The many polarizations of powers of maximal ideals

Abstract

In this paper, we study different polarizations of powers of the maximal ideal, and polarizations of their related square-free versions.For n = 3, we show that every minimal free cellular resolution of m d comes from a certain polarization of the ideal m d . This result is not true for n = 4. When I is a square-free ideal, we show that the Alexander dual of any polarization of I is a polarization of the Alexander dual ideal of I. We apply this theorem, and study different polarizations of the ideals m d sq.fr and their Alexander duals m n−d+1 sq.fr simultaneously, by giving a combinatorial description corresponding to such polarizations, with a natural dualization. We apply this theory, and study the case of d = 2 and d = n − 1 in more detail. Here, we show that there is a oneto-one correspondence between spanning trees of Kn and the maximal polarizations of these ideals.

Deformations of polarizations of powers of the maximal ideal

“…Remark 5.10. In [6], we show that the box polarization corresponds to the line graph, while the standard polarization corresponds to the star graph. One might therefore assume that these two polarizations would give the highest and lowest dimensional tangent space in the Hilbert scheme.…”

“…However, when d = 2, then all maximal polarizations lie in the same polynomial ring. In this case, we also have that every maximal polarization of m 2 is also a maximal polarization of m sq.fr 2 (see [6,Proposition 4.7]). We will study the deformations of the ideals in this case.…”

“…. This ideal have generally many more polarizations than the ideal m d , and the combinatorics describing these polarizations are studied in [6]. In this setting, and when d ≥ 3, we don't have such a nice choice of polynomial ring where all the polarizations lie.…”

Bi-Cohen-Macaulay graphs

“…In order to state the main result of this paper we recall the concept of inseparability introduced by Fløystad et al in [7], see also [12].…”

“…Each monomial ideal admits an inseparable model, but in general not only one. For example, the separable models of the powers of the graded maximal ideal of S have been considered by Lohne [12].…”