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Powers of generic ideals and the weak Lefschetz property for powers of some monomial complete intersections. (English) Zbl 1386.13047

Let \(I = (f_1,\dots, f_r) \subset \mathbb S := C[x_1,\dots,x_n]\) be an ideal generated by forms of degree \(d\) (or more generally \(d_1,\dots,d_r\), but in this paper the \(d_i\) are equal). Fröberg’s conjecture deals with the question of how small the Hilbert function of \(S/I\) can be. It gives a precise prediction for the Hilbert function when the \(f_i\) are general, and if true, it is known that this is the smallest possible. Here instead the authors consider the Hilbert function of \(S/I^k\), when the generators are general, and ask the same question. This is known when \(r \leq n\) and \(I\) is a complete intersection, but it is difficult in general. For \(k=1\) the result is known when \(n \leq 3\) or when \(r = n+1\). Here the authors consider these cases (\(n \leq 3\) or \(r = n+1\)) for larger \(k\). In part they consider \(n=2,3\). A natural guess is that for \(k \gg 1\) and \(r>n\), the Hilbert series of \(S/I^k\) is the same as that of \(\binom{k+r-1}{r-1}\) general forms of degree \(dk\). The authors show that this is not always the case, using a geometric argument. They also study the related problem of the Weak Lefschetz Property for \(S/I^k\), for certain monomial complete intersection ideals \(I\). They give a lot of computer-generated data from Macaulay2.

MSC:

13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13A02 Graded rings
13E10 Commutative Artinian rings and modules, finite-dimensional algebras
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)

Software:

Macaulay2
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Full Text: DOI arXiv

References:

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