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Degree of the subspace variety

Paul Breiding, Pierpaola Santarsiero

Abstract

Subspace varieties are algebraic varieties whose elements are tensors with bounded multilinear rank. In this paper, we compute their degrees by computing their volumes.

Degree of the subspace variety

Abstract

Subspace varieties are algebraic varieties whose elements are tensors with bounded multilinear rank. In this paper, we compute their degrees by computing their volumes.
Paper Structure (9 sections, 13 theorems, 64 equations, 1 figure)

This paper contains 9 sections, 13 theorems, 64 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary notions
  3. The Stiefel manifold
  4. The coarea formula
  5. Expected values of polynomials evaluated at complex Gaussians
  6. Metric geometry of the subspace variety
  7. The normal Jacobian of $\varphi$
  8. The volume of the subspace variety
  9. Macaulay2 code computing $\deg(X_{\mathbf{k}})$

Key Result

Theorem 1.1

Let $\mathbf{k}=(k_1,\dots,k_d)$, $\mathbf{n}=(n_1,\dots,n_d)$, where $k_i\leq n_i$ for all $i=1,\dots,d$ and set $N=n_1\cdots n_d-1$. Suppose that $X_{\mathbf{k}}\subset \mathbb{P}^N$ contains a tensor of multilinear rank equal to $\mathbf{k}$. Then,

Figures (1)

  • Figure 1: Tucker decomposition $T = (A_1 \otimes A_2\otimes A_3)\cdot C$ for a $3$-order tensor $T\in\mathbb C^{n_1}\otimes \mathbb{C}^{n_2}\otimes \mathbb{C}^{n_3}$.

Theorems & Definitions (25)

  • Theorem 1.1
  • Example 1.2
  • Example 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • ...and 15 more