Prime exceptional divisors on holomorphic symplectic varieties and monodromy-reflections

Let X be a projective irreducible holomorphic symplectic manifold. The second integral cohomology of X is a lattice with respect to the Beauville-Bogomolov pairing. A divisor E on X is called a prime exceptional divisor, if E is reduced and irreducible and of negative Beauville-Bogomolov degree. Let E be a prime exceptional divisor on X. We first observe that associated to E is a monodromy involution of the integral cohomology of X, which acts on the second cohomology lattice as the reflection by the cohomology class of E (Theorem 1.1). We then specialize to the case that X is deformation equivalent to the Hilbert scheme of length n zero-dimensional subschemes of a K3 surface. We determine the set of classes of exceptional divisors on X (Theorem 1.11). This leads to a determination of the closure of the movable cone of X.Comment: v2: 53 pages, Latex. The main Conjecture 1.11 is now Theorem 1.11. Final version. To appear in KJM, Maruyama memorial volum

Integral generators for the cohomology ring of moduli spaces of sheaves over Poisson surfaces

Let M be a smooth and compact moduli space of stable coherent sheaves on a projective surface S with an effective (or trivial) anti-canonical line bundle. We find generators for the cohomology ring of M, with integral coefficients. When S is simply connected and a universal sheaf E exists over SxM, then its class [E] admits a Kunneth decomposition as a class in the tensor product of the topological K-rings K(S) and K(M). The generators are the Chern classes of the Kunneth factors of [E] in K(M). The general case is similarComment: v3: Latex, 27 pages. Final version, to appear in Advances in Math. The proof of Lemma 21 is corrected and several other minor changes have been made. v2: Latex, 26 pages. The paper was split. The new version is a rewrite of the first three sections of version 1. The omitted results, about the monodromy of Hilbert schemes of point on a K3 surface, constitute part of the new paper arXiv:math.AG/0601304. v1: Latex, 53 page

Generators of the cohomology ring of moduli spaces of sheaves on symplectic surfaces

Let M be a moduli space of stable sheaves on a K3 or Abelian surface S. We express the class of the diagonal in the cartesian square of M in terms of the Chern classes of a universal sheaf. Consequently, we obtain generators of the cohomology ring of M. When S is a K3 and M is the Hilbert scheme of length n subschemes, this set of generators is sufficiently small in the sense that there aren't any relations among them in the stable cohomology ring. When S is the cotangent bundle of a Riemann surface, we recover the result of T. Hausel and M. Thaddeus: The cohomology ring of the moduli spaces of Higgs bundles is generated by the universal classes.Comment: Latex, 23 pages. The introduction is expanded, the coefficient in part 3 of Theorem 1 is corrected, plus several other minor change

Disagreement and easy bootstrapping

ABSTRACTShould conciliating with disagreeing peers be considered sufficient for reaching rational beliefs? Thomas Kelly argues that when taken this way, Conciliationism lets those who enter into a disagreement with an irrational belief reach a rational belief all too easily. Three kinds of responses defending Conciliationism are found in the literature. One response has it that conciliation is required only of agents who have a rational belief as they enter into a disagreement. This response yields a requirement that no one should follow. If the need to conciliate applies only to already rational agents, then an agent must conciliate only when her peer is the one irrational. A second response views conciliation as merely necessary for having a rational belief. This alone does little to address the central question of what is rational to believe when facing a disagreeing peer. Attempts to develop the response either reduce to the first response, or deem necessary an unnecessary doxastic revision, or imply that rational dilemmas obtain in cases where intuitively there are none. A third response tells us to weigh what our pre-disagreement evidence supports against the evidence from the disagreement itself. This invites epistemic akrasia