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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
##### Keywords:
multiprojective spaces; points; linear systems
CoCoA
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##### References:
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