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Characterizations of zero-dimensional complete intersections

Abstract

Given a 0-dimensional subscheme \({\mathbb X}\) of a projective space \({\mathbb P}^n_K\) over a field K, we characterize in different ways whether \({\mathbb X}\) is the complete intersection of n hypersurfaces. Besides a generalization of the notion of a Cayley–Bacharach scheme, these characterizations involve the Kähler and the Dedekind different of the homogeneous coordinate ring of \({\mathbb X}\) or its Artinian reduction. We also characterize arithmetically Gorenstein schemes in novel ways and bring in further tools such as the module of regular differential forms, the fundamental class, and the Jacobian module of \({\mathbb X}\). Throughout we strive to work over an arbitrary base field K and keep the scheme \({\mathbb X}\) as general as possible, thereby improving several known characterizations.

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Acknowledgments

This paper is partially based on the L. N. Long’s dissertation (Long 2015). M. Kreuzer and L. N. Long thank Jürgen Herzog and Ernst Kunz for their encouragement to elaborate the results presented here. We are also grateful to Uwe Storch for pointing out the characterization in Scheja and Storch (1975). Moreover, we thank the Mathematics Department of IISc Bangalore (India), and in particular Dilip Patil, for their kind hospitality during part of the preparation of this paper. Last, but not least, we are extremely thankful to the referee for his very detailed and enlightening comments.

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Correspondence to Martin Kreuzer.

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Kreuzer, M., Long, L.N. Characterizations of zero-dimensional complete intersections. Beitr Algebra Geom 58, 93–129 (2017). https://doi.org/10.1007/s13366-016-0311-9

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  • DOI: https://doi.org/10.1007/s13366-016-0311-9

Keywords

  • Zero-dimensional scheme
  • Complete intersection
  • Kähler different
  • Dedekind different
  • Arithmetically Gorenstein scheme
  • Cayley–Bacharach scheme
  • Hilbert function

Mathematics Subject Classification

  • Primary 14M10
  • Secondary 13N05
  • 13C40
  • 13D40
  • 14M05