Moduli spaces of reflexive sheaves of rank 2

Let \sF be a coherent rank 2 sheaf on a scheme Y \subset \proj{n} of dimension at least two. In this paper we study the relationship between the functor which deforms a pair (\sF,\sigma), \sigma \in H^0(\sF), and the functor which deforms the corresponding pair (X,\xi) given as in the Serre correspondence. We prove that the scheme structure of e.g. the moduli scheme M_Y(P) of stable sheaves on a threefold Y at (\sF), and the scheme structure at (X) of the Hilbert scheme of curves on Y are closely related. Using this relationship we get criteria for the dimension and smoothness of M_Y(P) at (\sF), without assuming Ext^2(\sF,\sF) = 0. For reflexive sheaves on Y = \proj{3} whose deficiency module M = H_{*}^1(\sF) satisfies Ext^2(M,M) = 0 in degree zero (e.g. of diameter at most 2), we get necessary and sufficient conditions of unobstructedness which coincide in the diameter one case. The conditions are further equivalent to the vanishing of certain graded Betti numbers of the free graded minimal resolution of H_{*}^0(\sF). It follows that every irreducible component of M_{\proj{3}}(P) containing a reflexive sheaf of diameter one is reduced (generically smooth). We also determine a good lower bound for the dimension of any component of M_{\proj{3}}(P) which contains a reflexive stable sheaf with "small" deficiency module M.Comment: 19 page

Quotients of E^n by A_{n+1} and Calabi-Yau manifolds

We give a simple construction, starting with any elliptic curve E, of an n-dimensional Calabi-Yau variety of Kummer type (for any n>1), by considering the quotient Y of the n-fold self-product of E by a natural action of the alternating group A_{n+1} (in n+1 variables). The vanishing of H^m(Y, O_Y) for 0<m<n follows from the non-existence of (non-zero) fixed points in certain representations of A_{n+1}. For n<4 we provide an explicit crepant resolution X in characteristics different from 2,3. The key point is that Y can be realized as a double cover of P^n branched along a hypersurface of degree 2(n+1).Comment: 9 page

Generalized chain surgeries and applications

We describe the Stein handlebody diagrams of Milnor fibers of Brieskorn singularities $x^p + y^q + z^r = 0$. We also study the natural symplectic operation by exchanging two Stein fillings of the canonical contact structure on the links in the case $p = q = r$, where one of the fillings comes from the minimal resolution and the other is the Milnor fiber. We give two different interpretations of this operation, one as a symplectic sum and the other as a monodromy substitution in a Lefschetz fibration

Evolution of the electronic excitation spectrum with strongly diminishing hole-density in superconducting Bi_{2}Sr_{2}CaCu_{2}O_{8+\delta}

A complete knowledge of its excitation spectrum could greatly benefit efforts to understand the unusual form of superconductivity occurring in the lightly hole-doped copper-oxides. Here we use tunnelling spectroscopy to measure the T\to 0 spectrum of electronic excitations N(E) over a wide range of hole-density p in superconducting Bi_{2}Sr_{2}CaCu_{2}O_{8+/delta}. We introduce a parameterization for N(E) based upon an anisotropic energy-gap /Delta (\vec k)=/Delta_{1}(Cos(k_{x})-Cos(k_{y}))/2 plus an effective scattering rate which varies linearly with energy /Gamma_{2}(E) . We demonstrate that this form of N(E) allows successful fitting of differential tunnelling conductance spectra throughout much of the Bi_{2}Sr_{2}CaCu_{2}O_{8+/delta} phase diagram. The resulting average /Delta_{1} values rise with falling p along the familiar trajectory of excitations to the 'pseudogap' energy, while the key scattering rate /Gamma_{2}^{*}=/Gamma_{2}(E=/Delta_{1}) increases from below ~1meV to a value approaching 25meV as the system is underdoped from p~16% to p<10%. Thus, a single, particle-hole symmetric, anisotropic energy-gap, in combination with a strongly energy and doping dependent effective scattering rate, can describe the spectra without recourse to another ordered state. Nevertheless we also observe two distinct and diverging energy scales in the system: the energy-gap maximum /Delta_{1} and a lower energy scale /Delta_{0} separating the spatially homogeneous and heterogeneous electronic structures.Comment: High resolution version available at: http://people.ccmr.cornell.edu/~jcdavis/files/Alldredge-condmat08010087-highres.pd