[Paper Review] Are all Secant Varieties of Segre Products Arithmetically Cohen-Macaulay?

Luke Oeding|arXiv (Cornell University)|Mar 29, 2016

Tensor decomposition and applications46 references3 citations

TL;DR

This paper investigates the arithmetically Cohen-Macaulay (aCM) property of secant varieties of Segre products of projective spaces, proposing an inductive lifting method (LW-lifting) adapted from Landsberg and Weyman to extend known aCM cases beyond the standard $n_i \geq r$ condition. It proves new families of secant varieties are aCM and arithmetically Gorenstein using resolution-by-small-partitions techniques and provides a new computation of the minimal free resolution for $3\times3\times4$ rank-4 tensors.

ABSTRACT

When present, the Cohen-Macaulay property can be useful for finding the minimal defining equations of an algebraic variety. It is conjectured that all secant varieties of Segre products of projective spaces are arithmetically Cohen-Macaulay. A summary of the known cases where the conjecture is true is given. An inductive procedure based on the work of Landsberg and Weyman (LW-lifting) is described and used to obtain resolutions of orbits of secant varieties from those of smaller secant varieties. A new computation of the minimal free resolution of the variety of border rank 4 tensors of format $3 imes 3 imes 4$ is given together with its equivariant presentation. LW-lifting is used to prove several cases where secant varieties are arithmetically Cohen-Macaulay and arithmetically Gorenstein.

Motivation & Objective

• To investigate the conjecture that all secant varieties of Segre products of projective spaces are arithmetically Cohen-Macaulay (aCM).
• To extend the applicability of Landsberg and Weyman’s LW-lifting method to cases where $n_i < r$ for some $i$, beyond the original $n_i \geq r$ condition.
• To provide a new computation of the minimal free resolution for the secant variety of rank-4 tensors of format $3\times3\times4$.
• To determine new families of secant varieties that are both aCM and arithmetically Gorenstein using the refined lifting procedure.

Proposed method

• Adapts Landsberg and Weyman’s LW-lifting technique to cases with $n_i < r$ by modifying the resolution-by-small-partitions condition.
• Uses a geometric construction involving orbit closures and partial desingularizations via subspace varieties to relate resolutions of smaller secant varieties to larger ones.
• Applies an iterated mapping cone construction on sheaf complexes to build a resolution of the coordinate ring of the larger orbit closure.
• Employs cohomological techniques, including Bott’s algorithm and the Borel-Weil theorem, to compute cohomology of vector bundles on flag varieties.
• Defines an $(s_j)$-small resolution via Schur modules indexed by partitions fitting within prescribed boxes, ensuring the resolution remains small under lifting.
• Derives a formula for the new smallness bounds $s_j$ based on the original resolution’s partitions and the dimensions of the ambient spaces.

Experimental results

Research questions

• RQ1 Are all secant varieties of Segre products arithmetically Cohen-Macaulay, as conjectured?
• RQ2 Can the LW-lifting method be extended to cases where $n_i < r$ for some $i$, thereby proving new aCM cases?
• RQ3 What is the minimal free resolution of the secant variety of rank-4 tensors in format $3\times3\times4$?
• RQ4 Which secant varieties of Segre products are both arithmetically Cohen-Macaulay and arithmetically Gorenstein?

Key findings

• The paper proves that if a $G'$-variety $Y$ is aCM and has a resolution that is $({\widehat{r_j}} - r_j)$-small for all $j$ with $r_j < a_j$, then the orbit closure $\overline{G.Y}$ is also aCM.
• A new, explicit computation of the minimal free resolution is provided for the secant variety of $3\times3\times4$ rank-4 tensors.
• The LW-lifting method is successfully extended to cases with $n_i < r$ for some $i$, yielding new families of aCM secant varieties.
• The paper identifies new cases where secant varieties are both aCM and arithmetically Gorenstein, using the refined lifting procedure.
• The resolution of $\overline{G.Y}$ obtained via the mapping cone is shown to be of length equal to the codimension of the variety, confirming its minimality in the derived sense.
• The construction provides a (not necessarily minimal) resolution of $\mathbb{C}[\overline{G.\widehat{Y}}]$ that is $(s_j)$-small, with $s_j$ explicitly computed from the original resolution’s partitions.