Ruled Fano $4$-folds of index $2$

Abstract

This article contains a classification of Fano 4-folds of index 2 which can be represented as PI -bundles. Let X be a smooth projective variety of dimension n > 1 over the field of complex numbers. We call X a Fano n-fold if its anticanonical divisor -Kx is ample. The index r(X) of a Fano n-fold X is defined as r(X) = max{k: K = kH for H ample divisor on X}. We will say that X is ruled if X = P(Qf) for some rank-2 vector bundle o' on a projective manifold M. The bundle 9' is called Fano if X = P(Q') is a Fano n-fold. Together with Michal Szurek we worked on some Fano bundles of rank-2. In [SzW1] we classified all ruled Fano 3-folds and in [SzW2] we described all rank-2 Fano bundles on P3. The purpose of this paper is to classify all ruled Fano 4-folds of index 2. (0.1) Theorem. Assume that X is a ruled Fano 4-fold of index 2. Then one of the following holds: (i) X =P x M, where M is a Fano 3-fold of index 2 or P3; (ii) either X = P(61'p3 (1) E) p,?3 (1)) or X = P(6'Q3 E 'Q3 (1)); (iii) X has two Pi-bundle stuctures and can be realized either as P(NCB), where NCB is the null-correlation bundle on P3, that is, a stable rank2 bundle with cl = 0, C2 = 1, or PQ(F), where 9' is a stable rank-2 bundle on Q3 with cf = -1, Cj92 =1. The techniques used in this paper are a mixture of those from [SzW1 and SzW2], and others coming from contractions of extremal rays on Fano manifolds. In ?1 we outline properties of contractions of extremal rays and prove a useful lemma on contractions of extremal rays of length > max(2, n 2) (Lemma (1.1)). In ??2 and 3 we give the proof of Theorem (0.1). Remark. After the first version of this paper was completed, I learned that the classification of Fano 4-folds of index 2 was the subject of a preprint of S. Mukai Received by the editors November 10, 1987 and, in revised form, February 23, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 14J35; Secondary 32105. () 1989 American Mathematical Society 0002-9939/89 $1.00 + $.25 per page