American Journal of Mathematics

American Journal of Mathematics 120.3, June 1998

Contents

    Damon, James.
  • On the legacy of free divisors: discriminants and Morse-type singularities PDF Acrobat file
    Abstract:
    We investigate when the freeness of a divisor (V, 0) is inherited by the discriminants for the versal deformations of nonlinear sections of V. We introduce Morse-type singularities for sections and give a criterion for freeness of the discriminant in terms of (V, 0) generically having Morse-type singularities. This criterion is applied to determine when the bifurcation sets of mappings and smoothings of space curves and complete intersections are free. It also explains the failure of freeness for discriminantal arrangements of hyperplanes. (Pages 453-492.) Abstract in TeX
    Doering, Luisa Rodrigues.
    Gunston, Tor.
    Vasconcelos, Wolmer V.
  • Cohomological degrees and Hilbert functions of graded modules PDF Acrobat file
    Abstract:
    Making use of the recent construction of cohomological degrees functions, we give several estimates on the relationship between number of generators and degrees of ideals and modules with applications to Hilbert functions. They extend results heretofore known from generalized Cohen--Macaulay local rings to nearly arbitrary local rings. The rules of computation these functions satisfy enables comparison with Castelnuovo--Mumford's regularity in the graded case. As application, we derive sharp improvements on predicting the outcome of effecting Noether normalizations in tangent cones. (Pages 493-504.) Abstract in TeX
    Gardner, R.J.
    Zhang, Gaoyang.
  • Affine inequalities and radial mean bodies PDF Acrobat file
    Abstract:
    We introduce for p> -1 the radial pth mean body RpK of a convex body K in . The distance from the origin to the boundary of RpK in a given direction is the pth mean of the distances from points inside K to the boundary of K in the same direction. The bodies RpK form a spectrum linking the difference body of K and the polar projection body of K, which correspond to and p=-1, respectively. We prove that RpK is convex when p>0. We also establish a strong and sharp affine inequality relating the volume of RpK to that of RpK when -1<p<q. When p=n and , this becomes the Rogers-Shephard inequality, and when p--> -1 and q=n, it becomes a reverse form of the Petty projection inequality proved previously by the second author. (Pages 505-528.) Abstract in TeX
    Joshi, Mark S.
    Barreto, Antônio Sá.
  • The generation of semilinear singularities by a swallowtail caustic PDF Acrobat file
    Abstract:
    We construct examples of bounded solutions to a semilinear system , f smooth, , with P a strictly hyperbolic differential operator of second order with smooth coefficients, such that for a time function t of P, the following properties are satisfied: (1) u is conormal to a smooth characteristic hypersurface in t<0 such that develops a swallowtail singularity at time t=0; (2) For t>0, u is singular, not only at , but also on the forward characteristic cone over the swallowtail tip. (Pages 529-550.) Abstract in TeX
    Shioda, Tetsuji.
  • Constructing curves with high rank via symmetry PDF Acrobat file
    Abstract:
    For any g > 1, we construct a curve of genus g defined over the rational function field of many variables Q(t1,..., tN), with rank at least 4g+7. An immediate consequence is that there exists an infinite family of (nonconstant) curves of genus g over Q with rank at least 4g+7, which will improve the bound 3g+7 which Néron claimed in 1954. (Pages 551-566.) Abstract in TeX
    Haagerup, Uffe.
    Winsløw, Carl.
  • The Effros-Maréchal topology in the space of von Neumann algebras PDF Acrobat file
    Abstract:
    New concepts of limes inferior and limes superior in the space of von Neumann algebras on a fixed Hilbert space are defined, and the topology corresponding to these notions is related to earlier works of Effros and Maréchal. A main technical result is that the commutant operation is a homeomorphism on the space of von Neumann algebras with this topology. Further, the topological properties of several classes and types of von Neumann factors (regarded as subspaces) are determined, and also continuity-type results for Tomita-Takesaki theory are proved. Some applications to subfactor theory are given. (Pages 567-617.) Abstract in TeX
    Edidin, Dan.
    Graham, William.
  • Localization in equivariant intersection theory and the Bott residue formula PDF Acrobat file
    Abstract:
    We prove the localization theorem for torus actions in equivariant intersection theory. Using the theorem we give another proof of the Bott residue formula for Chern numbers of bundles on smooth complete varieties. In addition, our techniques allow us to obtain residue formulas for bundles on a certain class of singular schemes which admit torus actions. This class is rather special, but it includes some interesting examples such as complete intersections and Schubert varieties. (Pages 619-636.) Abstract in TeX
    Kosarew, Siegmund.
  • Nonabelian duality on Stein spaces PDF Acrobat file
    Abstract:
    It is well known that a Stein complex space can be recovered from its algebra of holomorphic functions. Taking the infinite dimensional Lie group of holomorphic matrices instead of holomorphic functions, we show that a similar result holds. This may be interpreted as a biduality statement in a nonabelian situation. (Pages 637-648.) Abstract in TeX
    To, Wing-Keung.
    Weng, Lin.
  • Curvature of the L2-metric on the direct image of a family of Hermitian-Einstein vector bundles PDF Acrobat file
    Abstract:
    For a holomorphic family of simple Hermitian-Einstein holomorphic vector bundles over a compact Kähler manifold, the locally free part of the associated direct image sheaf over the parameter space forms a holomorphic vector bundle, and it is endowed with a Hermitian metric given by the L2 pairing using the Hermitian-Einstein metrics. Our main result in this paper is to compute the curvature of the L2-metric. In the case of a family of Hermitian holomorphic line bundles with fixed positive first Chern form and under certain curvature conditions, we show that the L2-metric is conformally equivalent to a Hermitian-Einstein metric. As applications, this proves the semi-stability of certain Picard bundles, and it leads to an alternative proof of a theorem of Kempf. (Pages 649-661.) Abstract in TeX



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