r-Nondefectivity in Secant Varieties

• r-Nondefectivity is the property ensuring that the r-th secant variety of a projective variety reaches its expected dimension of min{r(n+1)-1, N}.
• It is characterized by the sum of tangent spaces at general points and computed using tools like Terracini’s lemma and osculating projections.
• Recent advances extend nondefectivity criteria to Chow varieties, Grassmannians, and invariant or reducible cones, impacting generic rank and identifiability.

$r$-nondefectivity is the condition that the $r$th secant variety of a projective variety attains its expected dimension. For an irreducible projective variety $X\subset \mathbb P^N$ of dimension $n$, the $r$th secant variety is

$$\sigma_r(X)=\overline{\bigcup_{p_1,\dots,p_r\in X}\langle p_1,\dots,p_r\rangle}\subset \mathbb P^N,$$

and the naive parameter count gives

$\dim \sigma_r(X)\le \min\{r(n+1)-1,N\}.$

Equality defines nondefectivity; strict inequality defines defectivity (Torrance et al., 2020). In recent work, the notion has been developed along several axes: exact results for Chow varieties, asymptotic bounds for Grassmannians via osculating projections, representation-theoretic criteria for invariant secant varieties, and generalizations from irreducible varieties to reducible cones and $V$-embedded vector bundles (Massarenti et al., 2016, Blomenhofer et al., 2023, Blomenhofer et al., 12 Sep 2025).

1. Classical definition and equivalent formulations

For a non-degenerate, irreducible projective variety $X\subset \PP^N$ of dimension $n$, the expected dimension of the $r$0-secant variety is

$r$1

The variety is called $r$2-defective if

$r$3

and the $r$4-defect is

$r$5

When $r$6, one says that $r$7 is non-$r$8-defective (Massarenti et al., 2016).

The same notion is frequently expressed in affine-cone language. If $r$9 is an irreducible affine cone of dimension $X\subset \mathbb P^N$0, then $X\subset \mathbb P^N$1 is $X\subset \mathbb P^N$2-nondefective if for general $X\subset \mathbb P^N$3 one has

$X\subset \mathbb P^N$4

Otherwise $X\subset \mathbb P^N$5 is $X\subset \mathbb P^N$6-defective (Blomenhofer et al., 12 Sep 2025). This formulation is equivalent to the projective one after passage between a projective variety and its affine cone.

A recurrent notational difference is that some papers use $X\subset \mathbb P^N$7 and others use $X\subset \mathbb P^N$8 for the secant order. The underlying condition is the same: the span of general tangent spaces must grow at the rate predicted by the parameter count. This makes $X\subset \mathbb P^N$9-nondefectivity a geometric dimension statement rather than a statement about a particular decomposition algorithm.

2. Expected dimension, filling, and generic rank

The expected dimension arises from the observation that a general choice of $n$0 points on an $n$1-dimensional variety spans an $n$2-plane, so one expects

$n$3

This quantity is explicitly called the expected dimension in the Chow-variety analysis (Torrance et al., 2020). One therefore expects $n$4 to fill the ambient space as soon as

$n$5

(Torrance et al., 2020).

The smallest $n$6 such that $n$7 is the generic rank. A trivial parameter count gives

$n$8

(Blomenhofer et al., 2023). In settings where all secant varieties up to the filling threshold are nondefective, this lower bound becomes the actual generic rank.

A common source of confusion is the relation between nondefectivity and filling. Nondefectivity does not mean that $n$9 for every $r$0; it means only that $r$1 equals the expected value. Filling occurs only once the expected dimension reaches the ambient dimension. This distinction is central in applications to generic rank and identifiability.

3. Terracini’s lemma and projection-based criteria

The fundamental computational device is Terracini’s lemma. If $r$2 are general and $r$3 is a general point of the secant plane, then

$r$4

where hats denote affine cones. Consequently,

$r$5

(Torrance et al., 2020). In practice, $r$6-nondefectivity is therefore reduced to a statement about the dimension of a generic sum of tangent spaces.

A second approach uses tangential and osculating projections. Given general points $r$7, the $r$8-tangential projection is the linear projection with center $r$9. A proposition attributed to Chiantini–Ciliberto states that if the general $$\sigma_r(X)=\overline{\bigcup_{p_1,\dots,p_r\in X}\langle p_1,\dots,p_r\rangle}\subset \mathbb P^N,$$0-tangential projection is generically finite onto its image, then $$\sigma_r(X)=\overline{\bigcup_{p_1,\dots,p_r\in X}\langle p_1,\dots,p_r\rangle}\subset \mathbb P^N,$$1 is not $$\sigma_r(X)=\overline{\bigcup_{p_1,\dots,p_r\in X}\langle p_1,\dots,p_r\rangle}\subset \mathbb P^N,$$2-defective (Massarenti et al., 2016).

Osculating spaces refine this perspective. At a smooth point $$\sigma_r(X)=\overline{\bigcup_{p_1,\dots,p_r\in X}\langle p_1,\dots,p_r\rangle}\subset \mathbb P^N,$$3, the $$\sigma_r(X)=\overline{\bigcup_{p_1,\dots,p_r\in X}\langle p_1,\dots,p_r\rangle}\subset \mathbb P^N,$$4th osculating space $$\sigma_r(X)=\overline{\bigcup_{p_1,\dots,p_r\in X}\langle p_1,\dots,p_r\rangle}\subset \mathbb P^N,$$5 is the projective span of all partial derivatives of order at most $$\sigma_r(X)=\overline{\bigcup_{p_1,\dots,p_r\in X}\langle p_1,\dots,p_r\rangle}\subset \mathbb P^N,$$6, with

$$\sigma_r(X)=\overline{\bigcup_{p_1,\dots,p_r\in X}\langle p_1,\dots,p_r\rangle}\subset \mathbb P^N,$$7

An osculating projection is the linear projection from a span of several such spaces. For Grassmannians, Massarenti and Rischter prove a birationality criterion: if $$\sigma_r(X)=\overline{\bigcup_{p_1,\dots,p_r\in X}\langle p_1,\dots,p_r\rangle}\subset \mathbb P^N,$$8 is Plücker-embedded, $$\sigma_r(X)=\overline{\bigcup_{p_1,\dots,p_r\in X}\langle p_1,\dots,p_r\rangle}\subset \mathbb P^N,$$9 are coordinate points corresponding to disjoint $\dim \sigma_r(X)\le \min\{r(n+1)-1,N\}.$0-subspaces of $\dim \sigma_r(X)\le \min\{r(n+1)-1,N\}.$1, and

$\dim \sigma_r(X)\le \min\{r(n+1)-1,N\}.$2

then the osculating projection

$\dim \sigma_r(X)\le \min\{r(n+1)-1,N\}.$3

is birational onto its image, hence generically finite (Massarenti et al., 2016). This converts control of osculating behavior into nondefectivity bounds.

4. Inductive, combinatorial, and stationarity methods

For Chow varieties, Torrance and Vannieuwenhoven develop an inductive construction that combines Terracini’s lemma with Newton’s backward difference formula. They define backward differences with step size $\dim \sigma_r(X)\le \min\{r(n+1)-1,N\}.$4 by

$\dim \sigma_r(X)\le \min\{r(n+1)-1,N\}.$5

and use Newton’s formula

$\dim \sigma_r(X)\le \min\{r(n+1)-1,N\}.$6

to organize inclusion–exclusion calculations on specially constructed lattices of linear subspaces (Torrance et al., 2020). The induction reduces high-dimensional secant computations to finitely many base cases.

The computational part of that proof is explicit. In the main theorem, the authors take $\dim \sigma_r(X)\le \min\{r(n+1)-1,N\}.$7, $\dim \sigma_r(X)\le \min\{r(n+1)-1,N\}.$8, and $\dim \sigma_r(X)\le \min\{r(n+1)-1,N\}.$9. The hardest base case is the $V$0th secant variety of the degree-$V$1 Chow variety in $V$2. They construct a matrix whose columns span the sum of $V$3 tangent spaces plus $V$4 special linear spaces in $V$5, compute its rank over the finite field $V$6 with $V$7 by Gaussian elimination in optimized C++ libraries, and use semicontinuity to certify nondefectivity from a single full-rank instance (Torrance et al., 2020).

A different line of argument appears in the invariant and reducible settings. The key tool in the invariant case is the Stationarity Lemma: if

$V$8

for generic points, and if for some $V$9 one has

$X\subset \PP^N$0

then already

$X\subset \PP^N$1

(Blomenhofer et al., 2023). For reducible cones and $X\subset \PP^N$2-embedded vector bundles, this idea is extended to “rectangular” and “diagonal” stationarity, together with apex spaces

$X\subset \PP^N$3

to control joins, partial apices, and multi-index growth patterns (Blomenhofer et al., 12 Sep 2025).

These methods are structurally different but conceptually parallel. In each case, the problem is reduced to proving that dimension growth cannot stall prematurely without forcing a contradiction: either by combinatorial telescoping on lattices, or by stationarity that forces a span to become filling.

5. Established results for major classes of varieties

Several papers give either exact or asymptotic nondefectivity statements for specific families.

Variety or setting	Nondefectivity statement	Source
Chow variety of decomposable cubics $X\subset \PP^N$4, $X\subset \PP^N$5	For every $X\subset \PP^N$6, it is $X\subset \PP^N$7-nondefective for all $X\subset \PP^N$8; for $X\subset \PP^N$9, $n$0	(Torrance et al., 2020)
Chow variety of decomposable quaternary forms $n$1, $n$2	For every $n$3, it is $n$4-nondefective for all $n$5; for $n$6, $n$7	(Torrance et al., 2020)
Grassmannian $n$8	If $n$9, $r$00, $r$01, and $r$02, then $r$03 is not $r$04-defective for any $r$05	(Massarenti et al., 2016)
Irreducible $r$06-invariant cone $r$07 of dimension $r$08	For all $r$09, $r$10 is $r$11-nondefective; the generic rank satisfies $r$12	(Blomenhofer et al., 2023)
Reducible cone $r$13 with all components of dimension $r$14	For $r$15, $r$16 is nondefective; for $r$17, $r$18	(Blomenhofer et al., 12 Sep 2025)

For Grassmannians, the combinatorial function $r$19 is defined by writing $r$20 in binary,

$r$21

and setting

$r$22

The resulting bound improves the earlier Abo–Ottaviani–Peterson bound

$r$23

for every $r$24, except the sporadic small cases $r$25 (Massarenti et al., 2016).

For invariant secants, the same general theorem specializes to Grassmannians, Chow varieties, Segre–Veronese varieties, Gaussian moment varieties, the Lagrangian Grassmannian, and Spinor varieties by substituting the corresponding ambient-module and variety dimensions (Blomenhofer et al., 2023). The Chow case obtained this way yields broad lower-order nondefectivity ranges, whereas the Chow paper for cubics and quaternary forms gives complete nondefectivity for all secant orders in those two families (Torrance et al., 2020).

6. Consequences, extensions, and open directions

For Chow varieties, complete nondefectivity immediately determines generic Chow rank. A generic cubic in $r$26 variables decomposes as a sum of

$r$27

products of three linear forms, and a generic degree-$r$28 form in four variables decomposes as

$r$29

(Torrance et al., 2020). More generally, knowledge that all relevant secants are nondefective yields precise dimension formulas for joins of Chow varieties, with potential applications in tensor rank and algebraic complexity (Torrance et al., 2020).

In the invariant setting, $r$30-nondefectivity has identifiability consequences. The 2023 paper states that whenever the tangent map $r$31 is nondegenerate, $r$32-nondefectivity implies $r$33-identifiability, citing Massarenti–Mella (2022) (Blomenhofer et al., 2023). For Gaussian moment varieties, this is made explicit: for $r$34, the tangent map is known to be nondegenerate, so $r$35-nondefectivity implies $r$36-identifiability (Blomenhofer et al., 2023).

The reducible and bundle-theoretic generalization substantially broadens the scope of the subject. If

$r$37

is a reducible cone, then

$r$38

and $r$39 is defined to be $r$40-nondefective when every such join has its expected dimension (Blomenhofer et al., 12 Sep 2025). For a $r$41-embedded vector bundle $r$42, one introduces multi-index notions such as $r$43-nondefective and $r$44-filling; the bundle is $r$45-nondefective if it is $r$46-nondefective for every $r$47 with $r$48 (Blomenhofer et al., 12 Sep 2025). This framework is then applied to Fröberg’s conjecture, fat point schemes, partition rank, and identifiability of mixtures of Gaussians and Laplace distributions. In particular, the paper states that $r$49 Gaussians plus $r$50 Laplaces are identifiable as soon as $r$51 (Blomenhofer et al., 12 Sep 2025).

Several limitations and open questions remain explicit in the literature. The osculating-projection method raises the question of whether the exponents in the asymptotic Grassmannian bound can be improved, and whether a similar technique can settle the full conjecture that $r$52 is non-defective except for the four known sporadic cases (Massarenti et al., 2016). The Chow-variety induction suggests extensions to higher degrees $r$53 or dimensions $r$54, but only subject to computational verification of an enlarged finite list of base cases (Torrance et al., 2020). In the invariant framework, a “critical degree” phenomenon is identified: when the ambient module dimension grows like a polynomial of degree $r$55 and $r$56 grows with lower degree, the bound

$r$57

covers “almost all” interesting secant orders for $r$58, whereas for $r$59 most secants are known to be defective and the bound becomes vacuous (Blomenhofer et al., 2023). This suggests that $r$60-nondefectivity is both a local tangent-space problem and a large-scale asymptotic phenomenon tied to the growth of ambient representation spaces.