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The Kähler cone in families of quasi-Fano threefolds

The Kähler cone in families of quasi-Fano threefolds Math. Z. 227, 45–68 (1998) c Springer-Verlag 1998 The Kahler ¨ cone in families of quasi-Fano threefolds Roberto Paoletti Dipartimento di Matematica, Universita ´ di Pavia, Via Abbiategrasso 209, I-27100 Pavia, Italy Received: 19 September 1995; in final form 22 January 1996 1 Introduction Let X be a smooth projective 3-fold, and denote by H  N (X ) the Kahler ¨ cone of X , that is the convex cone generated by the numerical equivalence classes of ample divisors on X (see [KMM] for notation). Its closure, H  N (X ), is the convex cone of the numerically effective numerical classes of real divisors. The subject of this paper is the behaviour of the Kahler ¨ cone of X when X varies in an algebraic family, under the assumption that the canonical divisor of X is suitably negative. Definition 1.1 A smooth projective 3-fold is quasi-Fano (in short, qF) if its anticanonical divisor is big and nef. By a result of Ran ([R 92]), such a 3-fold is unobstructed if the linear series j− K j contains a smooth surface. We shall see in Sect. 2 that this is indeed always the case. The behaviour of the http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematische Zeitschrift Springer Journals

The Kähler cone in families of quasi-Fano threefolds

Paoletti, Roberto
Mathematische Zeitschrift , Volume 227 (1) – Jan 1, 1998
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References (5)

Publisher
Springer Journals
Copyright
Copyright © 1998 by Springer-Verlag Berlin Heidelberg
Subject
Legacy
ISSN
0025-5874
eISSN
1432-1823
DOI
10.1007/PL00004367
Publisher site
See Article on Publisher Site

Abstract

Math. Z. 227, 45–68 (1998) c Springer-Verlag 1998 The Kahler ¨ cone in families of quasi-Fano threefolds Roberto Paoletti Dipartimento di Matematica, Universita ´ di Pavia, Via Abbiategrasso 209, I-27100 Pavia, Italy Received: 19 September 1995; in final form 22 January 1996 1 Introduction Let X be a smooth projective 3-fold, and denote by H  N (X ) the Kahler ¨ cone of X , that is the convex cone generated by the numerical equivalence classes of ample divisors on X (see [KMM] for notation). Its closure, H  N (X ), is the convex cone of the numerically effective numerical classes of real divisors. The subject of this paper is the behaviour of the Kahler ¨ cone of X when X varies in an algebraic family, under the assumption that the canonical divisor of X is suitably negative. Definition 1.1 A smooth projective 3-fold is quasi-Fano (in short, qF) if its anticanonical divisor is big and nef. By a result of Ran ([R 92]), such a 3-fold is unobstructed if the linear series j− K j contains a smooth surface. We shall see in Sect. 2 that this is indeed always the case. The behaviour of the

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Mathematische Zeitschrift – Springer Journals

Published: Jan 1, 1998

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