Hilbert functions over toric rings (Q841539)

The present paper focuses on the interesting problem of extending important results on Hilbert functions to non-polynomial rings. As prominent examples of these results the authors cite \textit{F. S. Macaulay}'s theorem [Proceedings L. M. S. (2) 26, 531--555 (1927; JFM 53.0104.01)], Bigatti-Hullet-Pardue's extensions of it [\textit{A. M. Bigatti}, Commun. Algebra 21, No. 7, 2317--2334 (1993; Zbl 0817.13007); \textit{H. A. Hulett}, Commun. Algebra 21, No. 7, 2335--2350 (1993; Zbl 0817.13006); \textit{K. Pardue}, Ill. J. Math. 40, No. 4, 564--585 (1996; Zbl 0903.13004)] and \textit{R. Hartshorne}'s result on the connectedness of the Hilbert scheme [Publ. Math., Inst. Hautes Étud. Sci. 29, 5--48 (1966; Zbl 0171.41502)]. Some of these results have been studied in the Clements-Lidström ring and in the particular case \(\mathbf{k}[x_1,\dots,x_n]/(x_1^2,\dots,x_n^2)\). The goal of the paper is to build analogues of these results over projective toric rings. The first main result of the paper is Theorem 2.5 which states that for every homogeneous ideal \(P\) in a projective toric ring \(R\) there exists a monomial ideal \(M\) in \(R\) with the same Hilbert function and such that the Betti numbers of \(M\) are greater or equal to those of \(P\). In section 3 of the article the authors give a definition of lex ideal in a projective toric ring. For this definition it is crucial that every initial lex-segment generates an initial lex-segment in the next degree. This fact is proved in Theorem 3.4. Section 4 discusses several important open problems on the topic of the paper. The first one, Problem 4.1, is to identify projective toric rings for which Macaulay's theorem holds. The second problem, Problem 4.4, is the generalization of the extremal behaviour of the Betti numbers of lex ideals in the polynomial ring to projective toric rings. Problems 4.7 and 4.9 focus on the structure of minimal free resolutions and the Hilbert scheme respectively. The paper finishes with the example of rational normal curves, for which the authors prove that Macaulay's theorem holds (Theorem 5.1) and study the structure of minimal free resolutions over them (Theorems 5.5 and 5.7).