--- title: Grassmann Cactus Variety in Algebraic Geometry url: https://www.emergentmind.com/topics/grassmann-cactus-variety type: topic --- # Grassmann Cactus Variety in Algebraic Geometry Grassmann cactus variety is, in its standard algebraic-geometric sense, the Zariski closure in a Grassmannian of linear subspaces contained in spans of finite subschemes of bounded length on a projective scheme. It simultaneously generalizes the classical cactus variety and the Grassmann secant variety: the former is recovered when one asks for points rather than higher-dimensional linear spaces, while the latter is recovered by restricting to reduced finite schemes. In current research the notion is closely tied to simultaneous Waring problems, apolarity, Hilbert schemes of finite schemes, and determinantal rank conditions. The terminology is also used in distinct specialized ways in parts of the matroid and electrical-network literature, so its meaning depends on context [2507.21586]. ## 1. Definition and basic framework Let $X \subset \mathbb{P}^N=\mathbb{P}(V)$ be a quasiprojective scheme with a fixed locally closed embedding. For a finite subscheme $Z \subset X$, its linear span $\langle Z\rangle \subset V$ is the smallest linear subspace such that $Z \subset \mathbb{P}(\langle Z\rangle)$ as a subscheme. In standard projective notation, the Grassmannian of projective $k$-planes in $\mathbb{P}^N$ is $\operatorname{Gr}(k+1,N+1)$. For integers $k \ge 0$ and $r \ge 1$, the Grassmann cactus variety is $$ \operatorname{GCat}_{k,r}(X) = \overline{ \left\{ [\Lambda] \in \operatorname{Gr}(k+1,N+1) \;\middle|\; \exists \text{ finite subscheme } Z \subset X,\ \operatorname{length}(Z)\le r,\ \Lambda \subset \langle Z\rangle \right\} }. $$ The Grassmann secant variety is obtained by restricting to reduced subschemes of length $r$: $$ \operatorname{GSec}_{k,r}(X) = \overline{ \left\{ [\Lambda] \in \operatorname{Gr}(k+1,N+1) \;\middle|\; \exists \text{ reduced } Z \subset X,\ \operatorname{length}(Z)=r,\ \Lambda \subset \langle Z\rangle \right\} }. $$ One always has $$ \operatorname{GSec}_{k,r}(X)\subset \operatorname{GCat}_{k,r}(X). $$ If $X$ has at least $r$ distinct points of support, one may equivalently require $\operatorname{length}(Z)=r$ in the definition of $\operatorname{GCat}_{k,r}(X)$ [2507.21586]. This formulation is compatible with several notational conventions. In one common convention, especially in the Veronese literature, $\operatorname{Gr}(k,V)$ denotes the Grassmannian of $k$-dimensional vector subspaces of $V$; then $\operatorname{Gr}(1,V)=\mathbb{P}(V)$, so the case $k=1$ recovers the ordinary cactus variety. In projective language, the Grassmann cactus variety is therefore the natural higher-dimensional analogue of the ordinary cactus construction. ## 2. Socle dimension and reduction to better-behaved finite schemes A central structural result is that the definition of the Grassmann cactus variety can be simplified by restricting the allowed finite schemes. For a local Artin $k$-algebra $(A,\mathfrak m)$, the socle is $$ \operatorname{soc}(A)=\operatorname{Ann}_A(\mathfrak m), $$ and the socle dimension is $\operatorname{socdim}(A)=\dim_k\operatorname{soc}(A)$. For a finite scheme $R \simeq \bigsqcup \operatorname{Spec}A_i$, one says $\operatorname{socdim}(R)\le \ell$ if every local component satisfies $\operatorname{socdim}(A_i)\le \ell$. In particular, $R$ is Gorenstein if and only if $\operatorname{socdim}(R)\le 1$. The main theorem of Buczyńska–Buczyński–Gałązka states that no generality is lost by imposing a bound on local socle dimension: $$ K_{r,k}(X) = \overline{ \left\{ E\in \operatorname{Gr}(k,V) \;\middle|\; \exists R\subset X,\ \dim R=0,\ \deg R\le r,\ \operatorname{socdim}(R)\le k,\ E\subset \langle R\rangle \right\} }. $$ In standard projective notation, this says that $\operatorname{GCat}_{k,r}(X)$ is already obtained by using only finite subschemes whose local socle dimensions are at most $k$. The statement requires no smoothness hypothesis on $X$ and holds over an algebraically closed field of arbitrary characteristic [2507.21586]. The proof proceeds by minimality. If a $k$-plane lies in the span of some degree-$\le r$ finite scheme $R$, choose such an $R$ minimal by degree. If a local component of $R$ has socle dimension at least $k+1$, then every $k$-plane in $\langle R\rangle$ already lies in the span of a proper subscheme of degree $r-1$, contradicting minimality. Two lemmas drive this argument. First, if $R$ itself is finite of degree $r$, then its Grassmann cactus variety is simply the closed Grassmannian $\operatorname{Gr}(k,\langle R\rangle)$. Second, if $\operatorname{socdim}(R)\ge k+1$ at some local component, then $$ \operatorname{Gr}(k,\langle R\rangle)=\bigcup_{R'\subset R,\ \deg R'=r-1}\operatorname{Gr}(k,\langle R'\rangle). $$ This reduction is controlled by the Hilbert scheme of codimension-one subschemes. For a finite local scheme $R=\operatorname{Spec}A$ of degree $r$ with socle $s\subset A$, one has $$ (\operatorname{Hilb}^{r-1}R)_{\mathrm{red}}\simeq \mathbb{P}(s). $$ Thus degree-$(r-1)$ subschemes are parameterized, set-theoretically, by lines in the socle. The nonreduced structure of $\operatorname{Hilb}^{r-1}(R)$ can nevertheless be subtle: for example, if $R=\operatorname{Spec}k[t]/(t^2)$, then $\operatorname{Hilb}^1(R)=R\neq \mathbb{P}(s)=\operatorname{Spec}k$. Computationally, the theorem means that highly nonreduced schemes with large socle are unnecessary in describing $\operatorname{GCat}_{k,r}(X)$; for $k=1$ this recovers the familiar restriction to Gorenstein schemes [2507.21586]. ## 3. Veronese geometry, simultaneous Waring, and the first strict cactus phenomena The Veronese case is the principal testing ground. Let $V\simeq \mathbb{C}^{n+1}$, let $S^dV$ be the space of degree-$d$ forms, and let $$ \nu_d:\mathbb{P}^n\to \mathbb{P}(S^dV) $$ be the $d$-th Veronese embedding. For a $k$-dimensional vector subspace $W\subset S^dV$, the Grassmann rank and Grassmann cactus rank are defined by $$ r(W):=\min\left\{r\in \mathbb{N}\mid W\subset \langle L_1^d,\dots,L_r^d\rangle \text{ for some linear forms }L_i\in V^*\right\}, $$ and $$ cr(W):=\min\left\{r\in \mathbb{N}\mid W\subset \langle \nu_d(R)\rangle \text{ for some zero-dimensional }R\subset \mathbb{P}^n,\ \operatorname{length}(R)\le r\right\}. $$ Accordingly, $$ \sigma_{r,k}(\nu_d(\mathbb{P}^n)) = \{[W]\in \operatorname{Gr}(k,S^dV)\mid r(W)\le r\}, $$ and $$ \kappa_{r,k}(\nu_d(\mathbb{P}^n)) = \{[W]\in \operatorname{Gr}(k,S^dV)\mid cr(W)\le r\}. $$ For $k=1$, these reduce to the classical secant and cactus varieties [2007.16203]. The initial equality range is now known with some precision. For ordinary Veronese secants, $\sigma_r(\nu_d(\mathbb{P}^n))=\kappa_r(\nu_d(\mathbb{P}^n))$ for $r\le 13$. For Grassmann secants, $\sigma_{r,k}(\nu_d(\mathbb{P}^n))=\kappa_{r,k}(\nu_d(\mathbb{P}^n))$ for $r\le 7$. The first Grassmann case in which the inclusion is strict is therefore $\kappa_{8,3}$. For $n\ge 4$ and $d\ge 5$, $\kappa_{8,3}(\nu_d(\mathbb{P}^n))$ has two irreducible components. One is the Grassmann secant variety $\sigma_{8,3}(\nu_d(\mathbb{P}^n))$. The other, denoted $\mathfrak K_{8,3}(\nu_d(\mathbb{P}^n))$, is the closure of the locus $$ \left\{ [\ell^{d-2}\cdot U]\in \operatorname{Gr}(3,S^dV) \ \middle|\ \ell\in V^*,\ U\subset S^2V \text{ is a $3$-dimensional subspace, and after dehomogenizing by }\ell,\ \operatorname{Apolar}((U|_{\ell=1})^{\triangle}) \text{ has local Hilbert function }(1,4,3) \right\}. $$ For $n=4$, the Hilbert-function condition is automatic, so $\mathfrak K_{8,3}(\nu_d(\mathbb{P}^4))$ is exactly the locus of $3$-planes whose forms are all divisible by $\ell^{d-2}$. The mechanism behind $\kappa_{8,3}\neq \sigma_{8,3}$ is the appearance of a non-smoothable component $H_{(1,4,3)}$ of the Hilbert scheme of length-$8$ schemes; the extra cactus component is the image of that non-smoothable geometry under the universal span construction [2007.16203]. An explicit example exists already for $n=4$. Let $\ell=x_0$, and let $$ U=\operatorname{span}\{x_1x_3,\ x_2x_4,\ x_1x_4-x_2x_3\}\subset S^2\langle x_1,\dots,x_4\rangle. $$ Then $$ W=\operatorname{span}\{\ell^{d-2}q\mid q\in U\} $$ defines a point $[W]\in \kappa_{8,3}(\nu_d(\mathbb{P}^4))$ but $[W]\notin \sigma_{8,3}(\nu_d(\mathbb{P}^4))$. For smaller ambient dimensions, by contrast, the relevant Hilbert loci are irreducible, and the equality $\kappa=\sigma$ persists [2007.16203]. ## 4. Apolarity, catalecticants, and algorithmic membership tests Apolarity is the standard bridge between finite schemes and cactus rank. In the scalar case, for a nonzero $f\in S^dV$, classical apolarity gives $$ cr(f)\le k \iff \exists R\subset \mathbb{P}(V)\text{ of length }k\text{ with }I(R)\subset \operatorname{Ann}(f). $$ The Veronese paper extends this framework to the Grassmann setting and combines it with a homogenization or triangle operator that converts lower-degree data into degree-$d$ forms or subspaces with prescribed apolar algebra [2007.16203]. A basic warning is that catalecticant or flattening equations often detect cactus membership rather than secant membership. For a form $f\in S^dV$, the catalecticant map is $$ C_f^{(a)}:S^a(V^*)\to S^{d-a}V,\qquad \varphi\mapsto \varphi\!\,\lrcorner\, f. $$ In the Veronese setting, many classical rank conditions on these maps vanish on $\kappa_r$ rather than distinguishing $\sigma_r$. This is why points of the form $\ell^{d-3}g$ or $\ell^{d-2}U$ can satisfy all expected flattening conditions while remaining outside the corresponding secant variety. For $\kappa_{8,3}(\nu_d(\mathbb{P}^n))$ with $d\ge 5$ and $n\ge 4$, a decisive algorithm distinguishes cactus-only points from actual secant points. Starting with $[W]\in \kappa_{8,3}(\nu_d(\mathbb{P}^n))$, one computes $$ a=(\operatorname{Ann}(W))_{8n-7. $$ Here the tangent-space bound detects whether the associated Gorenstein point is smoothable; nonsmoothability is exactly what forces cactus-only membership [2007.16203]. A complementary, more constructive viewpoint comes from symbolic decomposition algorithms for symmetric tensors. In that setting, one computes a minimal apolar scheme $Z=\bigsqcup_i Z_i$ from generalized Hankel or moment matrices and multiplication operators. Common rank-$1$ eigenvectors recover support points, generalized eigenspaces recover multiplicities, and one obtains a cactus decomposition $$ f=\sum_i L_i^{d-k_i+1}N_i. $$ The output is not merely a decomposition of $f$: it also gives the span $\Lambda=\langle \nu_d(Z)\rangle$, hence a point of the Grassmannian model $$ \operatorname{GKC}_r(v_d(\mathbb{P}^n)) = \{\Lambda\in \operatorname{Gr}(r,N+1)\mid \exists Z\subset \mathbb{P}^n,\ \operatorname{length}(Z)=r,\ \langle Z\rangle=\Lambda\}, $$ thereby operationalizing the Grassmann cactus viewpoint through explicit apolar computation [1812.02612]. ## 5. Scheme structures and determinantal equations The set-theoretic cactus construction admits a scheme-theoretic refinement via relative linear spans. Given a family $\varphi:R\to Q$ of finite subschemes in $X$ over a possibly nonreduced base, one defines the family of linear spaces $\langle R\rangle\subset Q\times \mathbb{P}(W)$ using the degree-$\le 1$ part of the homogeneous ideal of $R$, and then takes the scheme-theoretic image in $\mathbb{P}(W)$. Running over all independent familiars of degree $\le r$ yields the cactus scheme $K_r(X)$; restricting to flat familiars yields $K_r^{\mathrm{flat}}(X)$. For Veronese embeddings, when $d\ge 2r$ and $r\le i\le d-r$, the $r$-th cactus scheme agrees on the dense open complement of the lower cactus stratum with the rank-$\le r$ catalecticant scheme: $$ K_r(\nu_d)\setminus K_{r-1}(\nu_d) = K_r^{\mathrm{flat}}(\nu_d)\setminus K_{r-1}^{\mathrm{flat}}(\nu_d) = \Upsilon_r^{i,d-i}\setminus \Upsilon_{r-1}^{i,d-i}. $$ Moreover, this open piece is identified with a Zariski open subset of a vector bundle over the Gorenstein Hilbert scheme of length-$r$ subschemes of $\mathbb{P}(V)$. The same paper states that its machinery adapts directly to a Grassmannian version: one forms incidences $$ I=\{(q,[\Lambda])\mid \Lambda\subset \langle R_q\rangle\} $$ over $\operatorname{Gr}(k+1,W)$ and defines a Grassmann cactus scheme by scheme-theoretic image. The resulting determinantal description on the Grassmannian is presented as a natural extension rather than as a fully established theorem for all parameters [2410.21908]. A separate determinantal theory for ordinary cactus varieties appears for sufficiently ample embeddings of arbitrary projective schemes. If $L=A\otimes B$ is a suitable splitting of a sufficiently ample line bundle and $$ \mu_{A,B}:H^0(X,A)\otimes H^0(X,B)\to H^0(X,L) $$ is the multiplication map, then in chosen bases it gives a matrix $M_{A,B}$ of linear forms. For every fixed $r'\in \mathbb N$, sufficiently ample $L$ can be chosen so that for all $r\le r'$, $$ \operatorname{Cact}_r(X)=V_{\mathrm{red}}(I_{r+1}(M_{A,B})). $$ The same data induce a Grassmannian bundle map $$ \Phi:H^0(X,B)^*\otimes \mathcal O_{\operatorname{Gr}} \longrightarrow H^0(X,A)\otimes \mathcal S^*, $$ where $\mathcal S$ is the tautological subbundle on $\operatorname{Gr}(r-1,N)$. Its degeneracy locus $$ D_r(\Phi)=\{\Lambda\mid \operatorname{rank}\Phi(\Lambda)\le r\} $$ contains the Grassmann cactus variety $$ \operatorname{GrCact}_r(X) = \overline{\{\Lambda\in \operatorname{Gr}(r-1,N)\mid \exists Z\subset X,\ \operatorname{length}(Z)=r,\ \Lambda=\langle Z\rangle\}}. $$ The paper stresses, however, that it does not assert $\operatorname{GrCact}_r(X)=D_r(\Phi)$ set-theoretically; the determinantal description is proved on the ambient projective space, while the Grassmannian statement is formulated as a natural induced containment [2412.00709]. ## 6. Terminological variants in matroid and network theory The phrase “Grassmann cactus variety” is not uniform across adjacent literatures. In the rank-three matroid literature on point-line configurations, one synthesis defines the Grassmann cactus variety of a cactus configuration $M$ as the subvariety cut out by the Grassmann–Cayley ideal $G_M$. In that setting, if $I_{\mathcal C(M)}$ is the circuit ideal, then for cactus configurations with acyclic high-valence locus one has $$ I_M=\sqrt{I_{\mathcal C(M)}+G_M}, \qquad V_M=V_{\mathcal C(M)}\cap V(G_M), $$ so Grassmann–Cayley concurrency equations together with circuit equations cut out the matroid variety set-theoretically. The same work emphasizes explicit bracket identities, such as $$ [1\,2\,3][4\,5\,6]-[1\,2\,4][3\,5\,6]=0 $$ for three concurrent lines [2508.14141]. A closely related paper uses the same phrase differently: there, a “Grassmann cactus variety” is the matroid variety $V_M$ itself for a cactus matroid $M$, embedded in $G(3,n)$ via Plücker coordinates. It proves that every cactus matroid is realizable and that its Grassmann cactus variety is irreducible. Under the same acyclicity condition on points of degree at least $3$, the defining ideal is generated up to radical by circuit equations and Grassmann–Cayley equations: $$ I(M)=\sqrt{I_C(M)+G_M}. $$ For Pascal and Pappus matroids one must add liftability equations $I_M^{\mathrm{lift}}$ [2506.07757]. In electrical-network theory, the phrase refers to a different Grassmannian object again. For cactus networks, Lam’s map produces a space $$ \mathcal X_n=H_n\cap \operatorname{Gr}_{\ge 0}(n+1,2n), $$ and this space is characterized intrinsically as $$ \mathcal X_n = IG^\Omega(n+1,2n)\cap \operatorname{Gr}_{\ge 0}(n+1,2n), $$ the totally nonnegative locus in the $\Omega$-isotropic Grassmannian for a specific skew-symmetric bilinear form $\Omega$ of corank $2$. Here the geometry is governed by total positivity, isotropy, grove measurements, and electrical duality rather than by spans of finite schemes [2106.15418]. These usages share a Grassmannian ambient space and a “cactus” organizing principle, but they are mathematically distinct. In contemporary algebraic geometry, the default meaning remains the finite-scheme formulation: closure of the locus of linear spaces contained in spans of finite subschemes of bounded length. Within that meaning, recent work has clarified three decisive features: restriction to low-socle schemes suffices, the first strict differences from Grassmann secants occur in the Veronese case at $\kappa_{8,3}$, and scheme-theoretic as well as determinantal descriptions are beginning to place the subject on the same footing as the classical theory of secant varieties. Source: https://www.emergentmind.com/topics/grassmann-cactus-variety