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Vanishing theorems, a theorem of Severi, and the equations defining projective varieties. (English) Zbl 0762.14012
The authors present a vanishing theorem about the cohomology of a smooth complex projective variety under a condition in terms of the codimension of the variety and the degrees of the generating hypersurfaces. They call it a “variant” of a theorem of F. Severi [Rend. Circ. Mat. Palermo 17, 73-103 (1903; JFM 34.0700.01)]. — They point out a surprising number of applications to questions involving the equations defining projective varieties like Cohen-Macaulayness, complete intersection, Castelnuovo-Mumford regularity, Hodge type and projectively normal embeddings.
The idea of the proof is to consider a blowing-up along the variety and to apply the Kawamata-Viehweg vanishing theorem. See also Y. Kawamata, Math. Ann. 261, 43–46 (1982; Zbl 0476.14007); E. Viehweg, J. Reine Angew. Math. 335, 1–8 (1982; Zbl 0485.32019); H. Esnault and E. Viehweg, Math. Ann. 263, 75-86 (1983; Zbl 0489.14016); and Invent. Math. 78, 445–490 (1984; Zbl 0532.10020)].

##### MSC:
 14F17 Vanishing theorems in algebraic geometry 14M10 Complete intersections 14N05 Projective techniques in algebraic geometry
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##### References:
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