Gotzmann squarefree ideals (Q384312)

Let \(S=\mathbb{K}[x_1,\ldots,x_n]\) be a polynomial ring over an arbitrary field \(\mathbb{K}\), \(I\subset S\) be an ideal of \(S\) and let \(R=S/(x_1^2,\ldots,x_n^2)\). The Hilbert function of \(I\) is defined as NEWLINE\[NEWLINEH(I,d)=\dim_{\mathbb{K}}I_d.NEWLINE\]NEWLINE A homogeneous ideal \(I\subset S\) is called Gotzmann if for each graded component \(I_d\) and every ideal \(J\) with \(H(I,d)=H(J,d)\) we have that NEWLINE\[NEWLINEH(mI,d+1)\leq H(mJ,d+1).NEWLINE\]NEWLINE The authors classify all squarefree monomial ideals which are Gotzmann. More especially, in their main theorem they are proving that a squarefree monomial ideal is Gotzmann if and only if the ideal \(I\) has the form NEWLINE\[NEWLINEI=m_1(x_{i_{1,1}},\ldots,x_{i_{1,r_1}})+m_1m_2(x_{i_{2,1}},\ldots,x_{i_{2,r_2}})+\cdots+m_1\ldots m_s(x_{s_{s,1}},\ldots,x_{i_{s,r_s}}),NEWLINE\]NEWLINE where \(m_1,\ldots,m_s\) are squarefree monomials and \(x_{i,j}\) are variables which all have pairwise disjoint support in \(S\).NEWLINENEWLINEIn the general case the problem of classifying Gotzmann monomial ideals of \(S\) is more difficult. In this case certain decomposition and reconstruction results are given by the authors. They conclude that a Gotzmann ideal has Gotzmann Alexander dual if and only if all of its degreewise components are lex in some order.