Connections between the geometry of a projective variety and of an ample section

Abstract

Let X be a smooth complex projective variety and let Z =(s =0) be a smooth submanifold which is the zero locus of a section of an ample vector bundle E of rank r with dim Z =dim X –r.We show with some examples that in general the Kleiman–Mori cones NE(Z) and NE(X) are different. We then give a necessary and sufficient condition for an extremal ray in NE(X) to be also extremal in NE(Z).We apply this result to the case r =1 and Z a Fano manifold of high index;in particular we classify all X with an ample divisor which is a Mukai manifold of dimension ≥4.In the last section we prove a general result in case Z is a minimal variety with 0 ≤κ (Z) <dim Z. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Some Remarks on Cones of Partially Ample Divisors

“…147 Corollary]. This provided the starting-block for much further work concerning weak Lefschetz for the ample cone ( [15], [18], [1], [2], [6], [24]).…”

“…Corollary. (=Corollary 25) Let X be a smooth projective complex variety such that either (1) −K X is ample, or (2) −K X is = 0 and nef and dim N 1 X ≥ 3. Then:…”

“…The answer is negative in general, as shown by Hassett-Lin-Wang [15]. On the other hand, the answer is positive for certain Fano varieties ( [26], [15], [18], [1], [6], [24]). Using the above Theorem, it turns out that the K X -visible part cuts out a part where weak Lefschetz holds for the nef cone:…”

Fano manifolds of coindex four as ample sections

“…In particular ÀðK X þ LÞ Á R ¼ ðÀD þ 2LÞ Á G > 0, so that, by [3,Theorem 3.2], R is extremal for NEðZÞ. that D Á C < 0 for some e¤ective curve C; in this case there exists an extremal ray R ¼ R þ ½G such that D Á R < 0.…”

“…r is the largest (positive) integer such that ÀK Z ¼ rH Z for an ample line bundle H Z on Z, with q c 3 were studied by adjunction theory; in particular, the case of projective space and hyperquadrics was considered in [6], the del Pezzo varieties were studied in [16] and the Mukai varieties were the object of the recent papers [8], [3] and [7]. r is the largest (positive) integer such that ÀK Z ¼ rH Z for an ample line bundle H Z on Z, with q c 3 were studied by adjunction theory; in particular, the case of projective space and hyperquadrics was considered in [6], the del Pezzo varieties were studied in [16] and the Mukai varieties were the object of the recent papers [8], [3] and [7].…”