Rigid ideals by deforming quadratic letterplace ideals

Abstract

We compute the deformation space of quadratic letterplace ideals L(2, P ) of finite posets P when its Hasse diagram is a rooted tree. These deformations are unobstructed. The deformed family has a polynomial ring as the base ring. The ideal J(2, P ) defining the full family of deformations is a rigid ideal and we compute it explicitly. In simple example cases J(2, P ) is the ideal of maximal minors of a generic matrix, the Pfaffians of a skew-symmetric matrix, and a ladder determinantal ideal.Date: September 21, 2018.

Letterplace and co-letterplace ideals of posets

“…These ideals therefore have the flavor of beeing "free" objects in the class of monomial ideals. We see this as accounting for many of their nice properties, see [8] and [16].…”

“…The notion of separation is further investigated in [24] and in [1], which shows that separation corresponds to a deformation of the monomial ideal, and identifies the deformation directions in the cotangent cohomology it corresponds to. In [16] deformations of letterplace ideals L(2, P ) are computed when the Hasse diagram has the structure of a rooted tree. The situation is remarkably nice.…”

Polarizations of powers of graded maximal ideals

“…Deformations of polarizations. In [13] the second author and A.Nematbakhsh showed that the letterplace ideals L(2, P ) (which are polarizations of quadratic Artinian monomial ideals) have unobstructed deformations when the Hasse diagram is a tree. Moreover we computed the full deformation family of these ideals.…”

“…We shall go through several steps in proving the above theorem. It turns out that we will be able to abstract the situation so our arguments will only involve a collection of isotone maps (13) χ…”

Poset Ideals of P-Partitions and Generalized Letterplace and Determinantal Ideals

“…When α is the constant function α(p) = n + 1, then the principal coletterplace ideal L(P, α) is the co-letterplace ideal L(P, n + 1) of [6] (note that we get n + 1 here since in [6] the chain [n] starts with 1 but here N starts with 0, so the chain [n + 1] is isomorphic to the chain [0, n]), and the Alexander dual L(α, P ) is the letterplace ideal L(n + 1, P ), see Proposition 4.7 below. In [7] the author and A. Nematbakhsh computes the full deformation families of the letterplace ideals L(2, P ) when the Hasse diagram of P is a rooted tree. The deformation theory, in analog with complete intersections, is extremely nice.…”