Obstructed Bundles of Rank Two on a Quintic Surface

Abstract

To the memory of Masaki MaruyamaIn this note we consider the moduli space of stable bundles of rank two on a very general quintic surface. We study the potentially obstructed points of the moduli space via the spectral covering of a twisted endomorphism. This analysis leads to generically nonreduced components of the moduli space, and components which are generically smooth of more than the expected dimension. We obtain a sharp bound asked for by O'Grady on when the moduli space is good. must be a singular point. The moduli space might on the other hand be smooth but overdetermined, i.e. having dimension bigger than the expected dimension. And of course it could also be overdetermined and singular too.As is well known (see [9;46;39, Sec. 1;24]), the dual of the space of obstructions is H 0 (X, End 0 (E) ⊗ K X ) by Serre duality. An element φ in this dual space is a trace-free morphism φ : E → E ⊗ K X . Such a φ corresponds, by Kuranishi theory, to an equation of the moduli space locally at E, and we call it a co-obstruction. A pair (E, φ) consisting of a bundle together with a nonzero co-obstruction, may be thought of as a K X -valued Hitchin pair on X [19]. These pairs are different from those considered in [41] for the surface X: the Higgs bundles corresponding to representations of π 1 are endomorphisms taking values in Ω 1 X . Over a curve these two notions coincide and indeed Hitchin used the notation K X in his original paper [19]. Generalizing his notation as written leads to the notion of a Higgs field E → E ⊗ K X which is exactly a co-obstruction, often called a "twisted endomorphism."A basic tool in the analysis of Hitchin pairs is the notion of spectral cover [19,8,3,46]. A twisted endomorphism φ : E → E ⊗ K X gives E the structure of coherent sheaf on the total space of the vector bundle K X , and the support of the coherent sheaf is the spectral covering associated to φ. It consists of the set of pairs (x, u) where x ∈ X and u ∈ K X,x such that u is an eigenvalue of φ x .In our rank-two case the spectral cover is particularly simple to describe: it is the divisor Z ⊂ K X determined by the equationWe investigate in a very basic way the possible classification of such spectral covers, and the implications for the locus of singularities of the moduli space. This follows Donaldson's original proof of generic smoothness [9], as it has been developed by Zuo in [46], and more recently by Langer [24].Many authors have shown that the moduli spaces of bundles of odd degree on abelian and K3 surfaces are smooth, going back to [11] and Mukai [32] (see the discussions and references in [43,44]). O'Grady has observed an important example of symplectic singularities in the moduli of rank-two bundles on a K3 surface [40], along the locus of reducible bundles. In view of these properties and examples, for understanding bundles on surfaces of general type it seems like a good idea to look at surfaces of general type which are as close as possible to K3 surfaces. This motivates our consideration of the example of a ver...

Irreducibility of the moduli space of stable vector bundles of rank two and odd degree on a very general quintic surface

“…See Corollary 4.3 below. From the previous papers [19,20], we know the dimensions of M (c ′ 2 ), so we can fill in the dimensions of the strata, as will be summarized in Table 2. Furthermore, by [19] and Li's theorem, the strata M (c 2 ; c ′ 2 ) are irreducible whenever c ′ 2 ≤ 9.…”

“…From the previous papers [19,20], we know the dimensions of M (c ′ 2 ), so we can fill in the dimensions of the strata, as will be summarized in Table 2. Furthermore, by [19] and Li's theorem, the strata M (c 2 ; c ′ 2 ) are irreducible whenever c ′ 2 ≤ 9. Nijsse [23] proves that M (c 2 ) is connected whenever c 2 ≥ 10, using O'Grady's techniques [24,25].…”

“…This is discussed in [21]. By [19], the moduli space M (c 2 ) is good, that is to say it is generically reduced of the expected dimension 4c 2 − 20, whenever c 2 ≥ 10. In particular, the dimension of the Zariski tangent space, minus the dimension of the space of obstructions, is equal to the dimension of the moduli space.…”

“…There we considered bundles of degree −1. The bundle of degree 1 denoted here by E is the same as the bundle denoted by E(1) in [19]. Thus [19,Lemma 5.2] speaks of h 1 (E) in our notation.…”

“…The bundle of degree 1 denoted here by E is the same as the bundle denoted by E(1) in [19]. Thus [19,Lemma 5.2] speaks of h 1 (E) in our notation. The present notation was already in effect in [20].…”