Another proof of a theorem of Caviglia and De Stefani concerning the Eisenbud-Green-Harris conjecture (Q6594890)

Let \(R = K[x_1,\dots,x_n]\) be the graded polynomial ring, where \(K\) is a field. Let \(I\) be a homogeneous ideal containing a regular sequence \(f_1,\dots,f_r\) of degrees \(2 \leq a_1 \leq \dots \leq a_r\). The Eisenbud-Green-Harris (EGH) Conjecture states that there is another homogeneous ideal \(J\) containing the polynomials \(x_1^{a_1}, \dots, x_r^{a_r}\) with the same Hilbert function. Usually \(J\) is taken to be a so-called Lex-Plus-Powers (LPP) ideal. Quoting the author, ``The conjecture is motivated by Cayley-Bacharach theorems and suggests a refinement of Macaulay's bound on the growth of the Hilbert function by involving the information of degrees of the regular sequence contained in the ideal.'' A very good reference is the paper [Assoc. Women Math. Ser. 29, 327--342 (2022; Zbl 1496.13026)] by \textit{S. Güntürkün}. The conjecture remains open, with only a few special cases known. In [Math. Res. Lett. 15, No. 2--3, 427--433 (2008; Zbl 1154.13001)], \textit{G. Caviglia} and \textit{D. Maclagan} proved the EGH Conjecture when \(a_i > \sum_{j=1}^{i-1} a_j -1\) for all \(2 \leq i \leq r\). In other words, each new element of the regular sequence has to have quite large degree compared to the earlier elements. Very recently Caviglia and De Stefani extended it by including the equality in the hypothesis for \(a_i\). The purpose of the paper under review is to give a new proof of this latter result. The main ingredients are the Clements-Lindström theorem, the fact that the EGH Conjecture remains true after linking via a complete intersection, and a result of the author that the conjecture is true when each polynomial of the regular sequence splits into linear factors.