Semi-stable vector bundles on fibred varieties

Abstract

Let π : Y →X be a surjective morphism between two irreducible, smooth complex projective varieties with dim Y > dim X>0. We consider polarizations of the form Lc = L+c·π * A on Y , with c > 0, where L, A are ample line bundles on Y, X respectively.For c sufficiently large, we show that the restriction of a torsion free sheaf F on Y to the generic fibre Φ of π is semi-stable as soon as F is Lc-semi-stable; conversely, if F ⊗ O Φ is L-stable on Φ, then F is Lc-stable. We obtain explicit lower bounds for c satisfying these properties. Using this result, we discuss the construction of semi-stable vector bundles on Hirzebruch surfaces and on P 2 -bundles over P 1 , and establish the irreducibility and the rationality of the corresponding moduli spaces.

On the rationality of the moduli space of instanton bundles on the projective 3-space

“…Proposition 2.3 implies that an isomorphism between the restrictions to D Y H of two Egen instantons actually comes from isomorphism over P 3 ; thus Θ is birational. It remains to observe that [8,Corollary 3.11] implies that MP 2 pr; nq λ is a rational variety. l…”

“…for any c ą 0. (See [8] for the terminology.) Indeed, let G Ă σ ˚F be a saturated subsheaf, so σ ˚F{ G is torsion-free; thus G is reflexive and its singularities (if any) are punctual.…”

“…The first author investigated [8] the geometry of the moduli space of semi-stable vector bundles on P 2 -bundles over P 1 . Our goal in here is to deepen those results in the case of the three-dimensional projective space and to address the rationality issue mentioned above.…”

“…-For studying the geometry of their moduli space, we restrict instantons on P 3 to (a pair of) planes. The End-general condition ensures that this restriction map is étale, so we can apply the results of [8] concerning the rationality of the moduli space of framed instantons on P 2 . We also highlight the brevity of our work, making the proofs easy to follow, which reflects the effectiveness of our methods: the arguments are not restricted only to the rank-2 case; we deduce the irreducibility of the relevant moduli space in a couple of lines (cf.…”

“…(ii) The results obtained in[8, Theorem 3.6] yield detailed descriptions of MP 2 pr; nq and MQ pr; 2nq: they are irreducible, rational varieties. Their general elements are, respectively, the kernels of surjective homomorphisms:O l j p1q,n " ar `ρ, 0 ď ρ ă r, and l 1 , .…”