[Paper Review] On the Eisenbud-Green-Harris Conjecture

Abed Abedelfatah|arXiv (Cornell University)|Dec 11, 2012

Commutative Algebra and Its Applications10 references4 citations

TL;DR

This paper proves the Eisenbud-Green-Harris (EGH) Conjecture for homogeneous ideals containing a regular sequence where each $ f_i $ splits into linear factors. Using a graded isomorphism to reduce to monomials, the author constructs a lex-plus-powers ideal with the same Hilbert function as the original ideal, confirming the conjecture in this special case via dimension growth analysis and Hilbert function recursion.

ABSTRACT

It has been conjectured by Eisenbud, Green and Harris that if $I$ is a homogeneous ideal in $k[x_1,...,x_n]$ containing a regular sequence $f_1,...,f_n$ of degrees $°(f_i)=a_i$, where $2\leq a_1\leq ... \leq a_n$, then there is a homogeneous ideal $J$ containing $x_1^{a_1},...,x_n^{a_n}$ with the same Hilbert function. In this paper we prove the Eisenbud-Green-Harris conjecture when $f_i$ splits into linear factors for all $i$.

Motivation & Objective

• To resolve the Eisenbud-Green-Harris Conjecture in the case where each $ f_i $ in the regular sequence splits into linear factors.
• To establish that every homogeneous ideal $ I i f_1,\dots,f_n $ with $ \deg(f_i) = a_i $ and $ f_i $ linear-factor-split has the same Hilbert function as an ideal containing $ x_1^{a_1},\dots,x_n^{a_n} $.
• To generalize previous results on Hilbert functions in quotient rings by extending Clements-Lindström's theorem to non-monomial regular sequences.
• To answer a question posed by Chen regarding the validity of the EGH Conjecture when $ f_i = x_i l_i $ with $ l_i \in S_1 $.

Proposed method

• Use a graded isomorphism $ \alpha: S \to S $ with $ \alpha(x_i) = p_i $, where $ p_i $ divides $ f_i $, to reduce the problem to the case where $ f_i = x_i l_i $, preserving Hilbert functions.
• Apply the fact that $ p_1,\dots,p_n $ are $ k $-linearly independent to assume $ f_i = x_i l_i $ with $ l_i \in S_1 $, simplifying the ideal structure.
• Analyze the dimension growth of ideals containing $ x_1 l_1,\dots,x_n l_n $ using recursive decomposition via $ I + \langle x_n \rangle $ and $ (I : x_n) $.
• Construct a target ideal $ K $ in $ S $ by lifting monomial generators from lower-dimensional rings $ R = k[x_1,\dots,x_{n-1}] $, using $ \operatorname{Mon}(L_0) \cup \{w x_n\} \cup \{w x_n^2\} \cup \{x_n^3\} $, where $ L_0, L_1, L_2 $ are lex-plus-powers ideals.
• Verify that the Hilbert function of $ K $ matches that of $ I $ via direct computation in Macaulay2, using the recursive formula $ H(S/I,t) = H(S/(I + \langle x_n \rangle),t) + H(S/(I : x_n),t-1) $.
• Leverage Clements-Lindström’s theorem to ensure that every Hilbert function in $ S/\langle x_1^{a_1},\dots,x_n^{a_n}\rangle $ is realized by a lex-plus-powers ideal.

Experimental results

Research questions

• RQ1 Does the Eisenbud-Green-Harris Conjecture hold when each $ f_i $ in the regular sequence splits into linear factors?
• RQ2 Can the Hilbert function of a homogeneous ideal $ I $ containing $ f_1,\dots,f_n $ with $ \deg(f_i) = a_i $ be matched by a lex-plus-powers ideal containing $ x_1^{a_1},\dots,x_n^{a_n} $ in the linear-factor case?
• RQ3 Is the EGH Conjecture valid for ideals generated by forms of the type $ x_i l_i $ with $ l_i \in S_1 $, as posed by Chen?
• RQ4 Can the recursive structure of $ I + \langle x_n \rangle $ and $ (I : x_n) $ be used to build a target ideal with the same Hilbert function as $ I $?

Key findings

• The EGH Conjecture is proven true when each $ f_i $ in the regular sequence splits into linear factors, extending prior results to this class.
• For $ I = \langle f_1,\dots,f_n \rangle $ with $ f_i = x_i l_i $, $ l_i \in S_1 $, there exists a lex-plus-powers ideal $ K $ such that $ H(S/I) = H(S/K) $, confirming the conjecture in this case.
• The Hilbert function of $ I $ is preserved under the graded isomorphism $ \alpha $, allowing reduction to the monomial case $ x_i l_i $, where $ l_i $ are linear forms.
• Explicit constructions of $ K $ using monomial lifts from lower-dimensional lex-plus-powers ideals $ L_0, L_1, L_2 $ yield the correct Hilbert function, as verified by Macaulay2 computations.
• In Example 4.6, $ H(S/I) = (1,5,8,3,0,\dots) $ is matched by $ H(S/K) $, and in Example 4.7, $ H(S/I) = (1,6,14,13,2,0,\dots) $ is matched by $ H(S/K) $, confirming the construction.
• The recursive structure $ H(S/I,t) = H(S/(I + \langle x_n \rangle),t) + H(S/(I : x_n),t-1) $ is used effectively to build $ K $ with matching Hilbert function.