--- title: r-Nondefectivity in Secant Varieties url: https://www.emergentmind.com/topics/r-nondefectivity type: topic --- # r-Nondefectivity in Secant Varieties $r$-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 [2005.12436]. 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 [1610.09332] [2312.12335] [2509.10443]. ## 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 $h$-secant variety is \[ \expdim\,\sigma_h(X)=\min\{hn+(h-1),N\}=\min\{h(n+1)-1,N\}. \] The variety is called $h$-defective if \[ \dim \sigma_h(X)<\expdim\,\sigma_h(X), \] and the $h$-defect is \[ \delta_h(X)=\expdim\,\sigma_h(X)-\dim\sigma_h(X). \] When $\delta_h(X)=0$, one says that $X$ is non-$h$-defective [1610.09332]. The same notion is frequently expressed in affine-cone language. If $X\subset V$ is an irreducible affine cone of dimension $N$, then $X$ is $r$-nondefective if for general $x_1,\dots,x_r\in X$ one has \[ \dim \langle T_{x_1}X,\dots,T_{x_r}X\rangle=\min\{rN,\dim V\}. \] Otherwise $X$ is $r$-defective [2509.10443]. 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 $r$ and others use $h$ 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 $r$-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 $r$ points on an $n$-dimensional variety spans an $(r-1)$-plane, so one expects \[ \dim \sigma_r(X)\le \min\{r\dim(X)+(r-1),N\}=\min\{r(n+1)-1,N\}. \] This quantity is explicitly called the expected dimension in the Chow-variety analysis [2005.12436]. One therefore expects $\sigma_r(X)$ to fill the ambient space as soon as \[ r(n+1)-1\ge N, \qquad\text{i.e.}\qquad r\ge r_{\exp}:=\left\lceil\frac{N+1}{n+1}\right\rceil \] [2005.12436]. The smallest $r^\circ$ such that $\sigma_{r^\circ}(X)=\mathbb P^N$ is the generic rank. A trivial parameter count gives \[ r^\circ\ge \left\lceil\frac{N+1}{n+1}\right\rceil \] [2312.12335]. 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 $\sigma_r(X)=\mathbb P^N$ for every $r$; it means only that $\dim \sigma_r(X)$ 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 $p_1,\dots,p_r\in X$ are general and $q\in \langle p_1,\dots,p_r\rangle$ is a general point of the secant plane, then \[ T_q\bigl(\widehat{\sigma_r(X)}\bigr)=\sum_{i=1}^r T_{p_i}(\widehat X), \] where hats denote affine cones. Consequently, \[ \dim \sigma_r(X)=\dim \langle T_{p_1}X,\dots,T_{p_r}X\rangle \] [2005.12436]. In practice, $r$-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 $x_1,\dots,x_h\in X$, the $h$-tangential projection is the linear projection with center $\langle T_{x_1}X,\dots,T_{x_h}X\rangle$. A proposition attributed to Chiantini–Ciliberto states that if the general $h$-tangential projection is generically finite onto its image, then $X$ is not $(h+1)$-defective [1610.09332]. Osculating spaces refine this perspective. At a smooth point $p\in X$, the $k$th osculating space $T_p^kX$ is the projective span of all partial derivatives of order at most $k$, with \[ T_p^0X=\{p\},\qquad T_p^1X=T_pX. \] An osculating projection is the linear projection from a span of several such spaces. For Grassmannians, Massarenti and Rischter prove a birationality criterion: if $G(r,n)\subset \PP^N$ is Plücker-embedded, $p_1,\dots,p_\ell$ are coordinate points corresponding to disjoint $(r+1)$-subspaces of $\CC^{n+1}$, and \[ k_1,\dots,k_\ell\le r-1,\qquad \ell\le \left\lfloor\frac{n+1}{r+1}\right\rfloor,\qquad \sum_{i=1}^{\ell}k_i\le r-1, \] then the osculating projection \[ \Pi_{k_1,\dots,k_\ell}:G(r,n)\dashrightarrow \PP^M \] is birational onto its image, hence generically finite [1610.09332]. 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 $\ell$ by \[ \Delta^0 f(t)=f(t),\qquad \Delta^k f(t)=\sum_{j=0}^{k}(-1)^j\binom{k}{j}f(t-j\ell), \] and use Newton’s formula \[ f(t)=\sum_{j=0}^{k}\binom{k}{j}\,\Delta^{\,k-j}f(t-j\ell) \] to organize inclusion–exclusion calculations on specially constructed lattices of linear subspaces [2005.12436]. 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 $\ell=27$, $K_0=3$, and $t_0=82$. The hardest base case is the $400$th secant variety of the degree-$82$ Chow variety in $\mathbb P^{98\,770-1}$. They construct a matrix whose columns span the sum of $400$ tangent spaces plus $3$ special linear spaces in $S^{82}\CC^4\cong \CC^{98\,770}$, compute its rank over the finite field $\mathbb F_p$ with $p=8191$ by Gaussian elimination in optimized C++ libraries, and use semicontinuity to certify nondefectivity from a single full-rank instance [2005.12436]. A different line of argument appears in the invariant and reducible settings. The key tool in the invariant case is the Stationarity Lemma: if \[ a_m=\dim\bigl(\langle T_{x_1},\dots,T_{x_m}\rangle\cap U_y\bigr) \] for generic points, and if for some $m$ one has \[ a_{m+1}=a_m>0, \] then already \[ \langle T_{x_1},\dots,T_{x_m}\rangle=\mathcal L \qquad\text{and}\qquad a_m=N' \] [2312.12335]. For reducible cones and $V$-embedded vector bundles, this idea is extended to “rectangular” and “diagonal” stationarity, together with apex spaces \[ \apex(X)=\{v\in V\mid X+v=X\}, \] to control joins, partial apices, and multi-index growth patterns [2509.10443]. 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 recent papers give either exact or asymptotic nondefectivity statements for specific families. | Variety or setting | Nondefectivity statement | Source | |---|---|---| | Chow variety of decomposable cubics $\mathcal C_{3,n}\subset \mathbb P^{\binom{n+3}{3}-1}$, $\dim \mathcal C_{3,n}=3n$ | For every $n\ge 1$, it is $r$-nondefective for all $r$; for $1\le r\le \left\lceil \binom{n+3}{3}/(3n+1)\right\rceil$, $\dim \sigma_r(\mathcal C_{3,n})=\min\{r(3n+1)-1,\binom{n+3}{3}-1\}$ | [2005.12436] | | Chow variety of decomposable quaternary forms $\mathcal C_{d,3}\subset \mathbb P^{\binom{d+3}{d}-1}$, $\dim \mathcal C_{d,3}=3d$ | For every $d\ge 1$, it is $r$-nondefective for all $r$; for $1\le r\le \left\lceil \binom{d+3}{d}/(3d+1)\right\rceil$, $\dim \sigma_r(\mathcal C_{d,3})=\min\{r(3d+1)-1,\binom{d+3}{d}-1\}$ | [2005.12436] | | Grassmannian $G(r,n)\subset \PP^N$ | If $r\ge 2$, $n\ge 2r+1$, $a=\left\lfloor\frac{n+1}{r+1}\right\rfloor$, and $H=a\,h_a(r-1)$, then $G(r,n)$ is not $(h+1)$-defective for any $h\cdots>\lambda_s\ge 0, \] and setting \[ h_a(k)=a^{\lambda_1-1}+\cdots+a^{\lambda_s-1}. \] The resulting bound improves the earlier Abo–Ottaviani–Peterson bound \[ h\le \frac{n-r}{3}+1 \] for every $r\ge 4$, except the sporadic small cases $(r,n)=(4,10),(5,11)$ [1610.09332]. 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 [2312.12335]. 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 [2005.12436]. ## 6. Consequences, extensions, and open directions For Chow varieties, complete nondefectivity immediately determines generic Chow rank. A generic cubic in $n+1$ variables decomposes as a sum of \[ \left\lceil \frac{\binom{n+3}{3}}{3n+1}\right\rceil \] products of three linear forms, and a generic degree-$d$ form in four variables decomposes as \[ f=\sum_{i=1}^{r}\ell_{i,1}\cdots \ell_{i,d}, \qquad r=\left\lceil \frac{\binom{d+3}{d}}{3d+1}\right\rceil \] [2005.12436]. 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 [2005.12436]. In the invariant setting, $m$-nondefectivity has identifiability consequences. The 2023 paper states that whenever the tangent map $x\mapsto T_xV$ is nondegenerate, $m$-nondefectivity implies $(m-1)$-identifiability, citing Massarenti–Mella (2022) [2312.12335]. For Gaussian moment varieties, this is made explicit: for $d=5,6,7,8,9$, the tangent map is known to be nondegenerate, so $r$-nondefectivity implies $(r-1)$-identifiability [2312.12335]. The reducible and bundle-theoretic generalization substantially broadens the scope of the subject. If \[ X=X_1\cup\cdots\cup X_k \] is a reducible cone, then \[ \sigma_r(X)=\bigcup_{i_1,\dots,i_r}J(X_{i_1},\dots,X_{i_r}), \] and $X$ is defined to be $r$-nondefective when every such join has its expected dimension [2509.10443]. For a $V$-embedded vector bundle $\pi:E\to X\subset V$, one introduces multi-index notions such as $\alpha$-nondefective and $\alpha$-filling; the bundle is $r$-nondefective if it is $\alpha$-nondefective for every $\alpha$ with $|\alpha|=r$ [2509.10443]. 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 $5$ Gaussians plus $7$ Laplaces are identifiable as soon as $n\ge 27$ [2509.10443]. 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 $G(r,n)$ is non-defective except for the four known sporadic cases [1610.09332]. The Chow-variety induction suggests extensions to higher degrees $d>3$ or dimensions $n>3$, but only subject to computational verification of an enlarged finite list of base cases [2005.12436]. In the invariant framework, a “critical degree” phenomenon is identified: when the ambient module dimension grows like a polynomial of degree $d_0$ and $\dim X$ grows with lower degree, the bound \[ r\le \frac{\dim \mathcal L}{\dim X}-\dim X \] covers “almost all” interesting secant orders for $d>2d_0$, whereas for $d<2d_0$ most secants are known to be defective and the bound becomes vacuous [2312.12335]. This suggests that $r$-nondefectivity is both a local tangent-space problem and a large-scale asymptotic phenomenon tied to the growth of ambient representation spaces. Source: https://www.emergentmind.com/topics/r-nondefectivity