Let (M,L) be a smooth 4-dimensional variety polarized by a very ample line bundle L. Let A be a smooth member of l$. Assume that A is a Fano threefold of index one, with $-K_A=H_A for some ample line bundle H_A on A. Let H be the line bundle on M which extends H_A. Up to sporadic examples and four special classes of pairs (M,L), and up to taking the first reduction (M',L') of (M,L), we show that M' is a Fano variety, and the cones of effective 1-cycles NE(A') and NE(M') coincide, where A' is the image of A under the first reduction map. We also show that there exists a new polarization H' on M' and our main result is proving that the usual adjunction process on (M',H') terminates, this leading to a coarse classification of (M,L).

Fano threefolds as hyperplane sections

FANIA, Maria Lucia
2005-01-01

Abstract

Let (M,L) be a smooth 4-dimensional variety polarized by a very ample line bundle L. Let A be a smooth member of l$. Assume that A is a Fano threefold of index one, with $-K_A=H_A for some ample line bundle H_A on A. Let H be the line bundle on M which extends H_A. Up to sporadic examples and four special classes of pairs (M,L), and up to taking the first reduction (M',L') of (M,L), we show that M' is a Fano variety, and the cones of effective 1-cycles NE(A') and NE(M') coincide, where A' is the image of A under the first reduction map. We also show that there exists a new polarization H' on M' and our main result is proving that the usual adjunction process on (M',H') terminates, this leading to a coarse classification of (M,L).
978-3-11-0181609
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11697/40835
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