A Fröberg type theorem for higher secant complexes

TL;DR

The paper addresses the problem of characterizing when the Stanley–Reisner ring of a $q$-secant complex $\sigma_q\Delta$ exhibits $(q+1)$-linear behavior via the property $N_{q+1,p}$, translating Fröberg-type results into higher secant combinatorics. It develops a graph-theoretic framework with forbidden induced subgraphs $\mathcal{F}_{q,1}$, $\mathcal{F}_{q,2}$ and $C_{2q+i}$ to classify when $S(\sigma_q\Delta)$ satisfies $N_{q+1,p}$, and provides explicit Betti-number formulas and projective-dimension criteria for $q$-secant chordal graphs. The main contributions include a precise equivalence between $N_{q+1,p}$ and forbidden-subgraph freeness, a Mayer–Vietoris–based regularity result for $2$-regular complexes, and a complete Cohen–Macaulay classification for forests via edge contractions, drawing deep parallels with varieties of minimal $q$-secant degree. Overall, the work offers a combinatorial pathway to understand syzygies of higher secant loci through graph operations and secant joins, with implications for both algebraic geometry and combinatorial commutative algebra.

Abstract

We generalize the celebrated Fröberg's theorem to embedded joins of copies of a simplicial complex, namely higher secant complexes to the simplicial complex, in terms of property $N_{q+1,p}$ due to Green and Lazarsfeld. Furthermore, we investigate combinatorial phenomena parallel to geometric ones observed for higher secant varieties of minimal degree.

Figures (5)

• Figure 1: Minimal free resolution
• Figure 2: The $4$-sunlet graph
• Figure 3: All elementary bipartite graphs on $6$ vertices
• Figure 4: All edge-minimal elementary bipartite graphs on $8$ vertices
• Figure 5: A graph that is $3$-secant chordal but not $2$-secant chordal

Theorems & Definitions (84)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Corollary 1.4
• Corollary 1.5
• Definition 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Definition 2.2