Secant Varieties of Gaussian Moment Varieties

Definitions and mathematical statement

The supplied record asks for the dimensions, degrees, and defining equations of Gaussian moment secant varieties for every triple of integers $k,n,d\geq 1$. It does not define $G_{n,d}$, its ground field, its ambient space, or its secant-indexing convention. The following gives a precise conditional formulation using complex projective moment varieties with unrestricted symmetric covariance matrices; these conventions require confirmation against the cited report.

Let $\mathbb N_0=\{0,1,2,\ldots\}$, let \[ A_{n,d}=\{\alpha\in\mathbb N_0^n:|\alpha|\leq d\},\qquad N=\binom{n+d}{d}-1, \] and give $\mathbb P^N_{\mathbb C}$ homogeneous coordinates $m_\alpha$, indexed by $A_{n,d}$. For $\mu\in\mathbb C^n$ and a symmetric matrix $\Sigma\in\operatorname{Sym}_n(\mathbb C)$, define the moment polynomials \[ M_\alpha(\mu,\Sigma)=\alpha!\,[t^\alpha]\exp\!\left(\sum_{i=1}^n\mu_i t_i+\frac12\sum_{i,j=1}^n\Sigma_{ij}t_it_j\right), \] where $|\alpha|=\sum_i\alpha_i$, $\alpha!=\prod_i\alpha_i!$, and coefficient extraction is in a formal power series. In particular, $M_0=1$. Under the stated convention, the Gaussian moment variety is the reduced projective Zariski closure \[ G_{n,d}=\overline{\left\{[M_\alpha(\mu,\Sigma)]_{\alpha\in A_{n,d}}:\mu\in\mathbb C^n,\ \Sigma\in\operatorname{Sym}_n(\mathbb C)\right\}}^{\mathrm{Zar}}\subseteq\mathbb P^N_{\mathbb C}. \] Here complex symmetric matrices are algebraic parameters; no positivity condition is imposed. Define the secant variety using spans of $k$ points: \[ \operatorname{Sec}_k(G_{n,d})=\overline{\bigcup_{x_1,\ldots,x_k\in G_{n,d}}\langle x_1,\ldots,x_k\rangle}^{\mathrm{Zar}}, \] with its reduced structure, so that $\operatorname{Sec}_1(G_{n,d})=G_{n,d}$.

Conditional problem. For every $k,n,d\in\mathbb Z_{\geq1}$, determine \[ \dim\operatorname{Sec}_k(G_{n,d}),\qquad \deg\operatorname{Sec}_k(G_{n,d}), \] and give generators for its homogeneous vanishing ideal \[ I\left(\operatorname{Sec}_k(G_{n,d})\right)\subseteq\mathbb C[m_\alpha:\alpha\in A_{n,d}]. \] Dimension and degree are projective dimension and degree in the displayed embedding. No restriction on the relative sizes of $k,n,d$ is supplied.

This conditional formulation uses moments of all total degrees at most $d$, full covariance parameters, and $k$-point secants. The supplied record does not establish these choices or whether defining equations are intended as ideal generators or merely set-theoretic equations. The homoscedastic reference does not itself establish a common-covariance restriction on the target problem. These qualifications remain unresolved.

Short English statement

For every positive number of components $k$, number of variables $n$, and moment cutoff $d$, determine the dimension, degree, and defining equations of the corresponding Gaussian moment secant variety. A precise formulation is given conditionally using complex projective closures, unrestricted covariance matrices, and spans of $k$ points; the source record does not specify these conventions. Its partially solved label is supplied curation, not an independently verified conclusion.

Sources

• Local problem registry: permanent UnsolvedMath ID $30003479$, Secant Varieties of Gaussian Moment Varieties, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2017.
• Supplied report citation: Algebraic Statistics (2018), Oberwolfach Reports, pp. 1207--1279, DOI 10.4171/owr/2017/20. https://doi.org/10.4171/owr/2017/20. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The citation's 2018 label and the record's 2017 year require reconciliation against the report.
• Supplied literature reference, labeled "Améndola Faugère Ranestad and Sturmfels": https://arxiv.org/abs/1510.04654.
• Supplied literature reference, labeled "homoscedastic Gaussian identifiability": https://doi.org/10.1007/s10208-020-09469-6.
• Supplied literature reference, labeled "nondefectivity of invariant secants": https://arxiv.org/abs/2312.12335.
• Literature assessment dated 2026-08-21: supplied dated curation, not independently verified. The record labels the problem partially solved and reports computations of many Gaussian moment and secant varieties in low-dimensional cases, identifiability in several regimes, and the absence of a uniform description for all $k,n,d$. Its background explicitly calls this dated literature triage derived from the supplied OWR curation and requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified, explaining that notation, typography, and sentence structure were repaired while mathematical content was preserved. This is a supplied editorial verification label, not mathematical verification performed for this draft.