexity. A simple, perfectly organized box has a low rank; a chaotic, jumbled one has a high rank. Recently, a new way of measuring this complexity was invented called geometric rank. Instead of just counting numbers, geometric rank looks at the shapes formed by the numbers. It asks: "If I look for all the parts of this box that are 'simple' (low rank), how big is the space they take up?" Usually, these simple parts are tiny, scattered specks. But sometimes, they form a surprisingly huge, unexpected cloud. When this happens, we say the tensor has degenerate geometric rank. It's like finding a whole ocean of water in a place where you expected only a few drops.

This paper is about hunting for these "oceans" and figuring out where they come from. The authors, Matej Doležalek, Paweł Pielasa, and Derek Wu, are trying to answer a big question: When we find a tensor with this weird, degenerate rank, is it just a piece of a bigger, simpler puzzle that we haven't seen yet? Or is it a unique, standalone monster that can't be broken down or "lifted" into something larger? They prove a set of rules to tell the difference. They also investigate a specific type of mystery: what happens when the "simple" parts are so simple they are just single lines (rank 1)? They show that if this causes the big surprise, the tensor is usually hiding a very specific, boring secret (like a flat sheet of paper or a symmetric pattern). Finally, they go on a treasure hunt to find examples of these degenerate ranks that aren't just flat sheets or simple patterns. They find several new, interesting examples where the "big puddle" includes curved, nonlinear shapes, but they also discover a surprising rule: even in these weird cases, the nonlinear shape is never the sole cause. There is almost always a boring, linear part hiding in the background that achieves the same degenerate rank, meaning the "weirdness" alone isn't enough to cause the whole problem.

The Story of the "Lift" and the "Neutral" Directions

The authors start by tackling the idea of liftability. Imagine you have a small, flat rug (a space of matrices) that has a strange, large puddle of water on it. You wonder, "Is this rug just a tiny corner of a much bigger rug that also has a big puddle?" If the answer is yes, the small rug is "lifted" to the big one. If the answer is no, the rug is "unliftable"—it's the whole story.

To figure this out, the authors invented a clever test involving rank-neutral directions. Picture the edge of the puddle on your rug. At every point on the edge, there is a specific direction you can push the rug without making the puddle get any bigger or smaller. These are the "neutral" directions. The authors proved a powerful rule: if you can find any direction to push the rug that isn't neutral, then the rug can be lifted to a bigger one. But, if every single possible direction you try is neutral, then the rug is stuck; it cannot be lifted. It is the final, unchangeable version of itself. They even built a computer program to help check this for specific cases, acting like a digital magnifying glass for these mathematical puddles.

The Case of the Rank-1 Locus

Next, the team focused on a specific type of puddle: the rank-1 locus. In the world of tensors, "rank 1" is the simplest possible state, like a single straight line. Usually, these lines are so rare they don't form a big shape. But sometimes, they bunch together to form a huge, degenerate rank.

The authors asked: "If the rank-1 lines are the only reason the rank is degenerate, what does the tensor look like?" They proved that there are really only two possibilities. Either the tensor is hiding a giant, flat sheet of rank-1 lines (a linear subspace), or it looks exactly like the space of 2x2 symmetric matrices (a specific, well-known shape in math). They showed that if it's the second case, the "rank-2" lines (which are slightly more complex) also cause the degeneracy. This means you can't have a situation where only the rank-1 lines cause the surprise in a truly interesting, complex way. If the rank-1 lines are the culprit, they are either boringly flat or part of a very specific, known pattern.

The Treasure Hunt: Nonlinear Surprises

Finally, the authors went on a hunt for the "holy grail": tensors where the degenerate rank comes from a nonlinear shape. A linear shape is like a straight line or a flat plane. A nonlinear shape is curved, like a sphere or a twisted ribbon. The hope was to find a tensor where the "big puddle" was a curved, complex shape that couldn't be explained by simple flat sheets.

They found several fascinating examples:

Matrix Multiplication: They looked at the tensor that represents multiplying two $n \times n$ matrices. They confirmed that for any size $n$ , this tensor is "unliftable" for certain ranks. It's a stubborn, standalone object.
Octonions: They examined the structure of the octonions, a weird, 8-dimensional number system that breaks the usual rules of multiplication. They found that the "rank-4" part of this system forms a curved surface (a hypersurface) that causes the degeneracy. They checked and confirmed this shape cannot be lifted to a bigger space.
Minimal Border Rank: They scanned a list of known "minimal" tensors (the most efficient ones) and found eight specific cases where the rank loci were curved and complex.

However, their hunt ended with a twist. While they found many examples of these curved, nonlinear shapes, they discovered that none of them were the sole cause of the degeneracy. In every single case they found, there was also a boring, flat, linear shape hiding in the same tensor that achieved the same degenerate rank. So, while nonlinear shapes can cause degenerate geometric rank, they never do it alone; they always have a linear partner in crime.

The paper concludes that while the world of degenerate geometric rank is full of interesting, curved surprises, the "unliftability" and the "degeneracy" are often still driven by the simpler, linear structures we already understand. The nonlinear shapes are the flashy decorations, but the linear ones are the foundation.

Technical Summary: On Degenerate Geometric Rank

Problem Statement
The paper investigates tensors of degenerate geometric rank (GR). For a tensor $T \in \mathbb{C}^m \otimes \mathbb{C}^m \otimes \mathbb{C}^m$ , the geometric rank $GR(T)$ is defined as the minimum of $\text{codim}(Y^r_T) + r$ , where $Y^r_T$ is the locus of points in the projective space of the tensor's flattening where the matrix rank is at most $r$ . A tensor has degenerate geometric rank if $GR(T) < m$.

While tensors with degenerate geometric rank are known to arise from linear spaces of matrices of bounded rank (a classical subject dating to Dieudonné and Flanders), the authors seek to understand the broader class of such tensors. Specifically, they address whether degeneracy can stem from nonlinear loci (e.g., hypersurfaces or higher-degree varieties) rather than just linear subspaces, and they investigate the structural properties of these tensors, particularly regarding their "liftability" (whether a tensor can be embedded into a larger tensor with the same rank behavior).

Methodology
The authors employ algebraic geometry and tensor rank theory, utilizing the following methodological tools:

Liftability Criteria: They generalize a criterion by Draisma [Dra06] for spaces of matrices of bounded rank. They define rank-neutral directions ( $RND_r(E)$ ) for a matrix space $E$ . If the space of neutral directions coincides with $E$ itself, the space is unliftable (cannot be extended while preserving the codimension of the rank locus).
Geometric Analysis of Rank Loci: They analyze the dimension and structure of the rank loci $Y^r_T$ . This involves studying the intersection of the tensor's image with secant varieties of the Segre variety.
Inductive Arguments and Projections: To characterize tensors where the rank 1 locus causes degeneracy, the authors use induction on the dimension of the intersection with the Segre variety, employing general hyperplane restrictions and Hopf's theorem on linear maps.
Computational Verification: The authors utilize SageMath to implement the unliftability criterion and verify specific examples, including matrix multiplication tensors, structure tensors of algebras, and tensors of minimal border rank.

Key Contributions and Results

Unliftability Criterion (Section 3): The paper establishes a sufficient condition (Theorem 4) for a tensor (viewed as a matrix space $E$ ) to be unliftable. If the set of rank-neutral directions $RND_r(E)$ equals $E$ , then $E$ cannot be lifted to a larger space $E'$ without increasing the codimension of the rank- $r$ locus. This generalizes Draisma's results from linear spaces of bounded rank to the broader context of geometric rank.
Characterization of Rank 1 Degeneracy (Section 4): The authors characterize tensors where the degenerate geometric rank is achieved specifically by the rank 1 locus. Theorem 10 states that if the rank 1 locus achieves $GR(T)$ and $T$ has degenerate geometric rank, then either:
- (a) The image of the flattening contains a linear subspace of the Segre variety; or
- (b) The image is equivalent (up to $GL$ action) to the space of $2 \times 2$ symmetric matrices.
  Crucially, in case (b), the rank 2 locus also achieves the geometric rank. This implies that degenerate geometric rank cannot be caused solely by a rank 1 locus in a "non-trivial" way (i.e., without the presence of a linear subspace or a symmetric matrix structure that also triggers the degeneracy at rank 2).
Examples of Nonlinear Degeneracy (Section 5): The paper provides new examples of tensors with degenerate geometric rank where the degeneracy arises from nonlinear loci, distinct from simple linear spaces of bounded rank:
- Matrix Multiplication ( $M_{\langle n \rangle}$ ): The authors confirm that the geometric rank of matrix multiplication tensors is degenerate and caused by high-degree loci (specifically, the rank $rn$ locus corresponds to rank $r$ matrices in the underlying $n \times n$ space). They prove these spaces are $rn$-unliftable.
- Octonion Structure Tensor ( $T_O$ ): The structure tensor of complexified octonions has a degenerate geometric rank ($GR=5$) caused by a rank 4 locus defined by the octonion norm (a hypersurface). This locus is verified to be unliftable.
- Minimal Border Rank Tensors: The authors analyze a list of tensors of minimal border rank for $m \le 5$ . They identify several cases (e.g., $T_{1,10}, T_{1,19}, T_{O58}$ ) where nonlinear loci (defined by ideals involving products of variables) contribute to the geometric rank.
- SL3-Invariant Tensor: An example constructed from the representation theory of $SL_3$ is presented, where the rank 4 locus is the second secant variety of a Veronese embedding.

Significance and Claims
The paper claims to expand the understanding of tensors with degenerate geometric rank beyond the classical setting of linear spaces of bounded rank.

It provides a rigorous tool (the $RND$ criterion) to determine when such tensors are "maximal" (unliftable), which is relevant for classifying tensors of degenerate geometric rank.
It clarifies the limitations of rank 1 loci in causing degeneracy, showing that "interesting" degeneracy solely from rank 1 is impossible without accompanying linear structures or symmetric matrix analogues.
It offers concrete, verified examples of nonlinear degeneracy, including the matrix multiplication tensor and the octonion structure tensor, demonstrating that such phenomena are not merely theoretical but occur in significant algebraic structures.

The authors note that while they found examples where nonlinear loci contribute to the geometric rank, they did not find any example where the geometric rank is achieved solely by a nonlinear locus; in all observed cases, a linear locus (or a locus equivalent to one) achieves the rank jointly with the nonlinear one. This observation aligns with their theoretical results in Section 4.

The Story of the "Lift" and the "Neutral" Directions

The Case of the Rank-1 Locus

The Treasure Hunt: Nonlinear Surprises

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