Fat Points, Inverse Systems, and Piecewise Polynomial Functions

Abstract

ŽWe explore the connection between ideals of fat points which correspond to n Ž . . subschemes of ‫ސ‬ obtained by intersecting mixed powers of ideals of points , and Ž . piecewise polynomial functions splines on a d-dimensional simplicial complex ⌬ d w x embedded in R . Using the inverse system approach introduced by Macaulay 11 , we give a complete characterization of the free resolutions possible for ideals in w x Ž k x, y generated by powers of homogeneous linear forms we allow the powers to . differ . We show how ideals generated by powers of homogeneous linear forms are related to the question of determining, for some fixed ⌬, the dimension of the vector space of splines on ⌬ of degree less than or equal to k. We use this relationship and the results above to derive a formula which gives the number of Ž . planar mixed splines in sufficiently high degree. ᮊ

Inverse systems, Gelfand-Tsetlin patterns and the weak Lefschetz property

“…by equation (3) and h 0 (S(I)(m)) = h 1 (D m ) = 0 by equation (9). Now, by equation (10), we have an exact sequence Lemma 4.6.…”

“…Whenever h 0 (S(I)(t + 1)) < h 0 (S(I )(t + 1)), we thus see that μ fails to be injective. This is precisely what occurs if (r, t, n) ∈ {(4, 3, 5), (5,3,9), (6,3,14), (6, 2, 7)}. For example, let (r, t, n) = (4, 3, 5) and consider the divi-…”

“…As a consequence of the results in Section 2, we obtain information on the WLP for quotients A by ideals of powers of general linear forms in cases where Theorem 3.1 applies. (5,3,9), (6,3,14), (6, 2, 7)}, then the map μ : A t → A t+1 has full rank.…”

“…As pointed out to us by Tony Iarrobino, Hochster-Laksov [14] obtain a similar result for general forms. (5,3,9), (6,3,14), (6, 2, 7)}.…”

Algebraic Stories from One and from the Other Pockets

“…In [Lundqvist et al (2017), Lemma 2.2], it was also shown that Conjecture 2.4 holds in the case of binary forms. The proof is obtained by specializing each one of the g i 's to be the d-th power of a generic linear form and applying the fact that generic power ideals in two variables have the generic Hilbert series Geramita and Schenck (1998). It is worth to mention that the same idea as in [Lundqvist et al (2017), Lemma 2.2] gives a positive answer to Problem F in the case of binary forms, by specializing i,1 = .…”