Ordinary and symbolic Rees algebras for ideals of Fermat point configurations

Abstract

Abstract. Fermat ideals define planar point configurations that are closely related to the intersection locus of the members of a specific pencil of curves. These ideals have gained recent popularity as counterexamples to some proposed containments between symbolic and ordinary powers [DST]. We give a systematic treatment of the family of Fermat ideals, describing explicitly the minimal generators and the minimal free resolutions of all their ordinary powers as well as many symbolic powers. We use these to study the ordinary and the symbolic Rees algebra of Fermat ideals. Specifically, we show that the symbolic Rees algebras of Fermat ideals are Noetherian. Along the way, we give formulas for the Castelnuovo-Mumford regularity of powers of Fermat ideals and we determine their reduction ideals.

Fermat-type configurations of lines in P3 and the containment problem

“…Both these beautiful books are focused on surfaces of general type arising as ball quotients. In the area of commutative algebra the name Fermat arrangement seems more customary, see [14].…”

“…A completely different proof based on a more general theoretical framework has been presented by Seceleanu, see [15,Proposition 4.2]. Additionally, idealtheoretic properties of these configurations have been treated by Nagel and Seceleanu in [14].…”

On Codimension Two Flats in Fermat-Type Arrangements

“…Note that we suppress the notation and write x i rather than x i . We will now analyze how the monomial m appears on the right hand side of (11). To this end we run the following procedure starting with the variables with least powers in m.…”

“…Fermat arrangements of lines have attracted recently considerable attention, see e.g. [11], because of their appearance on the border line of the following fundamental problem.…”

A homological criterion for the containment between symbolic and ordinary powers of some ideals of points in P2

“…The idea of using Rees algebra techniques to provide an explicit description of minimal free resolutions of all ordinary powers of a uniformly 3-generated ideal of points in P 2 will be exploited further in [19]. In particular, a formula for the minimal free resolutions and for the Castelnuovo-Mumford regularity of all ordinary powers of a uniformly 3-generated ideal of points in P 2 will appear in full detail in [19]. For the purposes of this note, we only require knowledge of the resolutions of I 2 and I 3 and a good command of the maps appearing therein, as illustrated in Proposition 2.3.…”