Geometry of the Severi variety
Steven
Diaz;
Joe
Harris
1-34
Abstract: This paper is concerned with the geometry of the Severi variety $ W$ parametrizing plane curves of given degree and genus, and specifically with the relations among various divisor classes on $ W$. Two types of divisor classes on $W$ are described: those that come from the intrinsic geometry of the curves parametrized, and those characterized by extrinsic properties such as the presence of cusps, tacnodes, hyperflexes, etc. The goal of the paper is to express the classes of the extrinsically defined divisors in terms of the intrinsic ones; this, along with other calculations such as the determination of the canonical class of $W$, is carried out by using various enumerative techniques. One corollary is that the variety of nodal curves of given degree and genus in the plane is affine.
Generating combinatorial complexes of polyhedral type
Egon
Schulte
35-50
Abstract: The paper describes a method for generating combinatorial complexes of polyhedral type. Building blocks ${\mathbf{B}}$ are implanted into the maximal simplices of a simplicial complex ${\mathbf{C}}$, on which a group operates as a combinatorial reflection group. Of particular interest is the case where $ {\mathbf{B}}$ is a polyhedral block and $ {\mathbf{C}}$ the barycentric subdivision of a regular incidence-polytope ${\mathbf{K}}$ together with the action of the automorphism group of $ {\mathbf{K}}$.
A random graph with a subcritical number of edges
B.
Pittel
51-75
Abstract: A random graph ${G_n}(\operatorname{prob} (\operatorname{edge} ) = p)\;(p = c/n,\,0 < c < 1)$ on $n$ labelled vertices is studied. There are obtained limiting distributions of the following characteristics: the lengths of the longest cycle and the longest path, the total size of unicyclic components, the number of cyclic vertices, the number of distinct component sizes, and the middle terms of the component-size order sequence. For instance, it is proved that, with probability approaching ${(1 - c)^{1/2}}\exp (\sum\nolimits_{j = 1}^l {{c^j}/2j)}$ as $n \to \infty$, the random graph does not have a cycle of length $> l$. Another result is that, with probability approaching $1$, the size of the $\nu$th largest component either equals an integer closest to $a\;\log (bn/\nu \,{\log ^{5/2}}n)$, $a = a(c)$, $b = b(c)$, or is one less than this integer, provided that $\nu \to \infty $ and $\nu = o(n/{\log ^{5/2}}n)$.
Odd primary periodic phenomena in the classical Adams spectral sequence
Paul
Shick
77-86
Abstract: We study certain periodic phenomena in the cohomology of the $ \bmod \;p$ Steenrod algebra which are related to the polynomial generators $ {v_n} \in {\pi _{\ast}}BP$. A chromatic resolution of the ${E_2}$ term of the classical Adams spectral sequence is constructed.
Carleson measures and multipliers of Dirichlet-type spaces
Ron
Kerman;
Eric
Sawyer
87-98
Abstract: A function $ \rho$ from $[0,\,1]$ onto itself is a Dirichlet weight if it is increasing, $\rho '' \leqslant 0$ and ${\lim _{x \to 0 + }}x/\rho (x) = 0$. The corresponding Dirichlet-type space, ${D_\rho }$, consists of those bounded holomorphic functions on $U = \{ z \in {\mathbf{C}}:\,\vert z\vert < 1\}$ such that $M({D_\rho }) = \{ g:\,U \to {\mathbf{C}}:gf \in {D_\rho },\forall f \in {D_\rho }\}$.
Zelevinski algebras related to projective representations
M.
Bean;
P.
Hoffman
99-111
Abstract: We define $ L$- $\operatorname{PSH} $-algebras, and prove a classification theorem for such objects. The letters refer respectively to a ground ring $L$ and to the positivity, selfadjointness and Hopf structures on an algebra, the basic example of which occurred in the study of projective representations of ${S_n}$. This is analogous to an idea over ${\mathbf{Z}}$ due to Zelevinski in connection with linear representations.
A sparse Graham-Rothschild theorem
Hans Jürgen
Prömel;
Bernd
Voigt
113-137
Abstract: The main result of this paper is a sparse version of the Graham-Rothschild partition theorem for $n$-parameter sets [R. L. Graham and B. L. Rothschild, Ramsey's theorem for $n$-parameter sets, Trans. Amer. Math. Soc. 159 (1971), 257-292]. In particular, a sparse version of Hales-Jewett's theorem is proved. We give several applications, e.g., for arithmetic progressions and finite sums of integers, confirming conjectures of J. Spencer and of J. Nešetřil and V. Rödl. We also consider graphs defined on parameter sets and prove a sparse and restricted induced partition theorem for such graphs, extending results from [H. J. Prömel, Induced partition properties of combinatorial cubes, J. Combin. Theory Ser. A 39 (1985), 177-208] and [P. Frankl, R. L. Graham, and V. Rödl, Induced restricted Ramsey theorems for spaces, J. Combin. Theory Ser. A 44 (1987), 120-128].
Function spaces generated by blocks associated with spheres, Lie groups and spaces of homogeneous type
Aleš
Založnik
139-164
Abstract: Functions generated by blocks were introduced by M. Taibleson and G. Weiss in the setting of the one-dimensional torus $ T$ [TW1]. They showed that these functions formed a space "close" to the class of integrable functions for which we have almost everywhere convergence of Fourier series. Together with S. Lu [LTW] they extended the theory to the $n$-dimensional torus where this convergence result (for Bochner-Riesz means at the critical index) is valid provided we also restrict ourselves to $ L\log L$. In this paper we show that this restriction is not needed if the underlying domain is a compact semisimple Lie group (or certain more general spaces of a homogeneous type). Other considerations (for example, these spaces form an interesting family of quasi-Banach spaces; they are connected with the notion of entropy) guide one in their study. We show how this point of view can be exploited in the setting of more general underlying domains.
Pseudoconvex classes of functions. III. Characterization of dual pseudoconvex classes on complex homogeneous spaces
Zbigniew
Slodkowski
165-189
Abstract: Invariant classes of functions on complex homogeneous spaces, with properties similar to those of the class of plurisubharmonic functions, are studied. The main tool is a regularization method for these classes, and the main theorem characterizes dual classes of functions (where duality is defined in terms of the local maximum property). These results are crucial in proving a duality theorem for complex interpolation of normed spaces, which is given elsewhere.
Umbral calculus, binomial enumeration and chromatic polynomials
Nigel
Ray
191-213
Abstract: We develop the concept of partition categories, in order to extend the Mullin-Rota theory of binomial enumeration, and simultaneously to provide a natural setting for recent applications of the Roman-Rota umbral calculus to computations in algebraic topology. As a further application, we describe a generalisation of the chromatic polynomial of a graph.
Trace Paley-Wiener theorem in the twisted case
J. D.
Rogawski
215-229
Abstract: A version of the trace Paley-Wiener theorem for a reductive $p$-adic group in the context of twisted harmonic analysis with respect to an outer automorphism is proved.
Valuations on meromorphic functions of bounded type
Mitsuru
Nakai
231-252
Abstract: The primary purpose of this paper is to show that every valuation on the field of meromorphic functions of bounded type on a finitely sheeted unlimited covering Riemann surface is a point valuation if and only if the same is true on its base Riemann surface. The result is then applied to concrete examples and some related results are obtained.
The Bergman spaces, the Bloch space, and Gleason's problem
Ke He
Zhu
253-268
Abstract: Suppose $ f$ is a holomorphic function on the open unit ball ${B_n}$ of $ {{\mathbf{C}}^n}$. For $1 \leqslant p < \infty$ and $m > 0$ an integer, we show that $f$ is in $ {L^p}({B_n},\,dV)$ (with $ dV$ the volume measure) iff all the functions $ {\partial ^m}f/\partial {z^{\alpha \,}}\;(\vert\alpha \vert\, = m)$ are in ${L^p}({B_n},\,dV)$. We also prove that $ f$ is in the Bloch space of ${B_n}$ iff all the functions $ {\partial ^m}f/\partial {z^\alpha }\;(\vert\alpha \vert\, = m)$ are bounded on ${B_n}$. The corresponding result for the little Bloch space of ${B_n}$ is established as well. We will solve Gleason's problem for the Bergman spaces and the Bloch space of ${B_n}$ before proving the results stated above. The approach here is functional analytic. We make extensive use of the reproducing kernels of $ {B_n}$. The corresponding results for the polydisc in ${{\mathbf{C}}^n}$ are indicated without detailed proof.
Representations of Hecke algebras
Eugene
Gutkin
269-277
Abstract: We find all operators of a certain type that satisfy the braid relations corresponding to any generalized Cartan matrix.
Kazhdan-Lusztig polynomials for Hermitian symmetric spaces
Brian D.
Boe
279-294
Abstract: A nonrecursive scheme is presented to compute the Kazhdan-Lusztig polynomials associated to a classical Hermitian symmetric space, extending a result of Lascoux-Schützenberger for grassmannians. The polynomials for the exceptional Hermitian domains are also tabulated. All the Kazhdan-Lusztig polynomials considered are shown to be monic.
On the dual of an exponential solvable Lie group
Bradley N.
Currey
295-307
Abstract: Let $G$ be a connected, simply connected exponential solvable Lie group with Lie algebra $\mathfrak{g}$. The Kirillov mapping $\eta :\,\,\mathfrak{g}{\ast}/\operatorname{Ad} {\ast}(G) \to \hat G$ gives a natural parametrization of $\hat G$ by co-adjoint orbits and is known to be continuous. In this paper a finite partition of $ \mathfrak{g}{\ast}/\operatorname{Ad} {\ast}(G)$ is defined by means of an explicit construction which gives the partition a natural total ordering, such that the minimal element is open and dense. Given $ \pi \in \hat G$, elements in the enveloping algebra of ${\mathfrak{g}_c}$ are constructed whose images under $\pi$ are scalar and give crucial information about the associated orbit. This information is then used to show that the restriction of $\eta$ to each element of the above-mentioned partition is a homeomorphism.
On the canonical rings of some Horikawa surfaces. I
Valentin
Iliev
309-323
Abstract: This paper is devoted to finding necessary and sufficient conditions for a graded ring to be the canonical ring of a minimal surface of general type with ${K^2} = 2{p_g} - 3$, $ {p_g} \geqslant 3$, and such that its canonical linear system has one base point.
A Diophantine problem on elliptic curves
Robert
Tubbs
325-338
Abstract: This paper examines simultaneous diophantine approximations to coordinates of certain points on a product of elliptic curves. Specifically, let $\wp (z)$ be a Weierstrass elliptic function with algebraic invariants and complex multiplication. Suppose that $ \beta$ is cubic over the "field of multiplications" of $\wp (z)$ and that $u \in \mathbb{C}$ such that $\zeta = (\wp (u),\,\wp (\beta u),\,\wp ({\beta ^2}u))$ is defined. We study approximations to $ \zeta$ by points which lie on curves defined over $ \mathbb{Z}$.
Boundary behavior of invariant Green's potentials on the unit ball in ${\bf C}\sp n$
K. T.
Hahn;
David
Singman
339-354
Abstract: Let $p(z) = \int_B {G(z,\,w)\,d\mu (w)}$ be an invariant Green's potential on the unit ball $ B$ in $ {{\mathbf{C}}^n}\;(n \geqslant 1)$, where $G$ is the invariant Green's function and $ \mu$ is a positive measure with $\int_B {{{(1 - \vert w{\vert^2})}^n}\,d\mu (w) < \infty }$. In this paper, a necessary and sufficient condition on a subset $E$ of $B$ such that for every invariant Green's potential $ p$, $\displaystyle \mathop {\lim }\limits_{z \to e} \,\inf {(1 - \vert z{\vert^2})^n}p(z) = 0,\qquad e = (1,\,0,\, \ldots ,\,0)\; \in \partial B,\;z \in E,$ is given. The condition is that the capacity of the sets $ E \cap \{ z \in B\vert\;\vert z - e\vert < \varepsilon \}$, $\varepsilon > 0$, is bounded away from 0. The result obtained here generalizes Luecking's result, see [L], on the unit disc in ${\mathbf{C}}$.
Symmetry diffeomorphism group of a manifold of nonpositive curvature
Patrick
Eberlein
355-374
Abstract: Let $\tilde M$ denote a complete simply connected manifold of nonpositive sectional curvature. For each point $p \in \tilde M$ let $ {s_p}$ denote the diffeomorphism of $\tilde M$ that fixes $p$ and reverses all geodesics through $ p$. The symmetry diffeomorphism group $ {G^{\ast}}$ generated by all diffeomorphisms $\{ {s_p}:\,p \in \tilde M\} $ extends naturally to group of homeomorphisms of the boundary sphere $\tilde M(\infty )$. A subset $X$ of $ \tilde M(\infty )$ is called involutive if it is invariant under ${G^{\ast}}$. Theorem. Let $X \subseteq \tilde M(\infty )$ be a proper, closed involutive subset. For each point $p \in \tilde M$ let $ N(p)$ denote the linear span in $ {T_p}\tilde M$ of those vectors at $p$ that are tangent to a geodesic $ \gamma$ whose asymptotic equivalence class $\gamma (\infty )$ belongs to $ X$. If $ N(p)$ is a proper subspace of $ {T_p}\tilde M$ for some point $p \in \tilde M$, then $ \tilde M$ splits as a Riemannian product ${\tilde M_1} \times {\tilde M_2}$ such that $N$ is the distribution of $ \tilde M$ induced by ${\tilde M_1}$. This result has several applications that include new results as well as great simplifications in the proofs of some known results. In a sequel to this paper it is shown that if $ \tilde M$ is irreducible and $ \tilde M(\infty )$ admits a proper, closed involutive subset $X$, then $\tilde M$ is isometric to a symmetric space of noncompact type and rank $ k \geqslant 2$.
On zeros of a system of polynomials and application to sojourn time distributions of birth-and-death processes
Ken-iti
Sato
375-390
Abstract: Zeros of the following system of polynomials are considered: $\displaystyle \left\{ \begin{gathered}{P_0}(x) = 1, {P_1}(x) = {B_0} ... ..._{n - 1}}(x)\quad {\text{for}}\;n \geqslant 1. \end{gathered} \right.$ Numbers of positive and negative zeros are determined and a separation property of the zeros of $ {P_m}(x)$ and $ {P_n}(x)$ is proved under the condition that ${C_n} > 0$ and $ {P_n}(0) > 0$ for every $ n$. No condition is imposed on ${A_n}$. These results are applied to determination of the distribution of a sojourn time with general (not necessarily positive) weight function for a birth-and-death process up to a first passage time. Unimodality and infinite divisibility of the distribution follow.
Local $H$-maps of $B{\rm U}$ and applications to smoothing theory
Timothy
Lance
391-424
Abstract: When localized at an odd prime $p$, the classifying space $PL/O$ for smoothing theory splits as an infinite loop space into the product $C \times N$ where $C = {\text{Cokernel}}\,(J)$ and $N$ is the fiber of a $p$-local $H$-map $BU \to BU$. This paper studies spaces which arise in this latter fashion, computing the cohomology of their Postnikov towers and relating their $ k$-invariants to properties of the defining self-maps of $BU$. If $Y$ is a smooth manifold, the set of homotopy classes $ [Y,\,N]$ is a certain subgroup of resmoothings of $Y$, and the $k$-invariants of $N$ generate obstructions to computing that subgroup. These obstructions can be directly related to the geometry of $Y$ and frequently vanish.
The fundamental module of a normal local domain of dimension $2$
Yuji
Yoshino;
Takuji
Kawamoto
425-431
Abstract: The fundamental module $E$ of a normal local domain $(R,\,\mathfrak{m})$ of dimension $2$ is defined by the nonsplit exact sequence $0 \to K \to E \to \mathfrak{m} \to 0$, where $K$ is the canonical module of $R$. We prove that, if $R$ is complete with $R/\mathfrak{m} \simeq \mathbb{C}$, then $ E$ is decomposable if and only if $R$ is a cyclic quotient singularity. Various other properties of fundamental modules will be discussed.