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Interpolation and the weak Lefschetz property. (English) Zbl 1426.13008

In the paper under review, the authors study several problems which are related, in a broad sense, with the Hermite interpolation. Assume that \(\mathbb{K}\) is a field. In the mentioned interpolation problem one wants to determine the least degree of a homogeneous form that, given a set of \(s\) points \(Z\) in \(\mathbb{P}^{N}\) and positive integers \(m_{1},\dots,m_{s}\), vanishes to order \(m_{i}\) at the point \(P_{i}\) for every \(i\).
This problem is very difficult and it remains open if we restrict our attention to the case \(k=m_{1} = \dots = m_{s}\). In this case, the polynomials vanishing to order \(k\) at every point of \(Z\) form an ideal \(I^{(k)}_{Z}\) that is called \(k\)-th symbolic power of the ideal \(I_{Z}\) of \(Z\), and the least degree of a non-zero element from \(I^{(k)}_{Z}\) is denoted by \(\alpha(I^{(k)}_{Z})\). In order to formulate the main results let us recall that a finite set \(Z = \{P_{0}, \dots, P_{s}\} \subset \mathbb{P}^{N}\) of points is said to be in linearly general position if any subset of \(Z\) with \(r+1 \leq N+1\) points spans an \(r\)-dimensional linear subspace. We also say that a set of linear forms in \(R = \mathbb{K}[x_{0}, \dots, x_{N}]\) is in linearly general position if the set of dual points in \(\mathbb{P}^{N}\) has this property.
Theorem A. Let \(Z \subset \mathbb{P}^{N}\) be a subset of \(N+2\) points spanning \(\mathbb{P}^{N}\). Denote by \(\ell_{1}, \dots, \ell_{N+2}\) linear forms that are dual to the given points. Let \(t\) be the least integer such that some subset of \(t+2\) points \(Z\) is linearly dependent, so \(1 \leq t \leq N\). Then for every \(d > 0\) and \(k > 0\) one has \[\operatorname{reg} R/(\ell_{1}^{d}, \dots, \ell_{N+2}^{d}) = \bigg\lfloor \frac{(2N+2-t)(d-1)}{2}\bigg\rfloor,\] \[\alpha(I_{Z}^{(k)}) = \bigg\lceil \frac{(2N+2-t)k}{2N-t}\bigg\rceil,\] where \(\operatorname{reg}(\cdot)\) denotes the Castelnuovo-Mumford regularity.
An analogous result was obtained for subsets of \(N+3\) points in \(\mathbb{P}^{N}\) in linearly general position, but a formulation of that result is much more complicated.
The next result is devoted to Demailly’s conjecture. Let us recall that the Waldschmidt constant is defined as \[ \hat{\alpha}(I) = \lim_{k \rightarrow \infty} \frac{\alpha(I^{(k)})}{k}, \] where \(I\) is a non-zero homogeneous ideal.
Conjecture. Any finite set of points \(Z \subset \mathbb{P}^{N}\) satisfies \[ \hat{\alpha}(I_{Z}) \geq \frac{\alpha(I_{Z}^{(k)})+N-1}{N+k-1} \] for every \(k \geq 1\).
Theorem B. Let \(Z \subset \mathbb{P}^{N}\) be a set of at most \(N+3\) points spanning \(\mathbb{P}^{N}\). If \(|Z| = N+3\), suppose that \(Z\) is in linearly general position. Then Demailly’s conjecture is true for \(Z\) and every \(k\geq 1\) unless \(|Z|=N+3\) and \(N\) is odd. In the latter case, Demailly’s conjecture holds whenever \(k \geq \frac{(N^{2}+N+1)(N^{2}+2N-1)}{2(N+2)}\).
The last part of the paper is devoted to the weak Lefschetz property. Let us recall that a graded Artinian algebra \(A\) is said to have the weak Lefschetz property (WLP) if it has a linear form \(\ell\) such that multiplication by \(\ell\) on \(A\) has maximal rank from each degree to the next.
Theorem C. The algebra \(R/(x_{0}^{d}, \dots, x_{N}^{d}, L^{d})\), where \(L \in R\) is a general linear form, fails to have the WLP if \(N\geq 8\) is even and \(d-2 \gg 0\) is divisible by \(N+1\).
Theorem D. Let \(N \gg 0\) be an integer and consider the ideal \(I(x_{0}^{d}, \dots, x_{N}^{d},L^{d})\) of the polynomial ring \(R = \mathbb{K}[x_{0}, \dots, x_{N}]\), where \(L \in R\) is a general linear form. Then the ring \(R/I\) fails the WLP for every \(d \gg 0\).

MSC:

13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
14C20 Divisors, linear systems, invertible sheaves
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13D02 Syzygies, resolutions, complexes and commutative rings
14N20 Configurations and arrangements of linear subspaces
05A10 Factorials, binomial coefficients, combinatorial functions
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References:

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