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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.)
Software:
CoCoA; Macaulay2
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References:
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