ABSTRACT
Contents 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 6.2 Maximal Growth of the Hilbert Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 6.3 UPP and WLP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 6.4 Setting the Stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 6.5 General Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 6.6 Results on Uniform Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 6.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
6.1 Introduction The title of this chapter, “The geometry of Hilbert functions,” might better
be suited for a multivolume treatise than for a single short chapter. Indeed, a large part of the beauty of, and interest in, Hilbert functions derives from their ubiquity in all of commutative algebra and algebraic geometry, and the unexpected information that they can give, very much of it expressible in a geometric way. Most of this chapter is devoted to describing just one small facet of this theory, which connects results of Davis (see [Davis]) in the 1980s, of Bigatti, Geramita, and myself (see [Bigatti-Geramita-Migliore]) in the 1990s, and of Ahn and myself (see [Ahn-Migliore]) very recently. On the other hand, we have an alphabet soup of topics that play a role here: UPP, WLP, SLP, ACM, at the very least. It is interesting to see the ways in which these properties interact, and we also try to illustrate some aspects of this.
