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A criterion for quasi-complete intersections and related embedding questions

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Riassunto

Un sotto-schemaX⊂Y èintersezione semplice delle ipersuperfici (divisori di Cartier effettivi)F 1,…,F m diY seX è il sottoschema intersezione diF 1, …,F m. SeX è irriducibile di codimensioner, X èuna quasi-completa intersezione, seX è intersezione semplice dir+1 ipersuperfici. PerX, Y⊂ P N entrambi chiusi irriducibili lisci, il Teorema 1 dà un criterio affinchèX sia intersezione semplice dir+e ipersuperficie, dimX≧e>0. Il Teorema 2 è un criterio affinchèX sia una quasi-intersezione completa, quandor=2,Y=P N.

Summary

A subschemeX⊂Y is asimple intersection of hypersurfaces (effective Cartier divisors)F 1,…,F m ofY ifX is the intersection subscheme ofF 1, …,F m.

WhenX is irreducible of codimensionr, X is aquasi-complete intersection, ifX is simple intersection ofr+1 hypersurfaces.

AssumingX, Y⊂ P N both irreducible closed smooth over an algebraically closed field, Theorem 1 gives a criterion forX to be a simple intersection ofr+e hyper-surfaces, dimX≧e>0. Theorem 2 gives a criterion forX to be a quasi-complete intersection whenr=2,Y=P N.

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Author notes

  1. Alexandru T. Lascu

    Present address: Mario Fiorentini Ist. Matematico, Univ. Ferrara, via Macchiavelli 35

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    1. Mario Fiorentini
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    2. Alexandru T. Lascu
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    AlexandruT. Lascu, Département de Mathématiques, Université de Montréal.

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    Fiorentini, M., Lascu, A.T. A criterion for quasi-complete intersections and related embedding questions. Ann. Univ. Ferrara 28, 153–166 (1982). https://doi.org/10.1007/BF02900762

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