The \(h\)-vector of the union of two sets of points in the projective plane (Q2904281)

Let \(X\) be a finite set of points in the projective plane \(\mathbb{P}^2_k\) (\(k\) is an algebraically closed field). One considers the associated reduced zero-dimensional subscheme of \(\text{Proj}~k[x_0,x_1,x_2]\). Its homogeneous coordinate ring \(S\) is \(1\)-dimensional, hence by the Hilbert-Serre's theorem, the Hilbert polynomial of \(S\) is constant. Consequently, one has the sequence \(h(X)=(h_0,h_1,\dots)\) where \(h_i=\dim_k S_i-\dim_k S_{i-1}\) where \(i\geq 0\), is eventually zero. If \(\tau(X)=\max\{i:h_i\neq 0\}\), we call \(h(X)=(h_0,\dots,h_{\tau(X)})\) the \textit{\(h\)-vector of \(X\)}.NEWLINENEWLINEThe \textit{length of \(h(X)\)} is \(\tau(X)+1\) and the \textit{height of \(h(X)\)} is \(b(X)=\max\{h_0,\dots,h_{\tau(X)}\}\). In general, \(h(X)\) depends on the geometry of \(X\). For instance, \(X\) has the \(h\)-vector \((1,\dots,1)\) if and only if all the points of \(X\) belong to a line in \(\mathbb{P}^2\). On the other hand, \(h(X)\) restricts the geometric properties of \(X\). For example, the number of points of \(X\) is \(\sum_{j=0}^{\tau(X)}h_j\), and \(\eta(h(X),d)=\sum_{j=0}^{\tau(X)}\min\{h_j,d\}\) is an upper bound for the number of points of \(X\) that belong to a curve of degree \(d\), for each \(d\geq 1\) (Proposition 3.12).NEWLINENEWLINEIn the paper under review, the authors considers the following problem: given two disjoint set \(X, Y\) of finitely many points in \(\mathbb{P}^2\), characterize all \(h\)-vectors which are \(h\)-vector of a reduced scheme which is set-theoretically union of \(X\) and \(Y\). They are able to provide many interesting exclusion criteria for the \(h\)-vector of the union, e.g.~ by bounding the length and height of the \(h\)-vector of the union. They also give many concrete examples to illustrate the resulting exclusion techniques.NEWLINENEWLINEWe can define a partial order \(\leq_g\) on the \(h\)-vectors as follow: \(h=(h_0,\dots,h_m) \leq_g h'=(h'_0,\dots,h'_n)\) if and only if \(\sum_{j=0}^ih_j\leq \sum_{j=0}^ih'_j\) for all \(i=0,\dots,m\) (with the convention \(h'_j=0\) if \(j>n\)). Using this order, the main result states that (Proposition 5.1): For any two given \(h\)-vectors, we can construct the unique minimum (with respect to the introduced order) \(h\)-vector for the union among all the possible ones. This \(h\)-vector achieves the lower bound for the height and the upper bound for the length. Finally, a conjecture is stated to predict all the possible \(h\)-vectors of the union.