Inductively computable unions of fat linear subspaces
The paper introduces complexes of linear varieties, called inclics (for INductively Constructible LInear ComplexeS). An inclic \[ \mathcal{C}=\mathcal{C}(n,r,s;L_0,L_1,\dots, L_r,H_0,H_1,\dots, H_s) \] is a collection of linear subvarieties \(L_0,L_1,\dots, L_r,H_0,H_1,\dots, H_s\subset \mathbb{P}^n\) such that the following conditions hold: (C1) \(H_0,H_1,\dots, H_s\) are distinct hyperplanes; (C2) \(L_i\subseteq H_0\) for \(i>0\) but \(L_0\not\subseteq H_0\); (C3) if \(L_i\subseteq L_j\), then \(i=j\); (C4) for all \(i\geq 0\) and \(j>0\) we have \(L_i\not\subseteq H_i\). By assigning an order of vanishing (i.e., a multiplicity) to each member of the complex, the authors obtain fat linear varieties (fat points if all of the linear varieties are points). The scheme theoretic union of these fat linear varieties gives an inclic scheme \[ X=\sum_{i\geq 0}l_iL_i+\sum_{j>0}h_jH_j \] by which we mean the scheme defined by the ideal \(I_X=(\cap_{i\geq 0}I_{L_i}^{l_i})\cap(\cap_{j>0}I_{H_j}^{h_j})\), where \(l_i\) and \(h_j\) are non-negative integers and \(I_X\subset R=K[\mathbb{P}^n]\). Clearly it is possible to choose coordinates such that \(I_{H_0}=(x_0)\) so \(R'=K[H_0]=K[x_1,\dots, x_n]\). Since \(L_0\not\subseteq H_0\), we may also assume that \(I_{L_0}=(x_{k+1}, \dots, x_n)\) for some \(0\leq k<n\) and denote the linear forms defining \(H_j\) for \(j>0\) by \(\eta_j\). Then define \(Y=l_1L_1+\cdots l_rL_r\) of \(\mathbb{P}^n\), \(Y'=Y\cap H_0\) and \(Y_i'=Y_i\cap H_0\) for \(Y_i=l_1(i)L_1+\cdots +l_r(i)L_r\), where \(l_j(i)=\max(0, l_j-i)\). Thus \(Y_0'=Y'\) and \(I_{Y_i'}=I_{Y_i}\cap K[x_1, \dots, x_n]\). Moreover, \(I_{Y_i}=I_Y:(x_0^i)\). Then finally set \(Z=l_0L_0\), \(W=Y\cup Z\), \(X=W\cup (\bigcup_{j>0}h_jH_j)\), \(L_0'=L_0\cap H_0\), \(Z'=Z\cap H_0\), \(W'=W\cap H_0\) and \(X'=X\cap H_0\). For such a inclic scheme \(X\), the author show there is an inductive procedure for computing the Hilbert function and a resolution of its defining ideal \(I_X\), regardless of the choice of multiplicities. In particular, for the Hilbert function, they show the following result. Theorem. Let \(Y'\), \(Y_i'\), \(Z\), \(W\) and \(X\) be as above, let \(l'=\max(l_1,\dots, l_r)\). Then \(I_X=\eta_1^{h_1}\cdots \eta_s^{h_s}I_W\) and \(I_W\) is bi-homogeneous, decomposing as a direct sum of \(R'-\)moduls as \[ I_W=\bigoplus_jx_0^j(I_{Y'_j})^{(k,l_0)}=(\bigoplus_{0\leq j<l'}x_0^j(I_{Y'_j})^{(k,l_0)})\oplus \bigoplus_{j\geq l'}x_0^jI_{Z'}=(\bigoplus_{0\leq j<l'}x_0^j(I_{Y'_j})^{(k,l_0)})\oplus x_0^{l'}I_{Z} \] where for an ideal \(J\subset K[x_1, \dots, x_n]\) we set \(J^{(k,t)}=J\cap (I_{L_0'})^t\), keeping in mind \(I_{L_0}=(x_{k+1}, \dots, x_n)\). Moreover, \(h(I_X,t)=h(I_W, t-\sum_{j>0}h_j)\) and \(h(I_W,t)=\sum_{j=0}^{l'-1}h((I_{Y_j'})^{(k,l_0)},t-j)+h(I_Z,t-l')\), where \(h(I_Z,t-l')=0\) if \(t<l'+l_0\) and \(h(I_Z,t-l')={t-l'+n \choose n}-\sum_{0\leq i<l_0} {t-l'-i+k \choose k}{i+n-k-1 \choose n-k-1}\) for \(t\geq l'+l_0\). As an application, the authors show how their results allow the computation of the Hilbert functions and of a resolution of fat points with all but one point having support in a hyperplane. They also explicitly compute the Waldschmidt constants \(\hat{\alpha}(I_X)\) for galactic inclics \(X\); these are special inclics constructed starting from a star configuration to which we add general points in a larger projective space. In case \(I\) is of the form \(I=\bigcap_jI_{V_j}^{m_j}\) for a finite set of linear varieties \(V_j\), none of which contains the other, we define the \(m\)th symbolic power \(I^{(m)}\) of \(I\) as \(I^{(m)}=\bigcap_jI_{V_j}^{mm_j}\), We then define the Waldschmidt constant to be \[ \hat{\alpha}(I)=\lim_{m\to \infty}\frac{\alpha(I^{(m)})}{m}. \] Another asymptotic measure related to \(\hat{\alpha}(I)\) is the resurgence defined for any homogeneous ideal \((0)\not= I\subsetneq R= K[\mathbb{P}^n]\) as \[ \rho(I)=\sup\{\frac{m}{r}: I^{(m)}\nsubseteq I^r\}. \] To define a galaxy, we start with a star configuration \(S(n,e,u)\subset \mathbb{P}^n\), which is defined by a set of \(u\geq n\) distinct hyperplanes \(A_1, \dots, A_u\cong \mathbb{P}^{n-1}\) in \(\mathbb{P}^n\) such that, for each \(1\leq i \leq n\), the intersection of any \(i\) of the hyperplanes has dimension at most \(n-i\). The star configuration of codimension \(e\leq n\) is the set \(S(n,e,u)\) of the \({u \choose e}\) linear varieties arising as intersections of \(e\) arbitrary distinct choices \(A_{i_1}, \dots, A_{i_e}\) of the hyperplanes. Let \(N\geq 1\) be an integer and regard \(\mathbb{P}^n\) as a linear subvariety of \(\mathbb{P}^{n+N}\). The galaxy \(\mathcal{G}=\mathcal{G}(n,N,e,h)=\mathcal{G}(n,N,e,h;S(n,e,u),\mathcal{H})\) consists of \(S(n,e,u)\) and a choice of \(h\) general points \(\mathcal{H}=\{P_1, \dots, P_h\}\in \mathbb{P}^{n+N}\); in particular, for each \(i\), \(P_{i+1}\) is not in the span of \(\mathbb{P}^n\) and \(P_1, \dots, P_i\). When \(h=N\), then we get an inclic in which the components of \(S(n,e,u)\) and the points \(P_1, \dots, P_{h-1}\) are the linear varieties \(L_i\), and there is only one hyperplane, \(H_0\) (so \(s=0\)), this hyperplane being the hyperplane containing the linear span of \(\mathbb{P}^n\) and the points \(P_1, \dots, P_{h-1}\), but not containing \(P_h\). The main result in the paper related to galactic inclics is the following. Theorem Let \(G\) be a reduced galactic inclic scheme. (a) Then \[ \frac{2}{\hat{\alpha}(I_G)}\leq \rho(I_G) \] and if in addition \(e=n\) then \[ \rho(I_G)\leq \frac{u-n+1}{\hat{\alpha}(I_G)}. \] (b) If \(K\) has characteristic 0, then \[ \hat{\alpha}(I_G)= \frac{N(u-e)+u}{N(u-e)+e}. \]
scientific article; zbMATH DE number 6476974
scientific article