r-Defectivity in Geometry and Beyond

• r-Defectivity is a parameterized measure indicating when secant or join constructions fail to achieve their expected dimensions in geometric settings.
• It is extended to cover vector bundles on reducible varieties and analyzed through concise secant varieties, which aids in tensor rank stratification.
• The concept also spans homological, computational, and morphological frameworks, offering practical criteria for identifying deviations in expected behavior.

The term $r$-defectivity does not denote a single invariant across the literature represented here. In algebraic geometry it most commonly refers to failure of an $r$-th secant or join construction to attain its expected dimension; in adjacent work it is generalized to embedded vector bundles on reducible varieties, reinterpreted through concise secant varieties, and connected to discriminants of families of point configurations. In other domains, the same lexical pattern appears in looser or analogous senses: as a “tail versus origin” vanishing problem for linearity defect, as distance to matrix defectivity, as a graded notion of acceptable hardware defectiveness, and as a label for morphological gaps or defect distinguishability. The unifying motif is a parameterized measure of failure of a generic or expected property, but the precise object, ambient category, and role of the parameter vary substantially (Blomenhofer et al., 12 Sep 2025).

1. Classical secant-theoretic meaning

In the standard projective-geometric setting, let $X\subseteq \mathbb P^N$ be an irreducible non-degenerate projective variety. The $h$-secant variety is the Zariski closure of the union of all $(h-1)$-planes spanned by $h$ general points of $X$, and its expected dimension is

$$\operatorname{expdim}\big(\operatorname{Sec}_h(X)\big)=\min\{hn+h-1,\ N\}$$

when $\dim X=n$. The variety is called $h$-defective when the actual secant dimension is strictly smaller than the expected one; otherwise it is non-$r$0-defective (Araujo et al., 2016).

This notion governs tensor-rank and border-rank geometry. For Segre varieties, secant varieties parametrize tensors of border rank at most $r$1, and defectivity expresses a failure of the expected rank stratification (Gesmundo, 2012). For Segre–Veronese varieties, the same definition is used with the embedded variety

$r$2

where non-defectivity means that every secant variety has the expected dimension. A major recent result states that if $r$3 and all degrees satisfy $r$4, then the Segre–Veronese variety $r$5 is not defective (Abo et al., 2024).

Several asymptotic non-defectivity bounds are known. For Segre–Veronese varieties, one has the estimate that $r$6 is not $r$7-defective for

$r$8

and asymptotically for

$r$9

derived by combining osculating-space descriptions, osculating projections, degeneration of tangent spaces, and the Massarenti–Rischter criterion (Araujo et al., 2016). For Grassmannians, the osculating-projection method yields the bound

$X\subseteq \mathbb P^N$0

improving the earlier Abo–Ottaviani–Peterson estimate for $X\subseteq \mathbb P^N$1 (Massarenti et al., 2016).

The same vocabulary also appears in explicit classification problems. For the variety $X\subseteq \mathbb P^N$2 of reducible plane curves of type $X\subseteq \mathbb P^N$3, the secant line variety $X\subseteq \mathbb P^N$4 is defective exactly when

$X\subseteq \mathbb P^N$5

where $X\subseteq \mathbb P^N$6 and $X\subseteq \mathbb P^N$7; in that case the defect equals

$X\subseteq \mathbb P^N$8

(Catalisano et al., 2014). For Segre varieties, asymptotic lower bounds for the largest non-defective secant index are expressed in terms of the “expected rank scale”

$X\subseteq \mathbb P^N$9

with a positive ratio lower bound depending only on the number of factors (Gesmundo, 2012).

2. Reformulations via osculating and concise secant geometry

A major development in the theory is that defectivity can often be studied more effectively after replacing tangent-space configurations by higher osculating spaces. The general principle is that if a suitable osculating projection is generically finite, then the corresponding tangential projection is generically finite, and the Chiantini–Ciliberto criterion implies non-defectivity of the next secant variety (Massarenti et al., 2016). In Segre–Veronese geometry this is implemented by explicit combinatorial descriptions of osculating spaces at coordinate points and by birationality of projections from spans of selected osculating spaces (Araujo et al., 2016).

A more recent reformulation replaces the ordinary secant variety by the concise secant variety $h$0, whose points represent concise tensors of border rank $h$1. In the Segre setting there is a natural morphism

$h$2

and $h$3 always has the “correct” dimension

$h$4

Accordingly, defectivity is transferred from the source variety to the map $h$5: for $h$6,

$h$7

while identifiability corresponds to birationality of $h$8 (Jagiełła et al., 27 Apr 2026).

This viewpoint separates two issues that are entangled in the classical picture. The concise secant variety itself is never defective in the sense of dimension count; instead, the obstruction lies in failure of the concise-to-ordinary secant map to be generically finite. The paper further shows that the ordinary abstract secant variety sits as an open subset of $h$9, via a tame locus $(h-1)$0 with

$(h-1)$1

so that concise secant geometry provides a compactification in which the points remain actual concise tensors (Jagiełła et al., 27 Apr 2026).

The same paper records strong geometric properties for small $(h-1)$2: $(h-1)$3 and $(h-1)$4 are smooth, while $(h-1)$5 is normal, Gorenstein, Cohen–Macaulay, has canonical singularities, and singular locus of codimension $(h-1)$6 (Jagiełła et al., 27 Apr 2026). This suggests that concise secant varieties can serve as a technically cleaner ambient space for the study of defectivity and identifiability.

3. Generalized $(h-1)$7-defectivity for vector bundles and reducible varieties

A broad generalization is introduced for an embedded vector bundle

$(h-1)$8

where $(h-1)$9 is a quasi-affine cone or quasi-projective variety, $h$0 is a finite-dimensional ambient vector space, $h$1 is Zariski-locally trivial, and each fiber $h$2 is a linear subspace of $h$3. If $h$4 is reducible, the fiber dimension may vary by component, with ranks $h$5 on $h$6 (Blomenhofer et al., 12 Sep 2025).

For a sequence type $h$7, the expected span dimension of general fibers is

$h$8

The bundle is called $h$9-nondefective when the span of general fibers has the value $X$0, and $X$1-filling when the span equals all of $X$2. It is called $X$3-nondefective if it is $X$4-nondefective for every $X$5 with $X$6 (Blomenhofer et al., 12 Sep 2025).

This framework extends classical secant defectivity by taking $X$7, the embedded tangent bundle. It also separates two notions relevant for reducible varieties: an $X$8-join may be defective even when another irreducible component of the secant decomposition behaves as expected. The paper therefore distinguishes “$X$9-defective” from “geometrically $$\operatorname{expdim}\big(\operatorname{Sec}_h(X)\big)=\min\{hn+h-1,\ N\}$$0-defective” in the reducible setting (Blomenhofer et al., 12 Sep 2025).

The proof technology is based on apex spaces

$$\operatorname{expdim}\big(\operatorname{Sec}_h(X)\big)=\min\{hn+h-1,\ N\}$$1

over general sequences of type $$\operatorname{expdim}\big(\operatorname{Sec}_h(X)\big)=\min\{hn+h-1,\ N\}$$2, together with a stationarity lemma. From this, the authors derive explicit nondefectivity criteria. In the general $$\operatorname{expdim}\big(\operatorname{Sec}_h(X)\big)=\min\{hn+h-1,\ N\}$$3-component theorem, if $$\operatorname{expdim}\big(\operatorname{Sec}_h(X)\big)=\min\{hn+h-1,\ N\}$$4 and

$$\operatorname{expdim}\big(\operatorname{Sec}_h(X)\big)=\min\{hn+h-1,\ N\}$$5

$$\operatorname{expdim}\big(\operatorname{Sec}_h(X)\big)=\min\{hn+h-1,\ N\}$$6

through

$$\operatorname{expdim}\big(\operatorname{Sec}_h(X)\big)=\min\{hn+h-1,\ N\}$$7

then $$\operatorname{expdim}\big(\operatorname{Sec}_h(X)\big)=\min\{hn+h-1,\ N\}$$8 is $$\operatorname{expdim}\big(\operatorname{Sec}_h(X)\big)=\min\{hn+h-1,\ N\}$$9-nondefective (Blomenhofer et al., 12 Sep 2025). A complementary filling criterion states that if

$\dim X=n$0

then $\dim X=n$1 is $\dim X=n$2-filling (Blomenhofer et al., 12 Sep 2025).

This generalization has concrete consequences for secant varieties of reducible varieties, Fröberg-type Hilbert function problems, fat point schemes, mixture distributions, and partition-rank geometry. A plausible implication is that $\dim X=n$3-defectivity here functions as a unifying dimension-theoretic language across several problems that are not naturally phrased in terms of a single irreducible secant variety.

4. Discriminants, Cayley sums, and defectivity of point configurations

A different but closely related algebraic-geometric meaning of defectivity appears for families of lattice point configurations. For finite sets $\dim X=n$4, one considers Laurent polynomials $\dim X=n$5 supported on $\dim X=n$6, the discriminantal variety $\dim X=n$7, and the mixed discriminant $\dim X=n$8. The family is called defective when the discriminantal variety is not a hypersurface, equivalently when

$\dim X=n$9

(Borger et al., 2018).

The central combinatorial tool is the Cayley sum

$h$0

together with the reduction theorem that defectivity of the family implies defectivity of the Cayley sum (Borger et al., 2018). This allows the use of the Furukawa–Ito criterion: a spanning configuration $h$1 is defective if and only if there exist natural numbers $h$2 and a lattice projection

$h$3

such that $h$4, where the Cayley sum is of join type and each $h$5 (Borger et al., 2018).

For a spanning family of full-dimensional configurations $h$6 with $h$7, the main theorem gives a necessary condition for defectivity: $h$8 Thus defectivity forces the convex hull of the Minkowski sum to have no interior lattice points (Borger et al., 2018).

In the case of $h$9 full-dimensional configurations in $r$00, this leads to a sharp mixed-volume characterization: $r$01 and in this case the configurations are all translates of the vertex set of the same unimodular simplex (Borger et al., 2018). This is a different notion from secant $r$02-defectivity, but the formal mechanism is comparable: defectivity is converted into a combinatorial obstruction via a structural decomposition.

5. Homological and representation-theoretic analogues

Outside projective geometry, the expression is sometimes used in a more analogical way. For a commutative Noetherian local ring $r$03, the linearity defect of a finitely generated module $r$04 is

$r$05

where $r$06 is the linear part of a minimal free resolution. For the residue field $r$07, the central question is whether

$r$08

The paper explicitly frames this as an $r$09-defectivity-type statement: vanishing in high homological degrees may or may not force vanishing from the start (Şega, 2013).

The reformulation uses maps

$r$10

One has

$r$11

while

$r$12

(Şega, 2013). The problem thus becomes whether eventual vanishing implies total vanishing.

Positive answers are established in several cases: when $r$13; for complete intersections provided $r$14 is Cohen–Macaulay; and for Golod rings with Cohen–Macaulay associated graded ring (Şega, 2013). A sharper theorem later proves that if $r$15 is Artinian with $r$16 and $r$17, then $r$18 (Maleki, 2016).

A different analogue occurs in rational conformal field theory. There the paper “Defect Relative Entropy” introduces distinguishability measures for topological defects, defining

$r$19

and identifies defect relative sectors consisting of topological defects with zero defect relative entropy (Ghasemi, 29 Jan 2026). The paper explicitly notes that it does not introduce a notion called $r$20-defectivity, but it does define a sandwiched Rényi family $r$21, so a graded defect distinguishability spectrum is present (Ghasemi, 29 Jan 2026).

This suggests a broader pattern: in non-geometric settings, “$r$22-defectivity” often labels a hierarchy indexed by degree, stage, or Rényi parameter rather than a secant-dimension defect.

6. Computational, physical, and linguistic extensions

In numerical linear algebra, defectivity refers to matrices with a multiple eigenvalue whose algebraic multiplicity exceeds geometric multiplicity. For a matrix $r$23 with distinct eigenvalues, the distance to defectivity is

$r$24

with $r$25 or $r$26. The computational strategy of (Buttà et al., 2014) tracks the most ill-conditioned $r$27-pseudoeigenvalue by a differential equation on a low-rank manifold and combines this with a Newton-bisection iteration for the smallest $r$28 at which $r$29. The output is generally a local upper bound rather than a certified global minimum (Buttà et al., 2014).

In quantum error correction, the paper on boundaries of acceptable defectiveness (BADs) rejects an all-or-nothing notion of a defective qubit. BAD is defined as the maximum physical noise level for which the logical error rate remains at or below a target threshold, and depends on code distance, noise homogeneity, defect location, defect severity, and the target logical threshold (Palmer et al., 24 Oct 2025). For rotated surface codes under heterogeneous noise, the paper argues that defectiveness should be treated as a spectrum rather than a binary label, and that distributed heterogeneity can be more damaging than a single severe outlier (Palmer et al., 24 Oct 2025). This is not a secant-theoretic use of $r$30, but it preserves the idea that defectiveness is parameterized and threshold-dependent.

In computational morphology, morphological defectivity denotes inflectional gaps: a lexeme is defective when at least one expected inflected form is missing, unacceptable, or effectively unused. The study of Latin and Italian Wiktionary data customizes an mBERT-enhanced UDTube analyzer, annotates CC-100 corpora, and validates crowd-sourced lists of defective verbs against corpus evidence (Sakunkoo et al., 21 Jun 2025). The paper states that morphological defectivity is also called inflectional gaps or $r$31-defectivity in the broader literature, and operationalizes defectivity using absolute frequency thresholds and the log-odds statistic

$r$32

(Sakunkoo et al., 21 Jun 2025).

These examples show that the term can migrate far from its projective-geometric origin. A plausible implication is that “$r$33-defectivity” functions as a family resemblance term: the parameter $r$34 or an analogous threshold indexes how severely an object fails an expected generic property, but the underlying structure may be geometric, homological, spectral, physical, or linguistic.

7. Conceptual synthesis and common misconceptions

A common misconception is that defectivity always means singularity, degeneration, or low quality in an undifferentiated sense. In the surveyed literature, defectivity is instead a sharply context-dependent failure of an expected statement. For secant varieties, it is a dimension deficit relative to a parameter count (Araujo et al., 2016). For families of point configurations, it is triviality of the mixed discriminant (Borger et al., 2018). For local rings, it is persistence of nonlinear behavior in a minimal free resolution (Şega, 2013). For matrices, it is the presence of nontrivial Jordan blocks and the distance to that locus (Buttà et al., 2014). For topological defects in CFT, the issue is distinguishability of defect-induced density matrices rather than secant failure (Ghasemi, 29 Jan 2026).

Another misconception is that non-defectivity and identifiability are equivalent. In the concise secant framework they are explicitly separated: $r$35 whereas

$r$36

(Jagiełła et al., 27 Apr 2026). Generic finiteness is weaker than birationality.

A further misconception is that defectivity is necessarily rare. The evidence is mixed. Some papers prove extensive non-defectivity ranges, and one recent theorem shows that Segre–Veronese varieties are never secant defective when each degree is at least three (Abo et al., 2024). Yet explicit infinite defective families also occur, such as the unbalanced cases for secant line varieties of reducible plane curves with $r$37 and $r$38 (Catalisano et al., 2014).

Across these settings, the recurring structure is a comparison between an observed object and an expected one: actual secant dimension versus expected dimension, actual discriminantal codimension versus hypersurface behavior, eventual versus initial homological linearity, actual defect-induced information versus distinguishability, or observed error landscape versus an acceptable threshold. In that restricted but substantive sense, $r$39-defectivity is best understood not as a single definition, but as a family of parameterized defect theories whose most developed and technically rigid form remains the secant-geometric one.