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The minimal resolutions of double points in $$\mathbb {P}^1 \times \mathbb P^{1}$$ with ACM support. (English) Zbl 1126.13013
The aim of this paper is to study ideals of 0-dimensional schemes $$Z \in {{\mathbb P}^{1}}\times {{\mathbb P}^{1}}$$ which are unions of double points (here a “double point” at a point $$P$$ is the scheme defined by the ideal $$I_P^2$$). Namely an algorithm is given to recover the bigraded Betti numbers of a minimal resolution of $$I_Y \subset S = k[x_0,x_1,y_0,y_1]$$ whenever the support $$X$$ of $$Z$$ is arithmetically Cohen-Macaulay (ACM, i.e. the ring $$S/I_Z$$ is Cohen-Macaulay). When $$Z$$ itself is ACM, a minimal resolution of $$I_Z$$ (hence its bigraded Betti numbers) can be recovered from combinatorial data on the coordinates of the points.
The idea in the paper is to consider a scheme $$Y$$, the completion of $$Z$$, obtained in this way: Let $$W$$ be the minimum reduced complete intersection scheme containing $$X$$; then $$Y := Z\cup (W-X)$$. It is proved that $$Y$$ is ACM, and its resolution is computed (again it depends only on the data of the coordinates of the points). Then if $$I_Z = I_Y + (F_1,\dots,F_p)$$, the ideals $$I_j = (I_{j-1},F_j)$$, $$j=0,\dots,p$$, are considered (so that $$I_p=I_Z$$, $$I_0=I_Y)$$, and it is shown that each ideal $$I_{j-1}:F_j$$ is a complete intersection. Eventually, via repeated mapping cone constructions using the exact sequences: $0 \rightarrow S/(I_{j-1}:F_j) \rightarrow S/I_{j-1} \rightarrow S/I_j \rightarrow 0,$ the resolution of $$I_Z$$ is computed from the one of $$I_Y$$. It is then shown that the total Betti numbers in the resolution of $$Z$$ satisfy a bound related to the shifts appearing in the resolution as conjectured by T. Römer.

##### MSC:
 13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 14A15 Schemes and morphisms 13D02 Syzygies, resolutions, complexes and commutative rings 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
##### Keywords:
fat points; Betti numbers; resolution
CoCoA; Macaulay2
Full Text:
##### References:
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