From points to complexes: a concept of unexpectedness for simplicial complexes

TL;DR

This work develops an unexpected systems of parameters framework for squarefree monomial ideals, linking the existence of unexpected sops to failures of the weak Lefschetz property in monomial algebras $A_\Delta(a)=\mathbb{K}[x_1,\dots,x_n]/(I_\Delta+(x_1^a,\dots,x_n^a))$ and to balanced simplicial spheres. It establishes a close connection between 2-unexpected sops and balanced spheres, with the colored sop providing a canonical example via Macaulay duality, and it introduces analytic-spread methods and half-hollow edgewise subdivisions to study rank conditions of multiplication maps through Rees-algebra data. The paper then derives nonexistence results for unexpected sops in collapsible complexes and applies these ideas to characterize the WLP for 1-dimensional complexes (graphs) in terms of a related subdivision’s bipartiteness, tying combinatorial topology to Lefschetz properties. The results open avenues for further exploration of SLP in special parameter systems, completions of the g-conjecture landscape, and a deeper Rees-algebra perspective on subdivision based constructions.

Abstract

In 2018, Cook, Harbourne, Migliore and Nagel introduced the concept of unexpected hypersurfaces, which connects the study of Lefschetz properties of artinian algebras defined by powers of linear forms, to a family of interpolation problems. In this paper, inspired by the theory of unexpected hypersurfaces, we introduce the concept of unexpected systems of parameters for squarefree monomial ideals. Similarly to the setting of points, we show that the existence of an unexpected system of parameters causes a certain algebra to fail the weak Lefschetz property. We then explore combinatorial interpretations of unexpected systems of parameters, and show that this notion is intrinsically related to the theory of balanced complexes. A consequence of our results is that the theory of Rees algebras turns out to be a powerful tool for studying the existence of systems of parameters satisfying special properties.

Figures (4)

• Figure 1: A complex $\Delta$ on the left, and its incidence complex $\Delta(2)$ on the right.
• Figure 2: The $4$-fold half-hollow edgewise subdivision of a $2$-simplex
• Figure 3: The complex $\Delta$ on the left, and the complex $\mathop{\mathrm{hesd}}\nolimits(\Delta(2), 2)$ on the right. The lines from vertices to facets correspond to the injection from \ref{['t:collapsiblespread']} that shows the log matrix of $\mathcal{F}(\mathop{\mathrm{hesd}}\nolimits(\Delta(2), 2))$ is upper triangular. Note that for every line, the vertex $i$ is contained only in facets corresponding to lower values of $i$, for example, the vertex $3$ is also contained in the facet containing the vertex $1$.
• Figure 4: A labeling of the vertices and facets of $\mathop{\mathrm{hesd}}\nolimits(\Delta(2), 2)$, where $\Delta$ is the complex from \ref{['f:1']} and $\mathcal{F}(\Delta) = (abd, bde, bce, def)$, that shows the log-matrix of $\mathcal{F}(\mathop{\mathrm{hesd}}\nolimits(\Delta(2), 2))$ is the multiplication map $\times L: A_\Delta(3)_{4} \to A_\Delta(3)_5$. Note that applying $\times L^T$ to the monomial label of a triangle yields the sum of the labels of its vertices. As an example, the triangle labeled by $a^2 b^2 d$ corresponds to the vector $\mathbf{a} = \mathbf{e}_{\{a,b\}}$ and its vertices labeled by $a^2 b^2$, $a^2 b d$ and $a b^2 d$ correspond to the vectors $2\mathbf{e}_{\{a,b\}}$, $\mathbf{e}_{\{a,b\}} + \mathbf{e}_{\{a,d\}}$ and $\mathbf{e}_{\{a,b\}} + \mathbf{e}_{\{b,d\}}$.

Theorems & Definitions (54)

• Theorem 1.2: \ref{['t:coloredunexpected', 'c:unexpectedcolored']}
• Lemma 1.3: \ref{['l:macaulaydualcolored']}
• Corollary 1.4: \ref{['c:nonexistence']}
• Theorem 2.1: KN2016,J2002
• proof
• Theorem 2.2: S1996B
• Example 2.3
• Theorem 2.4: Macaulay duality
• Example 2.5
• Theorem 3.1: H2025B