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Separators of points in a multiprojective space. (English) Zbl 1145.13007
Given a set of points \({\mathbb X} = {P_1,...,P_s} \subset {\mathbb P}^1 \times ... \times {\mathbb P}^{n_r}\), a multihomogeneous form \[ F \in k[x_{1,0},...,x_{1,n_1}; ... ;x_{r,0},...,x_{r,n_r}] \] is a separator for \(P\in {\mathbb X}\) if \(F(P)\neq 0\) and \(F(Q)=0\), \(\forall Q\in {\mathbb X} - \{P\} \).
The degree of a point \(P \in {\mathbb X}\) is the set: \[ \deg _{\mathbb X}(P) = \{\deg F | F \, {\text{is a minimal separator of P}}\} \] where the partial order on \({\mathbb N}^r\) is given by \((i_1,..,i_r) \geq (j_1,...,j_r)\) iff \(i_t \geq j_t\), \(\forall t=1,...,r\).
This generalizes to multiprojective spaces the notion of separator and degree of a point in \({\mathbb P}^n\). A. V. Geramita, P. Maroscia and L. G. Roberts [J. Lond. Math. Soc. 28, 443–452 (1983; Zbl 0535.13012)] showed how the degree can be used to relate the Hilbert function of \({\mathbb X}\) and \({\mathbb X}-\{P\}\), while L. Bazzotti, S. Abrescia and L. Marino [Matematiche 56, No. 1, 129–148 (2001; Zbl 1172.13306)] also showed how this degree is related to the graded Betti numbers in a minimal resolution of \(I_{\mathbb X}\).
Those results are generilized in this paper to the multiprojective case for Hilbert functions, and to the multigraded resolution when \({\mathbb X}\) is a.C.M. More detailed results are given when \({\mathbb X}\subset {\mathbb P}^1\times{\mathbb P}^1\).

MSC:
13D02 Syzygies, resolutions, complexes and commutative rings
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
Software:
CoCoA
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References:
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