A numerical semigroup \(S\) is an additive subsemigroup of the non-negative integers with finite complement. Throughout, \(S\) is assumed to have minimal generating set \(\{n_1, \ldots, n_d\}\). For \(m \in S\), the squarefree divisor complex \(\Delta_m^S\) of \(m\) in \(S\) is the simplicial complex on \([d] = \{1, \ldots, d\}\) whose faces are \[ \{ F \in [d] \ \mid \ m - n_F \in S \}, \] where \(n_F = \sum\limits_{i \in F} n_i\). Squarefree divisor complexes, also called (upper) Koszul complexes, were introduced in [W. Bruns and J. Herzog, J. Pure Appl. Algebra 122, No. 3, 185–208 (1997; Zbl 0884.13006)] in the context of semigroup rings to study multigraded Betti numbers. In particular, for numerical semigroup rings, they appear in the Hilbert series of \(S\) via the formula \[ \mathcal{H}(S;t) = \sum\limits_{m \in S} t^m = \frac{\sum_{m \in S} \chi(\Delta_m) t^m}{(1-t^{n_1})\cdots(1-t^{n_d})}, \] where \(\chi(\Delta_m) = \sum\limits_{F \in \Delta_m} (-1)^{|F|}\) is the Euler characteristic of \(\Delta_m\).
This paper accomplishes two main tasks. First, it produces a family of simplicial complexes called fat forests that can be realized as squarefree divisor complexes [Corollary 2.8] via a novel iterative method. Fat trees are simplicial complexes whose facets \(F_1, F_2, \ldots\) can be ordered so that \(F_j \cap \bigcup_{i < j} F_i\) is a simplex; fat forests are disjoint unions of fat trees. The method is to show a method for creating fat trees as squarefree divisor complexes by adding specific elements to the semigroup \(S\) [Theorem 2.6], then showing how to combine semigroups \(S\) and \(S'\) to form a new semigroup \(T\) so that the squarefree divisor complex \(\Delta_{kk'}^T\) is a disjoint union of \(\Delta_k^S\) and \(\Delta_{k'}^{S'}\) [Theorem 2.4].
Second, the article investigates two particular classes of numerical semigroups: supersymmetric numerical semigroups and numerical semigroups with embedding dimension \(3\). For supersymmetric \(S\), the authors identify a special class of elements of \(S\) whose squarefree divisor complexes have nonzero Euler characteristic [Theorem 3.1], then show that these are in fact the only \(m\) where \(\Delta_m\) has nonzero Euler characteristic. For \(S = \langle n_1, n_2, n_d \rangle\), the authors classify the squarefree divisor complexes with nonzero Euler characteristic [Theorem 3.5].