American Journal of Mathematics 120.3, June 1998
Contents
Damon, James.

On the legacy of free divisors: discriminants and
Morsetype singularities
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 Morsetype singularities for sections and give a criterion
for freeness of the discriminant in terms of (V, 0) generically having
Morsetype 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 453492.)
Abstract in TeX
Doering, Luisa Rodrigues.
Gunston, Tor.
Vasconcelos, Wolmer V.

Cohomological degrees and Hilbert functions
of graded modules
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 CohenMacaulay local rings to nearly arbitrary local
rings. The rules of computation these functions satisfy enables comparison with
CastelnuovoMumford's regularity in the graded case. As application, we derive sharp
improvements on predicting the outcome of effecting Noether normalizations in tangent
cones.
(Pages 493504.)
Abstract in TeX
Gardner, R.J.
Zhang, Gaoyang.

Affine inequalities and radial mean bodies
Abstract:


We introduce for p> 1 the radial pth mean body R_{p}K of a
convex body K in _{}.
The distance from the origin to the boundary of R_{p}K 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 R_{p}K 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 R_{p}K
is convex when p>0. We also establish a strong and sharp affine inequality relating the
volume of R_{p}K to that of R_{p}K when 1<p<q.
When p=n and _{},
this becomes the RogersShephard 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 505528.)
Abstract in TeX
Joshi, Mark S.
Barreto, Antônio Sá.

The generation of semilinear singularities by a
swallowtail caustic
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 529550.)
Abstract in TeX
Shioda, Tetsuji.

Constructing curves with high rank via symmetry
Abstract:


For any g > 1, we construct a curve of genus g
defined over the rational function field of many variables
Q(t_{1},..., t_{N}), 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 551566.)
Abstract in TeX
Haagerup, Uffe.
Winsløw, Carl.

The EffrosMaréchal topology in the space
of von Neumann algebras
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 continuitytype results for
TomitaTakesaki theory are proved. Some
applications to subfactor theory are given.
(Pages 567617.)
Abstract in TeX
Edidin, Dan.
Graham, William.

Localization in equivariant intersection theory
and the Bott residue formula
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 619636.)
Abstract in TeX
Kosarew, Siegmund.

Nonabelian duality on Stein spaces
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 637648.)
Abstract in TeX
To, WingKeung.
Weng, Lin.

Curvature of the L^{2}metric on the direct image
of a family of HermitianEinstein vector bundles
Abstract:


For a holomorphic family of simple HermitianEinstein
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 L^{2} pairing using the HermitianEinstein metrics. Our main result in this
paper is to compute the curvature of the L^{2}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 L^{2}metric is
conformally equivalent to a HermitianEinstein metric. As applications, this
proves the semistability of certain Picard bundles, and it leads to an
alternative proof of a theorem of Kempf.
(Pages 649661.)
Abstract in TeX
