Let X subset of P-n be a smooth codimension 2 subvariety. We first prove a "positivity lemma" (Lemma 1.1) which is a direct application of the positivity of N-X(-1). Then we first derive two consequences:1) Roughly speaking the family of "biliaison classes" of smooth subvarieties of P-5 lying on a hypersurface of degree s is limited.2) The family of smooth codimension 2 subvarieties of P-6 lying on a hypersurface of degree s is limited.The result in 1) is not effective, but 2) is. Then we obtain precise inequalities connecting the usual numerical invariants of a smooth subcanonical subvariety X subset of P-n, n >= 5 (the degree d, the integer e such that omega(X) similar or equal to O-X(e), the least degree, s, of a hypersurface containing X). In particular we prove: s >= n + 1 if X is not a complete intersection.
Smooth divisors on projective hypersurfaces
ELLIA, Filippo Alfredo;
2008
Abstract
Let X subset of P-n be a smooth codimension 2 subvariety. We first prove a "positivity lemma" (Lemma 1.1) which is a direct application of the positivity of N-X(-1). Then we first derive two consequences:1) Roughly speaking the family of "biliaison classes" of smooth subvarieties of P-5 lying on a hypersurface of degree s is limited.2) The family of smooth codimension 2 subvarieties of P-6 lying on a hypersurface of degree s is limited.The result in 1) is not effective, but 2) is. Then we obtain precise inequalities connecting the usual numerical invariants of a smooth subcanonical subvariety X subset of P-n, n >= 5 (the degree d, the integer e such that omega(X) similar or equal to O-X(e), the least degree, s, of a hypersurface containing X). In particular we prove: s >= n + 1 if X is not a complete intersection.I documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
