Face vectors of two-dimensional Buchsbaum complexes (Q1028852)

Let \(\Delta\) be a simplicial complex on \([n]=\{1,2,\dots,n\}\). The link of \(\Delta\) with respect to \(F \in \Delta\) is the simplicial complex \(\text{lk}_{\Delta}(F)=\{G \subset [n] \setminus F: G \cup F \in \Delta\}\). The complex \(\Delta\) is said to be Buchsbaum if it is pure and \(\text{lk}_{\Delta}(v)\) is Cohen-Macaulay for every vertex \(v\) of \(\Delta\). An interesting open problem is to characterize all possible \(h\)-vectors of \((d-1)\)-dimensional Buchsbaum complexes. In 1996, \textit{N. Terai} [Hokkaido Math. J. 25, 137--148 (1996; Zbl 0867.13005)] proposed a conjecture on the characterization of \(h\)-vectors of Buchsbaum complexes of a special type including all \(2\)-dimensional connected Buchsbaum complexes, and proved the necessity of the conjecture. The main result of this paper is the proof of the sufficiency of Terai's conjecture for \(2\)-dimensional Buchsbaum complexes.