title:
Mechanism of Self-regulation in a Simple Model of Hierarchically
  Organized Market

abstract:
  We propose a model for a market which structure is of the tree form. Each
branch of the tree is composed by identical firms, its root (the branch of the
first level) is formed by the firms producing raw material, and the branches of
the last level are the retail outlets. The branching points (tree nodes) are
micromarkets for firms forming branches connected with a given node. The prices
and the production rate are controlled by the ballance in supply and demand,
the competition is assumed to be perfect. We show that such a market functions
perfectly: the prices are specified by the production expense only, whereas
demand determines the production rate.
  We construct an efficiency functional which extremal gives the governing
equations for the market. It turns out that this ideal market is degenerated
with respect to its structure. It is shown, that such market functions ideally:
the prices are determined by the costs on production of the goods, and the
level of production of the goods of any kind defined only by demand for the
goods of this sort.


title: Replication Error and Time Evolution of a Self-Replicating System

abstract:
  A self-replicating system where the elements belonging to a solution category
can replicate themselves by copying their own informations, is considered. The
information carried by each element is defined by an element of all the n
multiple tensor product of a base space that consists of m different base
elements. We assume that in the replication the processes of copying each base
information are the same and independent from one another and that the copying
error distribution in each process is characterized by a small variation with a
quite small mean value. Concentrating on the number fluctuation of the
informations in the copying process, we analyze the time evolution of the
system. We illustrate the change of averaged number of informations carried by
system objects and the variation of the number distribution as a function of
time. Especially, it is shown that the averaged number of information grows in
general after large number of generations.


title: Neutral Evolution of Mutational Robustness

abstract:
  We introduce and analyze a general model of a population evolving over a
network of selectively neutral genotypes. We show that the population's limit
distribution on the neutral network is solely determined by the network
topology and given by the principal eigenvector of the network's adjacency
matrix. Moreover, the average number of neutral mutant neighbors per individual
is given by the matrix spectral radius. This quantifies the extent to which
populations evolve mutational robustness: the insensitivity of the phenotype to
mutations. Since the average neutrality is independent of evolutionary
parameters---such as, mutation rate, population size, and selective
advantage---one can infer global statistics of neutral network topology using
simple population data available from {\it in vitro} or {\it in vivo}
evolution. Populations evolving on neutral networks of RNA secondary structures
show excellent agreement with our theoretical predictions.


title: Characterizing self-organization and coevolution by ergodic invariants

abstract:
  In addition to the emergent complexity of patterns that appears when many
agents come in interaction, it is also useful to characterize the dynamical
processes that lead to their self-organization. A set of ergodic invariants is
identified for this purpose, which is computed in several examples, namely a
Bernoulli network with either global or nearest-neighbor coupling, a
generalized Bak-Sneppen model and a continuous minority model.


title: Evolution in Minority Games I. Games with a Fixed Strategy Space

abstract:
  In this paper we study the minority game in the presence of evolution. In
particular, we examine the behavior in games in which the dimension of the
strategy space, m, is the same for all agents and fixed for all time. We find
that for all values of m, not too large, evolution results in a substantial
improvement in overall system performance. We also show that after evolution,
results obey a scaling relation among games played with different values of m
and different numbers of agents, analogous to that found in the
non-evolutionary, adaptive games. Best system performance still occurs, for a
given number of agents, at m_c, the same value of the dimension of the strategy
space as in the non-evolutionary case, but system performance is now nearly an
order of magnitude better than the non- evolutionary result. For m<m_c, the
system evolves to states in which average agent wealth is better than in the
random choice game,despite (and in some sense because of) the persistence of
maladaptive behavior by some agents. As m gets large, overall systems
performance approaches that of the random choice game.


title: Correlated random walks with a finite memory range

abstract:
  We study a family of correlated one-dimensional random walks with a finite
memory range M.These walks are extensions of the Taylor's walk as investigated
by Goldstein, which has a memory range equal to one. At each step, with a
probability p, the random walker moves either to the right or to the left with
equal probabilities, or with a probability q=1-p performs a move, which is a
stochastic Boolean function of the M previous steps. We first derive the most
general form of this stochastic Boolean function, and study some typical cases
which ensure that the average value <R_n> of the walker's location after n
steps is zero for all values of n. In each case, using a matrix technique, we
provide a general method for constructing the generating function of the
probability distribution of R_n; we also establish directly an exact analytic
expression for the step-step correlations and the variance <R_n^2> of the walk.
  From the expression of <R_n^2>, which is not straightforward to derive from
the probability distribution, we show that, for n going to infinity, the
variance of any of these walks behaves as n, provided p>0. Moreover, in many
cases, for a very small fixed value of p, the variance exhibits a crossover
phenomenon as $n$ increases from a not too large value. The crossover takes
place for values of $n$ around 1/p. This feature may mimic the existence of a
non-trivial Hurst exponent, and induce a misleading analysis of numerical data
issued from mathematical or natural sciences experiments.


title:
Totalistic two-dimensional cellular automata exhibiting global periodic
  behavior

abstract:
  We have determined families of two-dimensional deterministic totalistic
cellular automaton rules whose stationary density of active sites exhibits a
period two in time. Each family of deterministic rules is characterized by an
``average probabilistic totalistic rule'' exhibiting the same periodic
behavior.


title: Three sided complex adaptative systems

abstract:
  We introduce two three sided adaptative systems as toy models to mimic the
exchange of commodities between buyers and sellers. These models are simple
extensions of the minority game, exhibiting similar behaviour as well as some
new features. The main difference between our two models is that in the first
the three sides are equivalent while in the second, one choice appears as a
compromise between the two other sides. Both models are investigated
numerically and compared with the original minority game.


title: Evolution dynamics in terraced NK landscapes

abstract:
  We study populations of agents evolving in fitness landscapes constructed
according to the rules of a modified NK model with a tunable amount of neutral
paths. In the `punctuated equilibrium' regime evolutionary events are
identified as jumps in the mean population fitness, and the statistics of these
jumps in an ensemble of independently evolving populations is analyzed. We show
that, for a wide range of landscapes parameters, the number of events in time t
is Poisson distributed, with the time parameter replaced by the logarithm of
time. This simple log-Poisson statistics likewise describes the number n of
records in any sequence of t independently generated random numbers. The
implications of this behavior for evolution dynamics are discussed.


title: Lattice three-species models of the spatial spread of rabies among foxes

abstract:
  Lattice models describing the spatial spread of rabies among foxes are
studied. In these models, the fox population is divided into three-species:
susceptible, infected or incubating, and infectious or rabid. They are based on
the fact that susceptible and incubating foxes are territorial while rabid
foxes have lost their sense of direction and move erratically. Two different
models are investigated: a one-dimensional coupled-map lattice model, and a
two-dimensional automata network model. Both models take into account the
short-range character of the infection process and the diffusive motion of
rabid foxes. Numerical simulations show how the spatial distribution of rabies,
and the speed of propagation of the epizootic front depend upon the carrying
capacity of the environment and diffusion of rabid foxes out of their
territory.


title: Extremal Coupled Map Lattices

abstract:
  We propose a model for co-evolving ecosystems that takes into account two
levels of description of an organism, for instance genotype and phenotype.
Performance at the macroscopic level forces mutations at the microscopic level.
These, in turn, affect the dynamics of the macroscopic variables. In some
regions of parameter space, the system self-organises into a state with
localised activity and power law distributions.


title: Model of self-replicating cell capable of self-maintenance

abstract:
  We construct a simple model of a proto-cell that simulates a stochastic
dynamics of abstract chemicals on a two-dimensional lattice. We assume that
chemicals catalyze their reproduction through interaction with each other, and
that between some chemicals repulsion occurs. We have shown that chemicals
organize themselves into a cell-like structure that maintains its membranes
dynamically. Further, we have obtained cells that can divide themselves
automatically into daughter cells.


title: Noise-induced memory in extended excitable systems

abstract:
  We describe a form of memory exhibited by extended excitable systems driven
by stochastic fluctuations. Under such conditions, the system self-organizes
into a state characterized by power-law correlations thus retaining long-term
memory of previous states. The exponents are robust and model-independent. We
discuss novel implications of these results for the functioning of cortical
neurons as well as for networks of neurons.


title: Number-conserving cellular automaton rules

abstract:
  A necessary and sufficient condition for a one-dimensional q-state n-input
cellular automaton rule to be number-conserving is established. Two different
forms of simpler and more visual representations of these rules are given, and
their flow diagrams are determined. Various examples are presented and
applications to car traffic are indicated. Two nontrivial three-state
three-input self-conjugate rules have been found. They can be used to model the
dynamics of random walkers.


title:
A microsimulation of traders activity in the stock market: the role of
  heterogeneity, agents' interactions and trade frictions

abstract:
  We propose a model with heterogeneous interacting traders which can explain
some of the stylized facts of stock market returns. In the model
synchronization effects, which generate large fluctuations in returns, can
arise either from an aggregate exogenous shock or, even in its absence, purely
from communication and imitation among traders. A trade friction is introduced
which, by responding to price movements, creates a feedback mechanism on future
trading and generates volatility clustering.


title: Globally coupled bistable elements as a model of group decision making

abstract:
  A simple mathematical model is proposed to study the effect of the average
trend of a population on the opinion of each individual, when a group decision
has to be made by voting. It is shown that if such effect is strong enough a
transition to coherent behavior occurs, in which the whole population converges
to a single opinion. This transition has the character of a first-order
critical phenomenon.


title: Evolution in Minority Games II. Games with Variable Strategy Spaces

abstract:
  We continue our study of evolution in minority games by examining games in
which agents with poorly performing strategies can trade in their strategies
for new ones from a different strategy space. In the context of the games
discussed in this paper, this means allowing for strategies that use
information from different numbers of time lags, m. We find, in all the games
we study, that after evolution, wealth per agent is high for agents with
strategies drawn from small strategy spaces (small m), and low for agents with
strategies drawn from large strategy spaces (large m). In the game played with
N agents, wealth per agent as a function of m is very nearly a step function.
The transition is at m=mt, where mt~mc-1. Here mc is the critical value of m at
which N agents playing the game with a fixed strategy space (fixed m) have the
best emergent coordination and the best utilization of resources. We also find
that overall system-wide utilization of resources is independent of N.
Furthermore, although overall system-wide utilization of resources after
evolution varies somewhat depending on some other aspects of the evolutionary
dynamics, in the best cases, utilization of resources is on the order of the
best results achieved in evolutionary games with fixed strategy spaces. Simple
explanations are presented for some of our main results.


title: Minimum Energy Information Fusion in Sensor Networks

abstract:
  In this paper we consider how to organize the sharing of information in a
distributed network of sensors and data processors so as to provide
explanations for sensor readings with minimal expenditure of energy. We point
out that the Minimum Description Length principle provides an approach to
information fusion that is more naturally suited to energy minimization than
traditional Bayesian approaches. In addition we show that for networks
consisting of a large number of identical sensors Kohonen self-organization
provides an exact solution to the problem of combining the sensor outputs into
minimal description length explanations.


title:
An Exactly Soluble Hierarchical Clustering Model: Inverse Cascades,
  Self-Similarity, and Scaling

abstract:
  We show how clustering as a general hierarchical dynamical process proceeds
via a sequence of inverse cascades to produce self-similar scaling, as an
intermediate asymptotic, which then truncates at the largest spatial scales. We
show how this model can provide a general explanation for the behavior of
several models that has been described as ``self-organized critical,''
including forest-fire, sandpile, and slider-block models.


title: Noise-induced breakdown of coherent collective motion in swarms

abstract:
  We consider swarms formed by populations of self-propelled particles with
attractive long-range interactions. These swarms represent multistable
dynamical systems and can be found either in coherent traveling states or in an
incoherent oscillatory state where translational motion of the entire swarm is
absent. Under increasing the noise intensity, the coherent traveling state of
the swarms is destroyed and an abrupt transition to the oscillatory state takes
place.


title: Lyapunov Exponents in Random Boolean Networks

abstract:
  A new order parameter approximation to Random Boolean Networks (RBN) is
introduced, based on the concept of Boolean derivative. A statistical argument
involving an annealed approximation is used, allowing to measure the order
parameter in terms of the statistical properties of a random matrix. Using the
same formalism, a Lyapunov exponent is calculated, allowing to provide the
onset of damage spreading through the network and how sensitive it is to
minimal perturbations. Finally, the Lyapunov exponents are obtained by means of
different approximations: through distance method and a discrete variant of the
Wolf's method for continuous systems.


title:
Metastable Evolutionary Dynamics: Crossing Fitness Barriers or Escaping
  via Neutral Paths?

abstract:
  We analytically study the dynamics of evolving populations that exhibit
metastability on the level of phenotype or fitness. In constant selective
environments, such metastable behavior is caused by two qualitatively different
mechanisms. One the one hand, populations may become pinned at a local fitness
optimum, being separated from higher-fitness genotypes by a {\em fitness
barrier} of low-fitness genotypes. On the other hand, the population may only
be metastable on the level of phenotype or fitness while, at the same time,
diffusing over {\em neutral networks} of selectively neutral genotypes.
Metastability occurs in this case because the population is separated from
higher-fitness genotypes by an {\em entropy barrier}: The population must
explore large portions of these neutral networks before it discovers a rare
connection to fitter phenotypes.
  We derive analytical expressions for the barrier crossing times in both the
fitness barrier and entropy barrier regime. In contrast with ``landscape''
evolutionary models, we show that the waiting times to reach higher fitness
depend strongly on the width of a fitness barrier and much less on its height.
The analysis further shows that crossing entropy barriers is faster by orders
of magnitude than fitness barrier crossing. Thus, when populations are trapped
in a metastable phenotypic state, they are most likely to escape by crossing an
entropy barrier, along a neutral path in genotype space. If no such escape
route along a neutral path exists, a population is most likely to cross a
fitness barrier where the barrier is {\em narrowest}, rather than where the
barrier is shallowest.


title: Strategy Selection in the Minority Game

abstract:
  We investigate the dynamics of the choice of an active strategy in the
minority game. A history distribution is introduced as an analytical tool to
study the asymmetry between the two choices offered to the agents. Its
properties are studied numerically. It allows us to show that the departure
from uniformity in the initial attribution of strategies to the agents is
important even in the efficient market. Also, an approximate expression for the
variance of the number of agents at one side in the efficient phase is
proposed. All the analytical propositions are supported by numerical
simulations of the system.


title: A dynamical model of non regulated markets

abstract:
  The main focus of this work is to understand the dynamics of non regulated
markets. The present model can describe the dynamics of any market where the
pricing is based on supply and demand. It will be applied here, as an example,
for the German stock market presented by the Deutscher Aktienindex (DAX), which
is a measure for the market status. The duality of the present model consists
of the superposition of the two components - the long and the short term
behaviour of the market. The long term behaviour is characterised by a stable
development which is following a trend for time periods of years or even
decades. This long term growth (or decline) is based on the development of
fundamental market figures. The short term behaviour is described as a
dynamical evaluation (trading) of the market by the participants. The trading
process is described as an exchange between supply and demand. In the framework
of this model there the trading is modelled by a system of nonlinear
differential equations. The model also allows to explain the chaotic behaviour
of the market as well as periods of growth or crashes.


title: Functional Dynamics II : Syntactic Structure

abstract:
  Functional dynamics, introduced in a previous paper, is analyzed, focusing on
the formation of a hierarchical rule to determine the dynamics of the
functional value. To study the periodic (or non-fixed) solution, the functional
dynamics is separated into fixed and non-fixed parts. It is shown that the
fixed parts generate a 1-dimensional map by which the dynamics of the
functional values of some other parts are determined. Piecewise-linear maps
with multiple branches are generally created, while an arbitrary
one-dimensional map can be embedded into this functional dynamics if the
initial function coincides with the identity function over a finite interval.
Next, the dynamics determined by the one-dimensional map can again generate a
`meta-map', which determines the dynamics of the generated map. This hierarchy
of meta-rules can continue recursively. It is also shown that the dynamics can
produce `meta-chaos' with an orbital instability that is stronger than
exponential. The relevance of the generated hierarchy to biological and
language systems is discussed, in relation with the formation of syntax of a
network.


title: Functional Dynamics I : Articulation Process

abstract:
  The articulation process of dynamical networks is studied with a functional
map, a minimal model for the dynamic change of relationships through iteration.
The model is a dynamical system of a function $f$, not of variables, having a
self-reference term $f \circ f$, introduced by recalling that operation in a
biological system is often applied to itself, as is typically seen in rules in
the natural language or genes. Starting from an inarticulate network, two types
of fixed points are formed as an invariant structure with iterations. The
function is folded with time, until it has finite or infinite piecewise-flat
segments of fixed points, regarded as articulation. For an initial logistic
map, attracted functions are classified into step, folded step, fractal, and
random phases, according to the degree of folding. Oscillatory dynamics are
also found, where function values are mapped to several fixed points
periodically. The significance of our results to prototype categorization in
language is discussed.


title: Resource Sharing and Coevolution in Evolving Cellular Automata

abstract:
  Evolving one-dimensional cellular automata (CAs) with genetic algorithms has
provided insight into how improved performance on a task requiring global
coordination emerges when only local interactions are possible. Two approaches
that can affect the search efficiency of the genetic algorithm are coevolution,
in which a population of problems---in our case, initial configurations of the
CA lattice---evolves along with the population of CAs; and resource sharing, in
which a greater proportion of a limited fitness resource is assigned to those
CAs which correctly solve problems that fewer other CAs in the population can
solve. Here we present evidence that, in contrast to what has been suggested
elsewhere, the improvements observed when both techniques are used together
depend largely on resource sharing alone.


title: Autoregressive model of 1/f noise

abstract:
  An analytically solvable model is proposed exhibiting 1/f spectrum in any
desirably wide range of frequency (but excluding the point f=0). The model
consists of pulses whose recurrence times obey an autoregressive process with
very small damping.


title:
Cumulant Dynamics of a Population under Multiplicative Selection,
  Mutation and Drift

abstract:
  We revisit the classical population genetics model of a population evolving
under multiplicative selection, mutation and drift. The number of beneficial
alleles in a multi-locus system can be considered a trait under exponential
selection. Equations of motion are derived for the cumulants of the trait
distribution in the diffusion limit and under the assumption of linkage
equilibrium. Because of the additive nature of cumulants, this reduces to the
problem of determining equations of motion for the expected allele distribution
cumulants at each locus. The cumulant equations form an infinite dimensional
linear system and in an authored appendix Adam Prugel-Bennett provides a closed
form expression for these equations. We derive approximate solutions which are
shown to describe the dynamics well for a broad range of parameters. In
particular, we introduce two approximate analytical solutions: (1) Perturbation
theory is used to solve the dynamics for weak selection and arbitrary mutation
rate. The resulting expansion for the system's eigenvalues reduces to the known
diffusion theory results for the limiting cases with either mutation or
selection absent. (2) For low mutation rates we observe a separation of
time-scales between the slowest mode and the rest which allows us to develop an
approximate analytical solution for the dominant slow mode. The solution is
consistent with the perturbation theory result and provides a good
approximation for much stronger selection intensities.


title: Scaling in a Multispecies Network Model Ecosystem

abstract:
  A new model ecosystem consisting of many interacting species is introduced.
The species are connected through a random matrix with a given connectivity. It
is shown that the system is organized close to a boundary of marginal stability
in such a way that fluctuations follow power law distributions both in species
abundance and their lifetimes for some slow-driving (immigration) regime. The
connectivity and the number of species are linked through a scaling relation
which is the one observed in real ecosystems. These results suggest that the
basic macroscopic features of real, species-rich ecologies might be linked with
a critical state. A natural link between lognormal and power law distributions
of species abundances is suggested.


title: Phase Transition in Random Networks with Multiple States

abstract:
  The critical boundaries separating ordered from chaotic behavior in randomly
wired S-state networks are calculated. These networks are a natural
generalization of random Boolean nets and are proposed as on extended approach
to genetic regulatory systems, sets of cells in different states or collectives
of agents engaged into a set of S possible tasks. A order parameter for the
transition is computed and analysed. The relevance of these networks to
biology, their relationships with standard cellular automata and possible
extensions are outlined.


title: A Complete Enumeration and Classification of Two-Locus Disease Models

abstract:
  There are 512 two-locus, two-allele, two-phenotype, fully-penetrant disease
models. Using the permutation between two alleles, between two loci, and
between being affected and unaffected, one model can be considered to be
equivalent to another model under the corresponding permutation. These
permutations greatly reduce the number of two-locus models in the analysis of
complex diseases. This paper determines the number of non-redundant two-locus
models (which can be 102, 100, 96, 51, 50, or 48, depending on which
permutations are used, and depending on whether zero-locus and single-locus
models are excluded). Whenever possible, these non-redundant two-locus models
are classified by their property. Besides the familiar features of
multiplicative models (logical AND), heterogeneity models (logical OR), and
threshold models, new classifications are added or expanded: modifying-effect
models, logical XOR models, interference and negative interference models
(neither dominant nor recessive), conditionally dominant/recessive models,
missing lethal genotype models, and highly symmetric models. The following
aspects of two-locus models are studied: the marginal penetrance tables at both
loci, the expected joint identity-by-descent probabilities, and the correlation
between marginal identity-by-descent probabilities at the two loci. These
studies are useful for linkage analyses using single-locus models while the
underlying disease model is two-locus, and for correlation analyses using the
linkage signals at different locations obtained by a single-locus model.


title: Models of Extinction: A Review

abstract:
  We review recent work aimed at modeling species extinction over geological
time. We discuss a number of models which, rather than dealing with the direct
causes of particular extinction events, attempt to predict overall statistical
trends, such as the relative frequencies of large and small extinctions, or the
distribution of the lifetimes of species, genera or higher taxa. We also
describe the available fossil and other data, and compare the trends visible in
these data with the predictions of the models.


title: Colective Effects from Induced Behaviour

abstract:
  We present a solvable model for describing quantitatively situations where
the individual behaviour of agents in a group "percolates" to collective
behaviour of the group as a whole as a result of mutual influence between the
agents.


title: Social Percolation

abstract:
  We introduce the simplest model which relates the emergence of collective
social/economic phenomena to the existence of a (possibly self-organized)
percolation transition. We suggest a series of extensions to financial,
economic, political systems.


title: Statistical measures of complexity for strongly interacting systems

abstract:
  In recent studies, new measures of complexity for nonlinear systems have been
proposed based on probabilistic grounds, as the LMC measure (Phys. Lett. A {\bf
209} (1995) 321) or the SDL measure (Phys. Rev. E {\bf 59}
  (1999) 2). All these measures share an intuitive consideration: complexity
seems to emerge in nature close to instability points, as for example the phase
transition points characteristic of critical phenomena. Here we discuss these
measures and their reliability for detecting complexity close to critical
points in complex systems composed of many interacting units. Both a
two-dimensional spatially extended problem (the 2D Ising model) and a
$\infty$-dimensional (random graph) model (random Boolean networks) are
analysed. It is shown that the LMC and the SDL measures can be easily
generalized to extended systems but fails to detect real complexity.


title:
Analysis of the optimality principles responsible for vascular network
  architectonics

abstract:
  The equivalence of two optimality principles leading to Murray's law has been
discussed. The first approach is based on minimization of biological work
needed for maintaining the blood flow through the vessels at required level.
The second one is the principle of minimal drag and lumen volume.
Characteristic features of these principles are considered. An alternative
approach leading to Murray's law has been proposed. For that we model the
microcirculatory bed in terms of delivering vascular network with symmetrical
bifurcation nodes, embedded uniformly into the cellular tissue. It was shown
that Murray's law can be regarded as a direct consequence of the organism
capacity for controlling the blood flow redistribution over the
microcirculatory beds.


title: Measuring Mutual Information in Random Boolean Networks

abstract:
  During the last few years an area of active research in the field of complex
systems is that of their information storing and processing abilities. Common
opinion has it that the most interesting beaviour of these systems is found
``at the edge of chaos'', which would seem to suggest that complex systems may
have inherently non-trivial information proccesing abilities in the vicinity of
sharp phase transitions. A comprenhensive, quantitative understanding of why
this is the case is however still lacking. Indeed, even ``experimental'' (i.e.,
often numerical) evidence that this is so has been questioned for a number of
systems. In this paper we will investigate, both numerically and analitically,
the behavior of Random Boolean Networks (RBN's) as they undergo their
order-disorder phase transition. We will use a simple mean field approximation
to treat the problem, and without lack of generality we will concentrate on a
particular value for the connectivity of the system. In spite of the simplicity
of our arguments, we will be able to reproduce analitically the amount of
mutual information contained in the system as measured from numerical
simulations.


title: Beyond Hebb: Exclusive-OR and Biological Learning

abstract:
  A learning algorithm for multilayer neural networks based on biologically
plausible mechanisms is studied. Motivated by findings in experimental
neurobiology, we consider synaptic averaging in the induction of plasticity
changes, which happen on a slower time scale than firing dynamics. This
mechanism is shown to enable learning of the exclusive-OR (XOR) problem without
the aid of error back-propagation, as well as to increase robustness of
learning in the presence of noise.

categories: adap-org cond-mat.dis-nn cond-mat.stat-mech nlin.AO physics.bio-ph q-bio.NC


title: Performance of data networks with random links

abstract:
  We investigate simplified models of computer data networks and examine how
the introduction of additional random links influences the performance of these
net works. In general, the impact of additional random links on the performance
of the network strongly depends on the routing algorithm used in the network.
Significant performance gains can be achieved if the routing is based on
"geometrical distance" or shortest path reduced table routing. With shortest
path full table routing degradation of performance is observed.


title:
Self-Organized Criticality: Self-Organized Complexity? The Disorder and
  ``Simple Complexity'' of Power Law Distributions

abstract:
  The disorder and a simple convex measure of complexity are studied for rank
ordered power law distributions, indicative of criticality, in the case where
the total number of ranks is large. It is found that a power law distribution
may produce a high level of complexity only for a restricted range of system
size (as measured by the total number of ranks), with the range depending on
the exponent of the distribution. Similar results are found for disorder.
Self-organized criticality thus does not guarantee a high level of complexity,
and when complexity does arise, it is self-organized itself only if
self-organized criticality is reached at an appropriate system size.


title: Fractal Analysis for Social Systems

abstract:
  This is a brief introduction to fractals, multifractals and wavelets in an
accessible way, in order that the founding ideas of those strange and
intriguing newcomers to science as fractals may be communicated to a wider
public. Fractals are the geometry of the wildness of nature, where the
euclidian geometry fails. The structures of nonlinear dynamics associated with
chaos are fractal. Fractals may also be used as the geometry of social systems.
  Wavelets are introduced as a tool for fractal analysis. As an example of its
application on a social system, we use wavelet fractal analysis to compare
electrical power demand of two different places, a touristic city and a whole
country.


title: A Simple Model of the Evolution of Simple Models of Evolution

abstract:
  In the spirit of the many recent simple models of evolution inspired by
statistical physics, we put forward a simple model of the evolution of such
models. Like its objects of study, it is (one supposes) in principle testable
and capable of making predictions, and gives qualitative insights into a
hitherto mysterious process.


title: Simple models of evolution and extinction

abstract:
  This article gives a brief introduction to the mathematical modeling of
large-scale biological evolution and extinction. We give three examples of
simple models in this field: the coevolutionary avalanche model of Bak and
Sneppen, the environmental stress model of Newman, and the increasing fitness
model of Sibani, Schmidt, and Alstrom. We describe the features of real
evolution which these models are intended to explain and compare the results of
simulations against data drawn from the fossil record.


title: Mathematical description of the heat transfer in living tissue (Part I)

abstract:
  In the present monograph we formulate a simple model for heat transfer in
living tissue with self - regulation. The initial point of the model is the
governing equations describing heat transfer in living tissue at the mesoscopic
level, i.e. considering different vessels individually. Then, basing on the
well known equivalence of the diffusion type process and random walks, we
develop a certain regular procedure that enables us to average these mesoscopic
equations practically over all scales of the hierarchical vascular network. The
microscopic governing equations obtained in this way describe living tissue in
terms of an active medium with continuously distributed self - regulation. One
of the interesting results obtained in the present monograph is that there can
be the phenomena of ideal self - regulation in large active hierarchical
systems. Large hierarchical systems are characterized by such a great
information flow that none of its elements can possess whole information
required of governing the system behavior. Nevertheless, there exists a
cooperative mechanism of regulation which involves individual response of each
element to the corresponding hierarchical piece of information and leads to
ideal system response due to self - processing of information. The particular
results are obtained for bioheat transfer. However, self - regulation in other
natural hierarchical systems seems to be organized in a similar way. The
characteristics of large hierarchical systems occurring in nature are discussed
from the stand point of regulation problems. By way of example, some ecological
and economic systems are considered. An cooperative mechanism of
self-regulation which enables the system to function ideally is proposed.


title: Mathematical description of the heat transfer in living tissue (Part II)


title:
Amplifying Phenomenal Information: Toward a Fundamental Theory of
  Consciousness

abstract:
  Fundamental approaches bypass the problem of getting consciousness from
non-conscious components by positing that consciousness is a universal
primitive. For example, the double aspect theory of information holds that
information has a phenomenal aspect. How then do you get from phenomenal
information to human consciousness? This paper proposes that an entity is
conscious to the extent it amplifies information, first by trapping and
integrating it through closure, and second by maintaining dynamics at the edge
of chaos through simultaneous processes of divergence and convergence. The
origin of life through autocatalytic closure, and the origin of an
interconnected worldview through conceptual closure, induced phase transitions
in the degree to which information, and thus consciousness, is locally
amplified. Divergence and convergence of cognitive information may involve
phenomena observed in light e.g. focusing, interference, and resonance. By
making information flow inward-biased, closure shields us from external
consciousness; thus the paucity of consciousness may be an illusion.


title:
Criticality and Punctuated Equilibrium in a Spin System Model of a
  Financial Market

abstract:
  We describe a financial market model which shows a non-equilibrium phase
transition. Near the transition punctuated equilibrium behaviour is seen, with
avalanches occuring on all scales. This scaling is described by an exponent
very near 1. This system shows intermittent time development with bursts of
global synchronization reminiscent of a market rollercoaster.


title:
Modelling Collective Opinion Formation by Means of Active Brownian
  Particles

abstract:
  The concept of active Brownian particles is used to model a collective
opinion formation process. It is assumed that individuals in community create a
two-component communication field that influences the change of opinions of
other persons and/or can induce their migration. The communication field is
described by a reaction-diffusion equation, the opinion change of the
individuals is given by a master equation, while the migration is described by
a set of Langevin equations, coupled by the communication field. In the
mean-field limit holding for fast communication we derive a critical population
size, above which the community separates into a majority and a minority with
opposite opinions. The existence of external support (e.g. from mass media)
changes the ratio between minority and majority, until above a critical
external support the supported subpopulation exists always as a majority.
Spatial effects lead to two critical ``social'' temperatures, between which the
community exists in a metastable state, thus fluctuations below a certain
critical wave number may result in a spatial opinion separation. The range of
metastability is particularly determined by a parameter characterizing the
individual response to the communication field. In our discussion, we draw
analogies to phase transitions in physical systems.


title:
Modelling Migration and Economic Agglomeration with Active Brownian
  Particles

abstract:
  We propose a stochastic dynamic model of migration and economic aggregation
in a system of employed (immobile) and unemployed (mobile) agents which respond
to local wage gradients. Dependent on the local economic situation, described
by a production function which includes cooperative effects, employed agents
can become unemployed and vice versa. The spatio-temporal distribution of
employed and unemployed agents is investigated both analytically and by means
of stochastic computer simulations. We find the establishment of distinct
economic centers out of a random initial distribution. The evolution of these
centers occurs in two different stages: (i) small economic centers are formed
based on the positive feedback of mutual stimulation/cooperation among the
agents, (ii) some of the small centers grow at the expense of others, which
finally leads to the concentration of the labor force in different extended
economic regions. This crossover to large-scale production is accompanied by an
increase in the unemployment rate. We observe a stable coexistence between
these regions, although they exist in an internal quasistationary
non-equilibrium state and still follow a stochastic eigendynamics.


title: Frontiers of finance: Evolution and efficient markets

abstract:
  In this review article we explore several recent advances in the quantitative
modeling of financial markets. We begin with the Efficient Markets Hypothesis
and describe how this controversial idea has stimulated a number of new
directions of research, some focusing on more elaborate mathematical models
that are captable of rationalizing the empirical facrts, others taking a
completely different different tack in rejecting rationality altogether. One of
the most promising directions is to view financial markets from a biological
perspective and, specifically, with an evolutionary framework in which markets,
instruments, institutions, and investors interact and evolve dynamically
according to the "law" of economic selection. Under this view, financial agents
compete and adapt, but they do not necessarily do so in an optimal fashion.
Evolutionary and ecological models of financial markets is truly a new frontier
whose exploration has just begun.


title: Physicists attempt to scale the ivory towers of finance

abstract:
  Physicists have recently begun doing research in finance, and even though
this movement is less than five years old, interesting and useful contributions
have already emerged. This article reviews these developments in four areas,
including empirical statistical properties of prices, random-process models for
price dynamics, agent-based modeling, and practical applications.


title:
Democracy versus Dictatorship in Self-Organized Models of Financial
  Markets

abstract:
  Models to mimic the transmission of information in financial markets are
introduced. As an attempt to generate the demand process, we distinguish
between dictatorship associations, where groups of agents rely on one of them
to make decision, and democratic associations, where each agent takes part in
the group decision. In the dictatorship model, agents segregate into two
distinct populations, while the democratic model is driven towards a critical
state where groups of agents of all sizes exist. Hence, both models display a
level of organization, but only the democratic model is self-organized. We show
that the dictatorship model generates less volatile markets than the democratic
model.


title: Fitness versus Longevity in Age-Structured Population Dynamics

abstract:
  We examine the dynamics of an age-structured population model in which the
life expectancy of an offspring may be mutated with respect to that of the
parent. While the total population of the system always reaches a steady state,
the fitness and age characteristics exhibit counter-intuitive behavior as a
function of the mutational bias. By analytical and numerical study of the
underlying rate equations, we show that if deleterious mutations are favored,
the average fitness of the population reaches a steady state, while the average
population age is a decreasing function of the overall fitness. When
advantageous mutations are favored, the average population fitness grows
linearly with time t, while the average age is independent of fitness. For no
mutational bias, the average fitness grows as t^{2/3}.

categories: adap-org cond-mat.stat-mech nlin.AO q-bio.PE


title: The Importance of Being Discrete - Life Always Wins on the Surface

abstract:
  Many systems in chemistry, biology, finance and social sciences present
emerging features which are not easy to guess from the elementary interactions
of their microscopic individual components. In the past, the macroscopic
behavior of such systems was modeled by assuming that the collective dynamics
of microscopic components can be effectively described collectively by
equations acting on spatially continuous density distributions. It turns out
that quite contrary, taking into account the actual individual/discrete
character of the microscopic components of these systems is crucial for
explaining their macroscopic behavior. In fact, we find that in conditions in
which the continuum approach would predict the extinction of all the population
(respectively the vanishing of the invested capital or of the concentration of
a chemical substance, etc), the microscopic granularity insures the emergence
of macroscopic localized sub-populations with collective adaptive properties
which allow their survival and development. In particular it is found that in 2
dimensions "life" (the localized proliferating phase) always prevails.

categories: adap-org cond-mat.stat-mech nlin.AO nlin.CG q-bio


title: Rational Curves on Calabi-Yau Threefolds

abstract:
  The point is to compare the mathematical meaning of the ``number of rational
curves on a Calabi-Yau threefold'' to the meaning ascribed to the same notion
by string theorists.


title:
Gorenstein Threefold Singularities with Small Resolutions via Invariant
  Theory for Weyl Groups

abstract:
  We classify simple flops on smooth threefolds, or equivalently, Gorenstein
threefold singularities with irreducible small resolution. There are only six
families of such singularities, distinguished by Koll{\'a}r's {\em length}
invariant. The method is to apply invariant theory to Pinkham's construction of
small resolutions. As a by-product, generators of the ring of invariants are
given for the standard action of the Weyl group of each of the irreducible root
systems.


title: Complex Contact Threefolds and Their Contact Curves

abstract:
  A complex contact threefold is a threefold with a two-dimensional
non-integrable holomorphic distribution. A contact curve on a contact threefold
is an integrable curve of the distribution. This work was inspired by two
papers of Bryant, in which he used complex contact geometry to study
superminimal surfaces in four-sphere and to investigate exotic holonomies. The
present paper is devoted to systematical studies of contact threefold and
contact curves on them. We generalize a result of Bryant and answer a question
of his.


title:
Mirror symmetry and rational curves on quintic threefolds: a guide for
  mathematicians

abstract:
  We give a mathematical account of a recent string theory calculation which
predicts the number of rational curves on the generic quintic threefold. Our
account involves the interpretation of Yukawa couplings in terms of variations
of Hodge structure, a new $q$-expansion principle for functions on the moduli
space of Calabi-Yau manifolds, and the ``mirror symmetry'' phenomenon recently
observed by string theorists.


title: On the difference between real and complex arrangements

abstract:
  Let $B$ be an arrangement of linear complex hyperplanes in $C^d$. Then a
classical result by Orlik \& Solomon asserts that the cohomology algebra of the
complement can be constructed from the combinatorial data that are given by the
intersection lattice.
  If $B'$ is, more generally, a $2$-arrangement in $R^{2d}$ (an arrangement of
real subspaces of codimension $2$ with even-dimensional intersections), then
the intersection lattice still determines the cohomology {\it groups} of the
complement, as was shown by Goresky \& MacPherson. We prove, however, that for
$2$-arrangements the cohomology {\it algebra} is not determined by the
intersection lattice. It encodes extra information on sign patterns, which can
be computed from determinants of linear relations or, equivalently, from
linking coefficients in the sense of knot theory. This also allows us (in the
case $d=2$) to identify arrangements with the same lattice but different
fundamental groups.


title: Twistors, Kaehler Manifolds, and Bimeromorphic Geometry II

abstract:
  Using examples of compact complex 3-manifolds which arise as twistor spaces,
we show that the class of compact complex manifolds bimeromorphic to K\"ahler
manifolds is not stable under small deformations of complex structure.


title: The algebraic de Rham theorem for toric varieties

abstract:
  On an arbitrary toric variety, we introduce the logarithmic double complex,
which is essentially the same as the algebraic de Rham complex in the
nonsingular case, but which behaves much better in the singular case. Over the
field of complex numbers, we prove the toric analog of the algebraic de Rham
theorem which Grothendieck formulated and proved for general nonsingular
algebraic varieties re-interpreting an earlier work of Hodge-Atiyah. Namely,
for a finite simplicial fan which need not be complete, the complex cohomology
groups of the corresponding toric variety as an analytic space coincide with
the hypercohomology groups of the single complex associated to the logarithmic
double complex. They can then be described combinatorially as Ishida's
cohomology groups for the fan. We also prove vanishing theorems for Ishida's
cohomology groups. As a consequence, we deduce directly that the complex
cohomology groups vanish in odd degrees for toric varieties which correspond to
finite simplicial fans with full-dimensional convex support. In the particular
case of complete simplicial fans, we thus have a direct proof for an earlier
result of Danilov and the author.


title: On irregular links at infinity of algebraic plane curves

abstract:
  We give two proofs of a conjecture of the first author (Inv. Math. 98, 1989)
that a reduced algebraic plane curve is regular at infinity if and only if its
link at infinity is a regular toral link. This conjecture has also been proved
by Ha H.~V. using Lojasiewicz numbers at infinity. Our first proof uses the
polar invariant and the second proof uses linear systems of plane curve
singularities. The second approach (based on a paper in preparation) also
proves a stronger conjecture (loc. cit.) describing topologically the regular
link at infinity associated with an irregular link at infinity.


title:
On a Grauert-Riemenschneider vanishing theorem for Frobenius split
  varieties in characteristic p

abstract:
  This paper is about sheaf cohomology for varieties (schemes) in
characteristic $p>0$. We assume the presence of a Frobenius splitting. (See
V.B. Mehta and A. Ramanathan, Frobenius splitting and cohomology vanishing for
Schubert varieties, Annals of Math. 122 (1985), 27--40). The main result is
that a non-zero higher direct image under a proper map of the ideal sheaf of a
compatibly Frobenius split subvariety can not have a support whose inverse
image is contained in that subvariety. Earlier vanishing theorems for Frobenius
split varieties were based on direct limits and Serre's vanishing theorem, but
our theorem is based on inverse limits and Grothendieck's theorem on formal
functions. The result implies a Grauert--Riemenschneider type theorem.


title: Another Relation Between Approaches to the Schottky Problem

abstract:
  The recent extensive work on different approaches to the Schottky problem has
produced marked progress on several fronts. At the same time, it has become
apparent that there exist very close connections between the various
characterizations of Jacobian varieties described in Mumford's classic lectures
{\it Curves and Their Jacobians\/}. Until now, the approach via double
translation manifolds has seemed to be quite different from other approaches to
the Schottky problem. The purpose of this paper is to bring this last approach
``into the fold'' as it were, and to show precisely how it relates to
characterizations of Jacobians based on trisecants and flexes of the Kummer
variety, and the K.P. equation.


title: The Geometry and Arithmetic of Elliptic Surfaces

abstract:
  We survey some aspects of the theory of elliptic surfaces and give some
results aimed at determining the Picard number of such a surface. For the
surfaces considered, this will be equivalent to determining the Mordell-Weil
rank of an elliptic curve defined over a function field in one variable. An
interesting conjecture concerning Galois actions on the relative de~Rham
cohomology of these surfaces is discussed.


title: Cohomology of coherent sheaves on a proper scheme

abstract:
  (Makes a Gamma-acylic coherent resolution of a coherent sheaf on a projection
scheme.)


title: Deformations of Semi-Euler Characteristics

abstract: (Generalizes theorem of Atiyah and Mumford.)


title: Degenerations of generalized KNN algebras on tori

abstract:
  Degenerations of Lie algebras of meromorphic vector fields on elliptic curves
(i.e. complex tori) which are holomorphic outside a certain set of points
(markings) are studied. By an algebraic geometric degeneration process certain
subalgebras of Lie algebras of meromorphic vector fields on P^1 the Riemann
sphere are obtained. In case of some natural choices of the markings these
subalgebras are explicitly determined. It is shown that the number of markings
can change.
  AMS subject classification (1991): 17B66, 17B90, 14F10, 14H52, 30F30, 81T40


title: Note on curves in a Jacobian

abstract:
  For a curve C, viewed as a cycle in its Jacobian, we study its image n_*C
under multiplication by n on JC. We prove that the subgroup generated by these
cycles, in the Chow group modulo algebraic equivalence, has rank at most d-1,
where d is the gonality of C. We also discuss some general facts on the action
of n_* on the Chow groups.


title: Congruences for real algebraic curves on an ellipsoid

abstract:
  The problem of arrangement of a real algebraic curve on a real algebraic
surface is related to the 16th Hilbert problem. We prove in this paper new
restrictions on arrangement of nonsingular real algebraic curves on an
ellipsoid. These restrictions are analogues of Gudkov-Rokhlin,
Gudkov-Krakhnov-Kharlamov, Kharlamov-Marin congruences for plane curves (see
e.g. \cite{V} or \cite{V1}). To prove our results we follow Marin approach
\cite{Marin} that is a study of the quotient space of a surface under the
complex conjugation. Note that the Rokhlin approach \cite{R} that is a study of
the 2-sheeted covering of the surface branched along the curve can not be
directly applied for a proof of Theorem \ref{Rokhlin} since the homology class
of a curve of Theorem \ref{Rokhlin} can not be divided by 2 hence such a
covering space does not exist.


title: Extensions of Rokhlin congruence for curves on surfaces

abstract:
  The subject of this paper is the problem of arrangement of a real nonsingular
algebraic curve on a real non-singular algebraic surface. This paper contains
new restrictions on this arrangement extending Rokhlin and
Kharlamov-Gudkov-Krakhnov congruences for curves on surfaces.


title: Arithmetically Cohen-Macaulay Curves cut out by Quadrics

abstract:
  Addressing a question of M. Stillman, it had been shown by Ein, Eisenbud, and
the author that in a projective space of dimension at most 5, every
arithmetically Cohen-Macaulay curve which is cut out by quadrics scheme-
theoretically also has its homogeneous ideal generated by quadrics. In this
note it is shown that this is not the case in higher dimensional spaces.


title: The cohomological Brauer group of a toric variety

abstract:
  Toric varieties are a special class of rational varieties defined by
equations of the form {\it monomial = monomial}. For a good brief survey of the
history and role of toric varieties see [10]. Any toric variety $X$ contains a
cover by affine open sets described in terms of arrangements (called fans) of
convex bodies in $\Bbb R^r$. The coordinate rings of each of these affine open
sets is a graded ring generated over the ground field by monomials. As a
consequence, toric varieties provide a good context in which cohomology can be
calculated. The purpose of this article is to describe the second \'etale
cohomology group with coefficients in the sheaf of units of any toric variety
$X$. This is the so-called cohomological Brauer group of $X$.


title:
Disproving Hibi's Conjecture with CoCoA or Projective Curves with bad
  Hilbert Functions

abstract:
  In this paper we show how to combine different techniques from Commutative
Algebra and a systematic use of a Computer Algebra System (in our case mainly
CoCoA) in order to explicitly construct Cohen-Macaulay domains, which are
standard $k$-algebras and whose Hilbert function is ``bad". In particular we
disprove a well-known conjecture by Hibi.


title: Grobner bases and extension of scalars

abstract:
  This paper studies the behavior of Grobner bases with respect to extensions
of scalars. We prove that an extension of scalars commutes with taking Grobner
bases iff the extension is flat. We consider what information can be deduced
about fibers of a family, from the Grobner basis of the defining ideal of the
family itself. This information can be used to construct algorithms, such as to
find locii over which a map is finite, or an isomorphism.


title: Classical Polylogarithms

abstract:
  This paper is an introduction to classical polylogarithms and is an expanded
version of a talk given by the author at the Motives conference. Topics covered
include, monodromy; the polylogarithm local systems; Bloch's constructions of
regulators using the dilogarithm; polylog locals systems as variations of mixed
Hodge structre; the polylogarithm quotient of the fundamental group of C -
{0,}. It is intended as background for understanding recent work of Beilinson,
Deligne, and Goncharov.


title: Regulators and characteristic classes of flat bundles

abstract:
  This is a substantial revision of the older version of this paper. The main
result of the old version (the equality, up to a factor of 2 of the Beilinson
and Borel regulators) is now a conjecture. The main results give equality of
Beilinson chern classes and Cheeger-Simons-Chern classes in various situations
such as for flat bundles over quasi projective varieties. We also prove the
equality (up to a factor of two) of the Borel regulator element and the
universal Cheeger-Simons Chern class.


title: Sur la cohomologie de certains espaces de modules de fibr\'es vectoriels

abstract:
  Let M be a moduli space of stable vector bundles on a curve with rank and
degree fixed and coprime. We give a simple proof that the rational cohomology
of M is generated by the Kunneth components of the Chern classes of the
universal bundle. The proof applies also to some moduli spaces of vector
bundles over higher-dimensional varieties.


title:
The Singularities of the parameter surface of a minimal elliptic
  threefold

abstract:
  Let f : X -> S be any elliptic fibration. If X has dimension 3 and is not
uniruled, then X has a minimal model (with terminal singularities) [Mori]. In
earlier work we have shown that there exists a birationally equivalent elliptic
fibration p: Y -> T such that Y is minimal and a multiple of K_Y can be
expressed as the pullback of a divisor from T. Moreover T has at worst quotient
singularities; it is not difficult to find examples where T is actually
singular. In this paper we describe the singularities of this parameter surface
T in the case of no multiple fibers. Although T is not uniquely determined by
the birational equivalence class of the fibration, any two such T are related
by a particular kind of birational map.


title: Picard-Fuchs equations and mirror maps for hypersurfaces

abstract:
  We describe a strategy for computing Yukawa couplings and the mirror map,
based on the Picard-Fuchs equation. (Our strategy is a variant of the method
used by Candelas, de la Ossa, Green, and Parkes in the case of quintic
hypersurfaces.) We then explain a technique of Griffiths which can be used to
compute the Picard-Fuchs equations of hypersurfaces. Finally, we carry out the
computation for four specific examples (including quintic hypersurfaces,
previously done by Candelas et al.). This yields predictions for the number of
rational curves of various degrees on certain hypersurfaces in weighted
projective spaces. Some of these predictions have been confirmed by classical
techniques in algebraic geometry.


title: Cohomological and Cycle-theoretic connectivity

abstract:
  One of the themes in algebraic geometry is the study of the relation between
the ``topology'' of a smooth projective variety and a (``general'') hyperplane
section. Recent results of Nori produce cohomological evidence for a conjecture
that a general hypersurface of sufficently large degree should have no
``interesting'' cycles. We compute precise bounds for these results and show by
example that there are indeed interesting cycles for degrees that are not high
enough. In a different direction Esnault, Nori and Srinivas have shown
connectivity for intersections of small multidegree. We show analogous
cycle-theoretic connectivity results.


title:
Complex Divisors on Algebraic Curves and Some Applications to String
  Theory

abstract:
  These are notes of a talk to the International Conference on Algebra in honor
of A. I. Mal'tsev, Novosibirsk, USSR, 1989 (to appear in Contemporary
Mathematics). The concept of a divisor with complex coefficients on an
algebraic curve is introduced. We consider families of complex divisors, or,
equivalently, families of invertible sheaves and define Arakelov-type metrics
on some invertible sheaves produced from them on the base. We apply this
technique to obtain a formula for the measure on the moduli space that gives
tachyon correlators in string theory.


title: Exceptional linear systems on curves on Enriques surfaces

abstract:
  The main purpose in this paper is to study the gonality, the Clifford index
and the Clifford dimension on linearly equivalent smooth curves on Enriques
surfaces. The method is similar to techniques of M.Green $\&$ R.Lazarsfeld and
G.Pareschi.


title: Prym Varieties and Integrable Systems

abstract:
  A new relation between Prym varieties of arbitrary morphisms of algebraic
curves and integrable systems is discovered. The action of maximal commutative
subalgebras of the formal loop algebra of GL(n) defined on certain
infinite-dimensional Grassmannians is studied. It is proved that every
finite-dimensional orbit of the action of traceless elements of these
commutative Lie algebras is isomorphic to the Prym variety associated with a
morphism of algebraic curves. Conversely, it is shown that every Prym variety
can be realized as a finite-dimensional orbit of the action of traceless
diagonal elements of the formal loop algebra, which defines the multicomponent
KP system.


title: Computing the Cohomological Brauer Group of a Toric Variety

abstract:
  The purpose of this article is to show how one might compute the \'etale
cohomology groups $H^p_{\acute{e}t}(X,G_m)$ in degrees $p=0$, $1$ and $2$ of a
toric variety $X$ with coefficients in the sheaf of units. The method is to
reduce the computation down to the problem of diagonalizing a matrix with
integral coefficients. The procedure outlined in this article has been fully
implemented by the author as a program written in the ``C'' programming
language.


title: On the Abel-Jacobi Map for Divisors of Higher Rank on a Curve

abstract:
  The aim of this paper is to present an algebro-geometric approach to the
study of the geometry of the moduli space of stable bundles on a smooth
projective curve defined over an algebraically closed field $k$, of arbitrary
characteristic. This establishes a bridge between the arithmetic approach of
Harder, Narasimhan et al. and the gauge group approach of Atiyah and Bott. One
of the basic ideas is to consider a notion of divisor of higher rank and a
suitable Abel-Jacobi map generalizing the classical notions in rank one.


title: Homotopy groups of the complements to singular hypersurfaces,II

abstract:
  The homotopy group $\pi_{n-k} ({\bf C}^{n+1}-V)$ where $V$ is a hypersurface
with a singular locus of dimension $k$ and good behavior at infinity is
described using generic pencils. This is analogous to the van Kampen procedure
for finding a fundamental group of a plane curve. In addition we use a certain
representation generalizing the Burau representation of the braid group. A
divisibility theorem is proven that shows the dependence of this homotopy group
on the local type of singularities and behavior at infinity. Examples are given
showing that this group depends on certain global data in addition to local
data on singularities.


title:
Weierstrass Gap Sequence at Total Inflection Points of Nodal Plane
  Curves

abstract:
  Let $\Gamma$ be a plane curve of degree $d$ with $\delta$ ordinary nodes and
no other singularities. If $P$ is a smooth point on $\Gamma$ then the
Weierstrass gap sequence at $P$ is considered as that at the corresponding
point on the normalization of $\Gamma$. A smooth point $P\in\Gamma$ is called a
total inflection point if $i(\Gamma ,T;P)=d$ where $T$ is the tangent line to
$\Gamma$ at $P$. There are many possible Weierstrass gap sequences at total
inflection points. Our main results are: Among them (1) There exists a pair
$(P,\Gamma )$ such that the gap sequence at $P$ is the minimal (in the sense of
weight). (2) There exists a pair $(P,\Gamma )$ such that the gap sequence at
$P$ is the maximal (resp. up to 1 maximal). And we characterize these cases in
the sense of location of nodes.


title: Report on the Torsion of the Differential Module of an algebraic Curve

abstract:
  There is a conjecture, that the torsionfreeness of the module of
differentials in a point of an algebraic or algebroid curve should imply that
the curve is non singular at that point. A report on the main results is given.


title: Cohomology, Symmetry, and Perfection

abstract:
  One of the aims of this paper is to better explain the philosophy behind the
computations in [E.Bifet, C.De Concini,C.Procesi Cohomology of Regular
Embeddings ] and to place them in a wider conceptual setting. Another aim of
the paper is to outline in the last section an ``equivariant'' approach to some
key results in the theory of toric varieties. The text of the first three
sections follows closely a talk delivered at the University of Copenhagen in
July 1989 on the occasion of the Zeuthen Symposium. This paper is dedicated to
the memory of my friend Pere Menal and will appear in the Fall 1992 issue,
dedicated to his memory, of Publicacions Matem\`atiques, Universitat Aut\`onoma
de Barcelona.


title: Chern classes and degenerations of hypersurfaces and their lines

abstract: We study limiting lines on degenerations of generic hypersurfaces in $P^n$.


title: Notes on Kodaira energies of polarized varieties

abstract:
  The Kodaira energy of a polarized manifold (M,L) is defined by
\kappa\epsilon(M,L)=-Inf{t\in Q|\kappa(K+tL)\ge 0}. Here we propose a couple of
conjectures and announce several partial results. 3-dimensional cases are
mainly considered. A hard copy is available on request to the author.


title: Linear orbits of d-tuples of points in P^1

abstract:
  The group PGL(2) of linear transformations of the projective line acts
naturally on the d-dimensional projective space P^d parametrizing
configurations (`d-tuples') of points on the line. In this note we are
concerned with the orbits of this action of PGL(2) on P^d. The closure of each
orbit is a projective subvariety of P^d of which we determine the degree, the
`boundary'--i.e., the complement of an orbit in its closure--, and the
multiplicity at points of the boundary. These results are used to provide a
complete classification of the non-singular orbit closures, and criteria for an
orbit closure to be non-singular in codimension 1.


title: Upper Bounds for the Betti Numbers of a given Hilbert Function

abstract:
  From a Macaulay's paper it follows that a lex-segment ideal has the greatest
number of generators (the 0-th Betti number $\b_0$) among all the homogeneous
ideals with the same Hilbert function. In this paper we prove that this fact
extends to every Betti number, in the sense that all the Betti numbers of a
minimal free resolution of a lex segment ideal are bigger than or equal to the
ones of any homogeneous ideal with the same Hilbert function.


title: On the Tangent Space to the Universal Teichmuller Space

abstract:
  We find a remarkably simple relationship between the following two models of
the tangent space to the Universal Teichm\"uller Space: (1) The real-analytic
model consisting of Zygmund class vector fields on the unit circle; (2) The
complex-analytic model comprising 1-parameter families of schlicht functions on
the exterior of the unit disc which allow quasiconformal extension. Indeed, the
Fourier coefficients of the vector field in (1) turn out to be essentially the
same as (the first variations of) the corresponding power series coefficients
in (2). These identities have many applications; in particular, to conformal
welding, to the almost complex structure of Teichm\"uller space, to study of
the Weil-Petersson metric, to variational formulas for period matrices, etc.
These utilities are explored.


title: Linear orbits of smooth plane curves

abstract:
  The group PGL(3) of linear transformations of the projective plane acts
naturally on the projective space parametrizing curves of a given degree. In
this note we begin the study of the orbits of smooth curves under this action:
we construct a resolution of the closure of the orbit of a given curve, and we
use it to compute its degree. This turns out to depend only on the degree of
the curve, the order of its automorphism group, and on the number and type of
its flexes. This paper will appear on the Journal of Algebraic Geometry.


title:
String Branchings on Complex Tori and Algebraic Representations of
  Generalized Krichever-Novikov Algebras

abstract:
  The propagation differential for bosonic strings on a complex torus with
three symmetric punctures is investigated. We study deformation aspects between
two point and three point differentials as well as the behaviour of the
corresponding Krichever-Novikov algebras. The structure constants are
calculated and from this we derive a central extension of the Krichever-Novikov
algebras by means of b-c systems. The defining cocycle for this central
extension deforms to the well known Virasoro cocycle for certain kinds of
degenerations of the torus.
  AMS subject classification (1991): 17B66, 17B90, 14H52, 30F30, 81T40


title: Affine like Surfaces

abstract:
  We classify smooth surfaces whose higher cohomologies of i-forms for all i
vanish. We show that if such a surface is not affine, then it has essentially
two possibilities.


title: Extremal Rays and Null Geodesics on a Complex Conformal Manifold

abstract: This is a revised version of the paper submitted before.


title: Two Two-dimensional Terminations

abstract:
  Varieties with log terminal and log canonical singularities are considered in
the Minimal Model Program, see \cite{...} for introduction. In
\cite{shokurov:hyp} it was conjectured that many of the interesting sets,
associated with these varieties have something in common: they satisfy the
ascending chain condition, which means that every increasing chain of elements
terminates. Philosophically, this is the reason why two main hypotheses in the
Minimal Model Program: existence and termination of flips should be true and
are possible to prove.
  In this paper we prove that the following two sets satisfy the ascending
chain condition:
  1. The set of minimal log discrepancies for $K_X+B$ where $X$ is a surface
with log canonical singularities.
  2. The set of groups $(b_1,...b_s)$ such that there is a surface $X$ with log
canonical and numerically trivial $K_X+\sum b_jB_j$. The order on such groups
is defined in a natural way.


title: Corps quadratiques dont le 5-rang du groupe des classes est >=3

abstract:
  We prove the existence of infinitely many real and imaginary fields whose
5-rank of the class group is >=3.


title: Rang de courbes elliptiques d'invariant donne

abstract:
  We prove that there exist infinitely many elliptic curves over \Q with given
modular invariant, and rank >=2. Furthermore, there exist infinitely many
elliptic curves over $\Q$ with invariant equal at 0 (resp. 1728) and rank >=6
(resp. >=4).


title: The fibers of the Prym map

abstract:
  In this work we use the bigonal, trigonal and tetragonal constructions to
describe the fibers of the Prym map P : R_{g} ---->A_{g-1} inthe cases when it
is dominant, i.e. for g < 7. The most interesting cases are g = 5, where the
fiber is a double cover of the Fano surface of lines on a cubic threefold, and
g=6, where the map is generically finite (of degree 27) with Galois group
WE_{6}, so that the general fiber has the structure of the 27 lines on a cubic
surface. For g > 6, the map is known to be generically injective. The
tetragonal construction gives many counterexamples to injectivity, and we
conjecture that all noninjectivity is due to the tetragonal construction.


title:
The complex separation and extensions of Rokhlin congruence for curves
  on surfaces

abstract:
  The subject of this paper is the problem of arrangement of real algebraic
curves on real algebraic surfaces. In this paper we extend Rokhlin,
Kharlamov-Gudkov-Krakhnov and Kharlamov-Marin congruences for curves on
surfaces and give some applications of this extension.
  For some pairs consisting of a surface and a curve on this surface (in
particular for M-pairs) we introduce a new structure --- the complex separation
that is separation of the complement of curve into two surfaces. In accordance
with Rokhlin terminology the complex separation is a complex topological
characteristic of real algebraic varieties. The complex separation is similar
to complex orientations introduced by O.Ya.Viro (to the absolute complex
orientation in the case when a curve is empty and to the relative complex
orientation otherwise). In some cases we calculate the complex separation of a
surface (for example in the case when surface is the double branched covering
of another surface along a curve). With the help of these calculations
applications of the extension of Rokhlin congruence gives some new restrictions
for complex orientations of curves on a hyperboloid.


title: Completions of mapping class groups and the cycle $C - C^-$

abstract:
  In this paper we study the proalgebraic completion of mapping class relative
to their maps to the symplectic group. The main result is that the natural map
from the unipotent (a.k.a. Malcev) completion of the Torelli group to the
prounipotent radical of the Sp_g completion of the mapping class group is a non
trivial central extension with kernel isomorphic to Q, at least when g \ge 8.
The theorem is proved by relating the central extension to the line bundle
associated to the archemidean height of the cycle C - C- in the Jacobian of the
curve C. We also develop some of the basic theory of relative completions.


title: Open String Diagrams I: Topological Type

abstract:
  An arbitrary Feynman graph for string field theory interactions is analysed
and the homeomorphism type of the corresponding world sheet surface is
completely determined even in the non-orientable cases. Algorithms are found to
mechanically compute the topological characteristics of the resulting surface
from the structure of the signed oriented graph. Whitney's
permutation-theoretic coding of graphs is utilized.


title: Abelian varieties over Q and modular forms

abstract:
  This paper gives a conjectural characterization of those elliptic curves over
the field of complex numbers which "should" be covered by standard modular
curves. The elliptic curves in question all have algebraic j-invariant, so they
can be viewed as curves over Q-bar, the field of algebraic numbers. The
condition that they satisfy is that they must be isogenous to all their Galois
conjugates. Borrowing a term from B.H. Gross, "Arithmetic on elliptic curves
with complex multiplication," we say that the elliptic curves in question are
"Q-curves." Since all complex multiplication elliptic curves are Q-curves (with
this definition), and since they are all uniformized by modular forms
(Shimura), we consider only non-CM curves for the remainder of this abstract.
We prove:
  1. Let C be an elliptic curve over Q-bar. Then C is a Q-curve if and only if
C is a Q-bar simple factor of an abelian variety A over Q whose algebra of
Q-endomorphisms is a number field of degree dim(A). (We say that abelian
varieties A/Q with this property are of "GL(2) type.")
  2. Suppose that Serre's conjecture on mod p modular forms are correct (Ref:
Duke Journal, 1987). Then an abelian variety A over Q is of GL(2)-type if and
only if it is a simple factor (over Q) of the Jacobian J_1(N) for some integer
N\ge1. (The abelian variety J_1(N) is the Jacobian of the standard modular


title: Weierstrass weight of Gorenstein singularities with one or two branches

abstract:
  Let $X$ denote an integral, projective Gorenstein curve over an algebraically
closed field $k$. In the case when $k$ is of characteristic zero, C. Widland
and the second author have defined Weierstrass points of a line bundle on $X$.
In the first section, this definition is extended to linear systems in
arbitrary characteristic. This definition may be viewed as a generalization of
the definitions of Laksov and St\"ohr-Voloch to the Gorenstein case. In the
second section, an example is given to illustrate the definition. This example
is a plane curve of arithmetic genus 3 in characteristic 2 such that the gap
sequence at every smooth point (with respect to the dualizing bundle) is 1,2,5
and there are no smooth Weierstrass points. In the third section, the
Weierstrass weight of a unibranch singularity (on a not necessarily rational
curve) is computed in terms of its semigroup of values. In the final section,
the Weierstrass weight of a singularity with precisely two branches (again
assuming that the characteristic is zero) is computed.
  AMS Classification: Primary 14H55, Secondary 14H20.


title: Torsion Sections of Semistable Elliptic Surfaces

abstract:
  Let S be a torsion section of an elliptic surface with only I_n fibers. This
article addresses the question: which components of singular fibers can S pass
through? We give necessary criteria for the "component numbers", and show an
equidistribution result for torsion sections of prime order.


title: On the $(-1)$-curve conjecture of Friedman and Morgan

abstract:
  Main difference with previous version: we prove that every differentiably
embedded sphere with self intersection $-1$ in a simply connected algebraic
surface with $p_g >0$ is homologous to a $(-1)$-curve if $|K_{\min}|$ contains
a smooth irreducible curve of genus at least 2 and $p_g$ is even or $K_{\min}^2
\not\equiv 7 \pmod8$ (here $K_{\min}$ is the canonical class of the minimal
model).


title: Arrangements of Hyperplanes and Vector Bundles on $P^n$

abstract:
  Any arrangement of hyperplanes in general position in $P^n$ can be regarded
as a divisor with normal crossing. We study the bundles of logarithmic 1-forms
corresponding to such divisors` from the point of view of classification of
vector bundles on $P^n$. It turns out that all such bundles are stable. The
study of jumping lines of these bundles gives a unified treatment of several
classical constructions associating a curve to a collection of points in $P^n$.
The main result of the paper is "Torelli theorem" which says that the
collection of hyperplanes can be recovered from the isomorphism class of the
corresponding logarithmic bundle unless the hyperplanes ocsulate a rational
normal curve. In this latter case our construction reduces to that of secant
bundles of Schwarzenberger.


title: Chow quotients of Grassmannian I

abstract:
  We introduce a certain compactification of the space of projective
configurations i.e. orbits of the group $PGL(k)$ on the space of $n$ - tuples
of points in $P^{k-1}$ in general position. This compactification differs
considerably from Mumford's geometric invariant theory quotient. It is obtained
by considering limit position (in the Chow variety) of the closures of generic
orbits. The same result will be obtained if we study orbits of the maximal
torus on the Grassmannian $G(k,n)$. We study in detail the closures of the
torus orbits and their "visible contours" which are Veronese varieties in the
Grassmannian. For points on $P^1$ our construction gives the Grothemdieck -
Knudsen moduli space of stable $n$ -punctured curves of genus 0. The "Chow
quotient" interpretation of this space permits us to represent it as a blow up
of a projective space.


title:
The simple method of distinguishing the underlying differentiable
  structures of algebraic surfaces

abstract:
  The simplest version of the Spin-polynomial invariants of the underlying
differentiable structures of algebraic surfaces were considered and the
simplest arguments were used in order to distinguish the underlying smooth
structures of certain algebraic surfaces.


title: Points of Low Degree on Smooth Plane Curves

abstract:
  The purpose of this note is to provide some applications of Faltings' recent
proof of S. Lang's conjecture to smooth plane curves. Let $C$ be a smooth plane
curve defined by an equation of degree $d$ with integral coefficients. We show
that for $d\ge 7$, the curve $C$ has only finitely many points whose field of
definition has degree $\le d-2$ over $Q$, and that for $d\ge 8$, all but
finitely many points of $C$ whose field of definition has degree $\le d-1$ over
$Q$ arise as points of intersection of rational lines through rational points
of $C$.


title: Degrees of Curves in Abelian Varieties

abstract:
  The degree of a curve $C$ in a polarized abelian variety $(X,\lambda)$ is the
integer $d=C\cdot\lambda$. When $C$ generates $X$, we find a lower bound on $d$
which depends on $n$ and the degree of the polarization $\lambda$. The smallest
possible degree is $d=n$ and is obtained only for a smooth curve in its
Jacobian with its principal polarization (Ran, Collino). The cases $d=n+1$ and
$d=n+2$ are studied. Moreover, when $X$ is simple, it is shown, using results
of Smyth on the trace of totally positive algebraic integers, that if $d\le
1.7719\, n$, then $C$ is smooth and $X$ is isomorphic to its Jacobian. We also
get an upper bound on the geometric genus of $C$ in terms of its degree.


title: Reductive group actions on K\"ahler manifolds

abstract:
  We prove that the action of a reductive complex Lie group on a K\"ahler
manifold can be linearized in the neighbourhood of a fixed point, provided that
the restriction of the action to some compact real form of the group is
Hamiltonian with respect to the K\"ahler form.


title: Stable pairs, linear systems and the Verlinde formula

abstract:
  We study the moduli problem of pairs consisting of a rank 2 vector bundle and
a nonzero section over a fixed smooth curve. The stability condition involves a
parameter; as it varies, we show that the moduli space undergoes a sequence of
flips in the sense of Mori. As applications, we prove several results about
moduli spaces of rank 2 bundles, including the Harder-Narasimhan formula and
the SU(2) Verlinde formula. Indeed, we prove a general result on the space of
sections of powers of the ideal sheaf of a curve in projective space, which
includes the Verlinde formula.


title:
Erratum to "The Homogeneous Coordinate Ring of a Toric Variety", along
  with the original paper

abstract:
  This submission consists of two papers: 1) an erratum that corrects an error
in the proof of Proposition 4.3 in my paper "The Homogeneous Coordinate Ring of
a Toric Variety", and 2) the original (unchanged) version of the paper,
published in 1995. The original paper introduced the homogeneous coordinate
ring of a toric variety (now called the total coordinate ring or Cox ring) and
gave a quotient construction. The paper also studied sheaves on a toric
variety, and in Section 4 described its automorphism group. The error in the
proof of Proposition 4.3 resulted from the faulty assumption that a certain set
of graded endomorphisms forms a ring; rather, it is a monoid under composition.
The erratum notes this error and gives a correct proof of the proposition.


title: Elliptic Three-folds I: Ogg-Shafarevich Theory

abstract:
  We calculate the Tate-Shafarevich group of an elliptic three-fold
$f:X\rightarrow S$ when $X$ and $S$ are regular and $f$ is flat, relating it to
the Brauer group of $X$ and $S$. We show that given certain hypotheses on $f$,
the Tate-Shafarevich group has the interpretation of isomorphism classes of
elliptic curves over the function field of $S$ which have the same jacobian as
the generic fibre of $f$, and for which there exists a relatively minimal model
which has no multiple fibres. We use this to give examples of elliptic
fibrations with isolated multiple fibres, and also to give a new counterexample
to the Luroth problem in dimension three. This is a revised, hopefully
improved, version with a few extra theorems and a few errors corrected.


title: Stable pairs on curves and surfaces

abstract:
  We describe stability conditions for pairs consisting of a coherent sheaf and
a homomorphism to a fixed coherent sheaf on a projective variety. The
corresponding moduli spaces are constructed for pairs on curves and surfaces.
We consider two examples. The fixed sheaf is the structure sheaf or is a vector
bundle on a divisor, i.e. Higgs pairs or framed bundles, resp. (unencoded
version)


title:
Algebraic approximations of holomorphic maps from Stein domains to
  projective manifolds

abstract:
  It is shown that every holomorphic map $f$ from a Runge domain $\Omega$ of an
affine algebraic variety $S$ into a projective algebraic manifold $X$ is a
uniform limit of Nash algebraic maps $f_\nu$ defined over an exhausting
sequence of relatively compact open sets $\Omega_\nu$ in $\Omega$. A relative
version is also given: If there is an algebraic subvariety $A$ (not necessarily
reduced) in $S$ such that the restriction of $f$ to $A\cap\Omega$ is algebraic,
then $f_\nu$ can be taken to coincide with $f$ on $A\cap\Omega_\nu$. The main
application of these results, when $\Omega$ is the unit disk, is to show that
the Kobayashi pseudodistance and the Kobayashi-Royden infinitesimal metric of a
quasi-projective algebraic manifold $Z$ are computable solely in terms of the
closed algebraic curves in $Z$. Similarly, the $p$-dimensional Eisenman metric
of a quasi-projective algebraic manifold can be computed in terms of the
Eisenman volumes of its $p$-dimensional algebraic subvarieties. Another
question addressed in the paper is whether the approximations $f_\nu$ can be
taken to have their images contained in affine Zariski open subsets of $X$. By
using complex analytic methods (pluricomplex potential theory and H\"ormander's
$L^2$ estimates), we show that this is the case if $f$ is an embedding (with
$\dim S<\dim X$) and if there is an ample line bundle $L$ on $X$ such that


title: On the stable rationality of $X/G$

abstract:
  Let $G$ be a connected, reductive algeraic group whose Dynkin diagram
contains no components of type $G_2,$ $F_4,$ $E_6,$ $E_7$ or $E_8.$ That is,
all the components are of classical type. Suppose $X$ is an affine variety, and
suppose $G$ acts freely on $X.$ Then $X$ and $X/G$ are stably birationally
equivalent.


title:
The enumeration of simultaneous higher-order contacts between plane
  curves

abstract:
  Using the Semple bundle construction, we derive an intersection-theoretic
formula for the number of simultaneous contacts of specified orders between
members of a generic family of degree $d$ plane curves and finitely many fixed
curves. The contacts counted by the formula occur at nonsingular points of both
the members of the family and the fixed curves.


title: Automorphisms and the K\"ahler cone of certain Calabi-Yau manifolds

abstract:
  For the Calabi-Yau threefolds $X$ constructed by C. Schoen as fiber products
of generic rational elliptic surfaces, we show that the action of the
automorphism group of $X$ on the K\"ahler cone of $X$ has a rationally
polyhedral fundamental domain. The second author has conjectured that this
statement will hold in general, the example presented here being the first
non-trivial case in which the statement has been checked. The conjecture was
motivated by the desire to use a construction of E. Looijenga to compactify
certain moduli spaces which arise in the study of conformal field theory and
``mirror symmetry.''


title:
Lines on Calabi Yau complete intersections, mirror symmetry, and Picard
  Fuchs equations

abstract:
  A relation between the number of rational curves of fixed degree on Calabi
Yau threefolds and the Picard Fuchs equations, which was suggested as part of
the study of mirror symmetry, is verified in the case of complete intersection
of two cubics and lines.


title: Minimal Cohomology Classes and Jacobians

abstract:
  We show that on the Jacobian $(JC,\theta)$ of a smooth curve $C$ of genus
$g$, any effective cycle in $JC$ with cohomology class $\theta^d/d!$ is a
translate of $W_{g-d}(C)$ or $-W_{g-d}(C)$. We then use this result to prove
that for $1<d<g$, the Jacobian locus (\resp the locus of intermediate Jacobians
of cubic threefolds) is an irreducible component of the set of principally
polarized abelian varieties of dimension $g$ for which $\theta^d/d!$ (\resp
$\theta^3/3!$) is the class of an effective algebraic cycle. Moreover, on the
intermediate Jacobian of a generic cubic threefold, $\theta^2/2!$ is not the
class of an effective algebraic cycle.


title: Non-trivial Linear Systems on Smooth Plane Curves

abstract:
  Let $C$ be a smooth plane curve of degree $d$ defined over an algebraically
closed field $k$. A base point free complete very special linear system $g^r_n$
on $C$ is trivial if there exists an integer $m\ge 0$ and an effective divisor
$E$ on $C$ of degree $md-n$ such that $g^r_n=|mg^2_d-E|$ and
$r=(m^2+3m)/2-(md-n)$. In this paper, we prove the following: Theorem Let
$g^r_n$ be a base point free very special non-trivial complete linear system on
$C$. Write $r=(x+1)(x+2)/2-b$ with $x, b$ integers satisfying $x\ge 1, 0\le b
\le x$. Then $n\ge n(r):=(d-3)(x+3)-b$. Moreover, this inequality is best
possible.


title: The Geometry of Bielliptic Surfaces in P^4

abstract:
  In 1988 Serrano \cite{Ser}, using Reider's method, discovered a minimal
bielliptic surface in $\PP^4$. Actually he showed that there is a unique family
of such surfaces and that they have degree 10 and sectional genus 6. In this
paper we describe, among other things, the geometry of the embedding of the
minimal bielliptic surfaces. A consequence of this description will be the
existence of smooth nonminimal bielliptic surfaces of degree 15 in $\PP^4$. We
also explain how to construct the degree 15 surfaces with the help of the
quadro-cubic Cremona transformation of $\PP^4$. Finally, we remark that the
quintic elliptic scroll and the abelian and bielliptic surfaces of degree 10
and 15 are essentially the only smooth irregular surfaces known in $\PP^4$ (all
others can be derived via finite morphisms $\PP^4\to\PP^4 $).


title: The topology of the space of rational curves on a toric variety

abstract:
  Let $X$ be a compact toric variety. Let $Hol$ denote the space of based
holomorphic maps from $CP^1$ to $X$ which lie in a fixed homotopy class. Let
$Map$ denote the corresponding space of continuous maps. We show that $Hol$ has
the same homotopy groups as $Map$ up to some (computable) dimension. The proof
uses a description of $Hol$ as a space of configurations of labelled points,
where the labels lie in a partial monoid determined by the fan of $X$.


title:
Rational curves on Calabi-Yau manifolds: verifying predictions of Mirror
  Symmetry

abstract:
  Mirror symmetry, a phenomenon in superstring theory, has recently been used
to give tentative calculations of several numbers in algebraic geometry. In
this paper, the numbers of lines and conics on various hypersurfaces which
satisfy certain incidence properties are calculated, and shown to agree with
the numbers predicted by Greene, Morrison, and Plesser using mirror symmetry in
every instance. This increases the number of verified predictions from 3 to 65.
Calculations are performed using the Maple package {\sc schubert} written by
Katz and Str{\o}mme.


title: Bounds of automorphism groups of genus 2 fibrations

abstract:
  For a complex surface of general type with a relatively minimal genus 2
fibration, the bounds of the orders of the automorphism group of the fibration,
of its abelian subgroups and of its cyclic subgroups are determined as linear
functions of $c^2_1$. Most of them are the best.


title: A note on non-vanishing and applications

abstract:
  Let $X$ be a normal variety over the field of complex numbers with log
terminal singularities and the canonical divisor $K_X$ being ${\bf
Q}$-Gorenstein. Assume that $L$ is an ample line bundle over $X$ and $\phi:
X\to Y$ is a morphism supported by $K_X+rL$ for some positive rational number
$r$. In the present paper we study the evaluation $\phi^*\phi_*(L)\to L$ and
the locus of points where it is not surjective which we call relative base
point locus of $L$. In particular, we prove that, if the dimension of a fiber
of $\phi$ is small with respect to $r$ then the relative base point locus does
not meet the fiber. Consequently, in this case, we discuss the structure of the
map $\phi$ for a smooth $X$.


title: On the surjectivity of Wahl maps on a general curve

abstract:
  This paper explores the geometric meaning of the failure of certain kinds of
Wahl maps to surject on a general curve. Sufficient conditions for surjectivity
are given. An approach used by Voisin to study canonical Wahl maps is applied
in this direction.


title: Lefschetz Fixed Point Theorem and Lattice Points in Convex Polytopes

abstract:
  A simple convex lattice polytope $\Box$ defines a torus-equivariant line
bundle $\LB$ over a toric variety $\XB.$ Atiyah and Bott's Lefschetz
fixed-point theorem is applied to the torus action on the $d''$-complex of
$\LB$ and information is obtained about the lattice points of $\Box$. In
particular an explicit formula is derived, computing the number of lattice
points and the volume of $\Box$ in terms of geometric data at its extreme
points. We show this to be equivalent the results of Brion \cite{brion} and
give an elementary convex geometric interpretation by performing Laurent
expansions similar to those of Ishida \cite{ishida}.


title: Intersection homology Betti numbers

abstract:
  A generalization of the formula of Fine and Rao for the ranks of the
intersection homology groups of a complex algebraic variety is given. The proof
uses geometric properties of intersection homology and mixed Hodge theory.


title:
A Finiteness Theorem for Quaternionic-Kaehler Manifolds with Positive
  Scalar Curvature

abstract:
  We study the topology and geometry of those compact Riemannian (4n)-manifolds
(M,g), n > 1, with positive scalar curvature and holonomy in Sp(n)Sp(1). Up to
homothety, we show that there are only finitely many such manifolds of any
dimension 4n.


title:
Existence and Deformation Theory for Scalar-Flat Kaehler Metrics on
  Compact Complex Surfaces

abstract:
  Let M be a compact complex surface which admits a Kaehler metric whose scalar
curvature has integral zero; and suppose the fundamental group of M does not
contain an Abelian subgroup of finite index. Then if M is blown up at
sufficiently many points, the resulting surface M' admits scalar-flat Kaehler
metrics.


title:
Semistable Minimal Models of Threefolds in Positive or Mixed
  Characteristic

abstract:
  We extend the minimal model theorem to the 3-dimensional schemes which are
projective and have semistable reduction over the spectrum of a Dedekind ring.


title: Projective varieties with many degenerate subvarieties

abstract:
  We study the problem of classifying the irreducible projective varieties $X$
of dimension $n\ge 2$ in $\Bbb P^N$ which contain an algebraic family $\Cal F$
of dimension $h+1$ ($h<n$) of subvarieties $Y$ of dimension $n-h$, each one
contained in a $\Bbb P^{N-h-1}$. We prove that one of the following happens:
(i) there exists an integer $r$, $r<N-n$ such that $X$ is contained in a
variety $V_r$ of dimension at most $N-r$ containing a family of dimension $h+1$
of subvarieties of dimension $N-h-r$, each one contained in a linear space of
dimension $N-h-1$; (ii) The degree of $Y$ is bounded by a function of $h$ and
$N-n$ (in this case $X$ is called of isolated type). Successively we study some
special cases; in particular we give a complete classification of surfaces in
$\Bbb P^5$ containing a family of dimension $2$ of curves of $\Bbb P^3$.