On the regularity index of \(n+3\) almost equimultiple fat points in \(\mathbb P^n\) (Q2846884)

The aim of this paper is to prove a well-known conjecture on the regularity index of \(n+3\) almost equimultiple fat points in \(\mathbb{P}^n\).NEWLINENEWLINELet \( R = k[x_0 ,\dots, x_n]\) be the standard graded polynomial ring over an algebraically closed field \(k\).NEWLINENEWLINELet \( X = \{P_1, \dots , P_s\} \subset \mathbb{P}^n\) be a set of \(s\) distinct points, and let \(m_1,\ldots,m_s\) be positive integers. Denote \(Z:=m_1P_1+\ldots+m_sP_s\) the zero-scheme corresponding to the ideal of all forms of \(R\) vanishing at \(P_i\) with multiplicity at least \(m_i\), for \(i=1,\ldots,s.\) This zero-scheme \(Z\) is called a set of fat points and its defining ideal is \(I_Z =\wp_1^{m_1} \cap \dots \cap \wp_s^{m_s}\).NEWLINENEWLINEIt is well known that the Hilbert function \(H_{R/I}(t) := \dim_{k}(R/I)_t=\dim_{k}R_t-\dim_{k}I_t\) is strictly increasing until it reaches the multiplicity \({e(R/I)}: =\sum_{i=1}^{s}{{m_i+n-1}\choose{n}}\) at which it stabilizes. The least integer \(t\) for which \(H_{R/I}(t)= {e(R/I)}\) is called the regularity index of \(Z\) and it is denoted by \(\mathrm{reg}(Z)\). It is well-known that \(\mathrm{reg}(Z)\) is equal to the Castelnuovo regularity index \(\mathrm{reg}(R/I)\).NEWLINENEWLINEIn 1996, N. V. Trung formulated the following conjecture (independently given also by G. Fatabbi and A. Lorenzini):NEWLINENEWLINEConjecture. Let \(Z=m_1P_1+\ldots+m_sP_s\) be a set of fat points in \(\mathbb{P}^n\). For \(j=1,\ldots,n\), let NEWLINE\[NEWLINET_j:=\max\{[{\frac {1}{n}}(m_{i_1}+m_{i_2}+\ldots+m_{i_q}+j-2)]| P_{i_1},P_{i_2},\dots,P_{i_q} \text{ lie~ on~ a \(j\)-plane}\}NEWLINE\]NEWLINE then \(\mathrm{reg}(Z)\leq T_{Z}:=\max\{T_j|j=1,\dots,n\}\).NEWLINENEWLINEA set of fat points \(Z\) is called almost equimultiple if the multiplicities of the points are equal to \(m\) or \(m-1\) for a given integer \(m\geq 2\). In this paper the authors prove the conjecture for any set of \(n+3\) almost equimultiple, non degenerate fat points in \(\mathbb{P}^n\). The authors also show several results which will be used to prove the conjecture, and explain why the case of \(n+3\) fat points is more complicated than the case of \(n+2\) fat points.