93 research outputs found
Addition of divisors on trigonal curves
In this paper we present a realisation of addition of divisors on the basis
of uniformisation of Jacobi varieties of plain algebraic curves. Addition is
realised through consecutive reduction of a degree divisor to a divisor
of degree . Representation of these divisors does not go beyond the
functions which give a solution of the Jacobi inversion problem -- the
functions of the smallest order. We focus on the simplest non-hyperelliptic
family which is the family of trigonal curves. The method can be generalised to
a curve of arbitrary gonality.Comment: 20 page
Mass in K\"ahler Geometry
We prove a simple, explicit formula for the mass of any asymptotically
locally Euclidean (ALE) K\"ahler manifold, assuming only the sort of weak
fall-off conditions required for the mass to actually be well-defined. For ALE
scalar-flat K\"ahler manifolds, the mass turns out to be a topological
invariant, depending only on the underlying smooth manifold, the first Chern
class of the complex structure, and the K\"ahler class of the metric. When the
metric is actually AE (asymptotically Euclidean), our formula not only implies
a positive mass theorem for K\"ahler metrics, but also yields a Penrose-type
inequality for the mass.Comment: 53 pages, minor corrections and improvements, final versio
On the shape of a pure O-sequence
An order ideal is a finite poset X of (monic) monomials such that, whenever M
is in X and N divides M, then N is in X. If all, say t, maximal monomials of X
have the same degree, then X is pure (of type t). A pure O-sequence is the
vector, h=(1,h_1,...,h_e), counting the monomials of X in each degree.
Equivalently, in the language of commutative algebra, pure O-sequences are the
h-vectors of monomial Artinian level algebras. Pure O-sequences had their
origin in one of Richard Stanley's early works in this area, and have since
played a significant role in at least three disciplines: the study of
simplicial complexes and their f-vectors, level algebras, and matroids. This
monograph is intended to be the first systematic study of the theory of pure
O-sequences. Our work, making an extensive use of algebraic and combinatorial
techniques, includes: (i) A characterization of the first half of a pure
O-sequence, which gives the exact converse to an algebraic g-theorem of Hausel;
(ii) A study of (the failing of) the unimodality property; (iii) The problem of
enumerating pure O-sequences, including a proof that almost all O-sequences are
pure, and the asymptotic enumeration of socle degree 3 pure O-sequences of type
t; (iv) The Interval Conjecture for Pure O-sequences (ICP), which represents
perhaps the strongest possible structural result short of an (impossible?)
characterization; (v) A pithy connection of the ICP with Stanley's matroid
h-vector conjecture; (vi) A specific study of pure O-sequences of type 2,
including a proof of the Weak Lefschetz Property in codimension 3 in
characteristic zero. As a corollary, pure O-sequences of codimension 3 and type
2 are unimodal (over any field); (vii) An analysis of the extent to which the
Weak and Strong Lefschetz Properties can fail for monomial algebras; (viii)
Some observations about pure f-vectors, an important special case of pure
O-sequences.Comment: iii + 77 pages monograph, to appear as an AMS Memoir. Several, mostly
minor revisions with respect to last year's versio
Multiple Dirichlet Series for Affine Weyl Groups
Let be the Weyl group of a simply-laced affine Kac-Moody Lie group,
excepting for even. We construct a multiple Dirichlet series
, meromorphic in a half-space, satisfying a group
of functional equations. This series is analogous to the multiple Dirichlet
series for classical Weyl groups constructed by Brubaker-Bump-Friedberg,
Chinta-Gunnells, and others. It is completely characterized by four natural
axioms concerning its coefficients, axioms which come from the geometry of
parameter spaces of hyperelliptic curves. The series constructed this way is
optimal for computing moments of character sums and L-functions, including the
fourth moment of quadratic L-functions at the central point via
and the second moment weighted by the number of divisors of the conductor via
. We also give evidence to suggest that this series appears as a
first Fourier-Whittaker coefficient in an Eisenstein series on the twofold
metaplectic cover of the relevant Kac-Moody group. The construction is limited
to the rational function field , but it also describes the
-part of the multiple Dirichlet series over an arbitrary global field
Threefold extremal contractions of type IA
Let be a germ of a threefold with terminal singularities along an
irreducible reduced complete curve with a contraction
such that C=f^{-1}(o)_{\red} and is ample. Assume that a general
member meets only at one point and furthermore is
Du Val of type A if index. We classify all such germs in terms of a
general member containing .Comment: LaTeX, 37 page
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