Instanton bundles on the Segre threefold with Picard number three

Abstract

We study instanton bundles on ℙ 1 × ℙ 1 × ℙ 1 . We construct two different monads which are the analog of the monads for instanton bundles on ℙ 3 and on the flag threefold (0, 1, 2). We characterize the Gieseker semistable cases and we prove the existence of -stable instanton bundles generically trivial on the lines for any possible 2 ( ). We also study the locus of jumping lines. K E Y W O R D SBeilinson spectral theorem, instanton bundles, jumping lines, Segre varieties M S C ( 2 0 1 0 ) (Primary) 14J60, (Secondary) 13C14, 14F05 1026

Even and odd instanton bundles on Fano threefolds

“…In [25], the author proves that for many families of Fano threefolds X with ̺ X = 1 the space SI X ε, ζ has a generically smooth irreducible component SI X ε, ζ . Similar results have been obtained in [41,14,4,17,3] for other families of Fano threefolds.…”

“…Example 3.3. Decomposable instanton bundles exist on the flag threefold (see [41,Proposition 2.5]): it is easy to check that that the same is true on P 1 × P 1 × P 1 (see [4]). On the other hand, the blow up of P 3 at a point does not support any decomposable instanton bundle (see [14,Proposition 6.5]).…”

“…When ̺ X = 1, then each instanton bundle E is automatically µ-stable because h 0 X, E = 0. In [41,4] the authors proved the existence of ordinary instanton bundles which are not µ-stable on F 6,2 and F 6,3 . When X ∼ = F 7 , then all instanton bundle with quantum number k ≤ 14 are µ-stable (see [14]).…”

“…More recently, the definition has been further generalized to each Fano threefold, regardless of i X or ̺ X : see [14,Definition 1.2]. Moreover, the existence of instanton bundles according to the latter definition has been settled on some Fano threefolds: see [14,17,4,3].…”

“…Generically trivial ordinary instanton bundles have been constructed on several Fano threefolds ( [25,41,4,17,3]). Thus it is quite natural to analyze the behaviour of the instanton bundles whose existence is guaranteed by Theorems 1.5 and 1.6 when restricted to general lines: we make such an analysis in Section 7 where we prove the following result.…”