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Fat points schemes on a smooth quadric. (English) Zbl 1044.14025
Summary: We study zero-dimensional fat points schemes on a smooth quadric \(Q\cong\mathbb{P}^1\times\mathbb{P}^1\), and we characterize those schemes which are arithmetically Cohen-Macaulay (aCM for short) as subschemes of \(Q\) giving their Hilbert matrix and bigraded Betti numbers. In particular, we can compute the Hilbert matrix and the bigraded Betti numbers for fat points schemes with homogeneous multiplicities and whose support is a complete intersection. Moreover, we find a minimal set of generators for schemes of double points whose support is a CM.

MSC:
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
14A15 Schemes and morphisms
14M10 Complete intersections
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13D02 Syzygies, resolutions, complexes and commutative rings
13C40 Linkage, complete intersections and determinantal ideals
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