Let \(k\) be an algebraically closed field, let \(S=k[x_0,\dots,x_n]\) and let \(Z\) be a general, non-degenerate set of \(s\) points in \(\mathbb{P}^n=\mathbb{P}_k^n\). Consider the minimal free resolution of the homogeneous ideal \(I_Z\) of \(Z\): \(0\to L_{n-1}\to\cdots\to L_0\to I_Z\to 0\). The generality of \(Z\) implies that \[ L_p=S(-d-p-1)^{a_p}\oplus S(-d-p)^{b_p}, \] \(a_p=h^1(\Omega^{p+1}(d+p+1)\otimes{\mathcal I}_Z)\) and \(b_p = h^0(\Omega^p(d+p)\otimes {\mathcal I}_Z)\), where \(d\) is the unique positive integer satisfying \({d+n-1\choose n}\leq s<{d+n\choose n}\). The “minimal resolution conjecture” (MRC) of A. Lorenzini [J.Algebra 156, No. 1, 5-35 (1993; Zbl 0811.13008)] states, briefly, that for \(Z\) sufficiently general, and for all \(p\geq 1\), we have \(a_{p-1} \cdot b_p = 0\). This conjecture has been verified in \(\mathbb{P}^2\) and \(\mathbb{P}^3\) in a number of papers. It has been proven false for some values of \(s\) beginning in \(\mathbb{P}^6\), first computationally by Schreyer and then proven false rigorously, and put in a more general framework by D. Eisenbud and Popescu [“Gale duality and free resolutions of ideals of points” (preprint)]. On the other hand, A. Hirschowitz and C. Simpson [Invent. Math. 126, No. 3, 467-503 (1996; Zbl 0877.14035)] have shown that for all \(n\), and for \(s\) large enough with respect to \(n\), a sufficiently general set \(Z\) of \(s\) points has the resolution predicted by the MRC.
Still, the problems with the MRC seem to come in the middle of the resolution, and one can try to prove it for either end of the resolution. In particular, the predicted rank of \(L_{n-1}\) is called the “Cohen-Macaulay type conjecture”. A proof was given by Ngo Viet Trung and G. Valla [J. Algebra 125, No. 1, 110-119 (1989; Zbl 0701.14042)], but a gap was pointed out by C. Walter. The paper under review proves this latter conjecture. The MRC is really a maximal rank statement, and this is reflected in the fact that the main result of this paper is the following:
Let \(Z\) be as above, and let \(T_{\mathbb{P}^n}\) be the tangent bundle of \(\mathbb{P}^n\). For every integer \(\ell\), the restriction map \(H^0(\mathbb{P}^n,T_{\mathbb{P}^n}(\ell)) \to H^0(Z,T_{{\mathbb{P}}^n} (\ell)_{| Z})\) has maximal rank.
The proof uses the so-called “méthode d’Horace” in different formulations, and the connection to the MRC is made.