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On Segre’s bound for fat points in \(\mathbb{P}^n\). (English) Zbl 1332.13016

Let \(S = \{p_1, \dots, p_s\}\) be a set of distinct points in \({\mathbb{P}}^n = {\mathbb{P}}^n_K\) and let \(\wp_1,\dots, \wp_s\) be the associated homogeneous prime ideals in the polynomial ring \(R := K[x_0, \dots , x_n]\), where \(K\) is an algebraically closed field. Given positive integers \(m_1,\dots ,m_s\) we denote by \(Z :=\sum^s_{i=1} m_i\wp_i\) the \(0\)-dimensional subscheme of \({\mathbb{P}}^n \) defined by the saturated ideal \(I_Z := \wp_1^{m_1}\cap\dots\cap \wp_s^{m_s}\). We denote by \(Z_{\mathrm{red}} := \sum^s_{i=1} p_i\) the support of \(Z\) and by \(w(Z) := \sum^s_{i=1}m_i\) its weight.
The regularity index \(\mathrm{reg}(Z)\) of \(Z\) is the smallest positive integer \(d\) such that \(h_1({\mathbb{P}}^n, I_Z(d)) = 0\) or, equivalently, \(h_1({\mathbb{P}}^n,{L}_{n,d}(m_1,\dots ,m_s)) = 0\) where \({L}_{n,d}(m_1,\dots ,m_s)\) is the linear systems of the degree-\(d\) hypersurfaces of \({\mathbb{P}}^n\) passing through each point \(p_i\) with multiplicity at least \(m_i\), for all \(d\geq 0\). This number corresponds to the Castelnuovo-Mumford regularity of the Cohen-Macaulay graded ring \(R/I_Z\).
For arbitrary fat points in \({\mathbb{P}}^n\), G. Fatabbi and A. Lorenzini [J. Pure Appl. Algebra 161, No. 1–2, 91–111 (2001; Zbl 0994.14028)] gave the following conjecture for the regularity index.
{ Conjecture 1}. For \(r = 1,\dots , n\) and for any linear \(r\)-subspace \(L\) of \({\mathbb{P}}^n\), set \(T(Z,L) := \lfloor\frac{ w_L(Z) + r - 2}{r}\rfloor\). Then \(\mathrm{reg}(Z) \leq \max\{T(Z, L) : L \subset{\mathbb{P}}^n\}\).
In Section 2 the authors prove that Conjecture 1 holds for schemes with \(n+3\) arbitrary fat points of \({\mathbb{P}}^n\).
In Section 3, Theorem 3.6 improves Segre’s bound \(\mathrm{reg}(Z)\leq \max\{m_1+m_2-1,\lfloor\frac{ w_L(Z) + n - 2}{n}\rfloor\) for fat points \(Z\) in general position in \({\mathbb{P}}^n\).
In Section 4, the authors pose a modification of the Segre conjecture for the regularity index of a scheme of fat points, \(\mathrm{reg}(Z)\), and prove it holds for \(n = 3\).

MSC:

13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
14C20 Divisors, linear systems, invertible sheaves

Citations:

Zbl 0994.14028

Software:

Macaulay2
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References:

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