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...

In this paper we continue our study of the moduli space of stable bundles of rank two and degree 1 on a very general quintic surface. The goal in this paper is to understand the irreducible components of the moduli space in the first case in the "good" range, which is c 2 = 10. We show that there is a single irreducible component of bundles which have seminatural cohomology, and conjecture that this is the only component for all stable bundles.This paper is the next in a series, starting with [13], in which we study the moduli spaces of rank two bundles of odd degree on a very general quintic hypersurface X ⊂ P 3 . This series is dedicated to Professor Maruyama, who brought us together in the study of moduli spaces, a subject in which he was one of the first pioneers.In the first paper, we showed that the moduli space M X (2, 1, c 2 ), of stable bundles of rank 2, degree 1 and given c 2 , is empty for c 2 ≤ 3, irreducible for 4 ≤ c 2 ≤ 9, and good (i.e. generically smooth of the expected dimension) for c 2 ≥ 10. On the other hand, Nijsse has shown that the moduli space is irreducible for c 2 ≥ 15 [15] using the techniques of O'Grady [16] [17]. This leaves open the question of irreducibility for 10 ≤ c 2 ≤ 14.Conjecture 0.1. The moduli space M X (2, 1, 10) is irreducible.We haven't yet formulated an opinion about the cases 11 ≤ c 2 ≤ 14.In the present paper, due to lack of time and for length reasons, we treat a special case of the conjecture: the case of bundles with seminatural cohomology, meaning that only at most one of h 0 (E(n)), h 1 (E(n)) or h 2 (E(n)) can be nonzero for each n. Let M sn X (2, 1, 10) denote the open subvariety of the moduli space consisting of bundles with seminatural cohomology. In Section 3 we show that the seminatural condition is a consequence of assuming just h 0 (E(1)) = 5. The main result of this paper is:Theorem 0.2. The moduli space M sn X (2, 1, 10) is irreducible. Recall from [13] that our inspiration to look at this question came from the recent results of Yoshioka, for the case of Calabi-Yau surfaces originating in [14]. Yoshioka shows that the moduli spaces are irreducible for all positive values of c 2 , when X is an abelian or K3 surface [20] [21]. His results apply for example when * This represents a change in notation from [13], where we considered bundles of degree −1. For the present considerations, bundles of degree 1 are more practical in terms of Hilbert polynomial. We apologize for this inconvenience, but luckily the indexation by second Chern class stays the same. Indeed, if E has degree 1 then c 2 (E) = c 2 (E(−1)) as can be seen for example on the bundle E = O X ⊕ O X (1) with c 2 (E) = c 2 (E(−1)) = 0. Thus, the moduli space of stable bundles M X (2, 1, c 2 ) we look at here is isomorphic to M X (2, −1, c 2 ) considered in [13].Proof. As discussed above, h 2 (E(1)) = 0 so the fact that χ(E(1)) = 5 gives the first statement. For the second statement, use the fact that H 1 (O X (n)) = 0 for all n, and the long exact sequences of cohomology for the extension E(1) ...

The moduli space M (c 2 ), of stable rank two vector bundles of degree one on a very general quintic surface X ⊂ P 3 , is irreducible for all c 2 ≥ 4 and empty otherwise.

Soient Mg l'espace des modules (grossier) des courbes de genre g (g_-> 3), kg son corps de fonctions et egg la courbe universelle sur kg. A Franchetta a conjectur6 (voir [4]) que le groupe de Picard Pic(C~e) est engendr6 par la ctasse du diviseur canonique.Cette conjecture est maintenant d6montr6e (voir [1]) comme cons6quence de r6sultats topologiques de J. Harer (voir [10]) si g vaut au moins 5, et parce que la courbe canonique g6n6rale de genre 4 (resp. 3) est intersection compl6te d'une quadrique et d'une cubique de p3 (resp. est plane de degr6 4).On d6montre ici (Corollaire 4.7) ce qu'on appelle la conjecture deFranchetta forte c'est-h-dire que le groupe PiC%/k~(kg ) des points rationnels du schema de Picard PiC%/k~ de egg (d6fini en [9]) est engendr6 par la classe du diviseur canonique. Autrement dit, l'injection naturelle du groupe de Picard de c~g dans le groupe des classes rationelles de diviseurs de egg est un isomorphisine. Ce n'est pas toujours le cas, rappelons que si C est une vari6t6 de BrauerSeveri sur un corps k sans point rationnet sur k alors Pic(C)r (cf.[9]); on peut aussi trouver deux autres exemples (dont l'un est obtenu avec des courbes hypereUiptiques de genre impair) dans [13].Pour tout entier r, notons J" la Jacobienne de degr6 r de ~g c'est-~-dire la composante de degr6 r de Pi___~%,/~ . On sait (voir [14] et 4.7) que le point trivial est le seul point rationnel de jo et que Jg-1 n'a pas de point rationnel, pour d6montrer la conjecture de Franchetta forte, il suffit donc de d6montrer que si jr a un point rationnel alors g-1 divise r. C'est le Corollaire 4.5 du Th6or6me 4.4 off on d6montre que s'il existe sur J' un faisceau coh6rent ~ de caract6ristique d'Euler-Poincar6 X(~) 6gale ~ d, alors g-1 divise r d.Soit X un sch6ma projectif sur un corps k de caract6ristique nulle, pas forc6ment alg6briquement clos. L'ensemble des entiers d tels qu'il existe sur X un faisceau ~ avec X(~)=d engendre un sous groupe E de Z. Soit n l'entier naturel tel que E=nZ, on dira que nest le nombre d'Euler de X et on le notera eul(X). Tout le travail consiste ~ prouver que g-1 divise r x eul(J').Parce que le nombre d'Euler se comporte bien par sp6cialisation (voir 1.12), au lieu de travailler avec Mg, on se ram6ne ~ travailler avec une famiUe de