The Gale transform and multi-graded determinantal schemes

Eisenbud and Popescu showed that certain finite determinantal subschemes of projective spaces defined by maximal minors of adjoint matrices of homogeneous linear forms are related by Veronese embeddings and a Gale transform. We extend this result to adjoint matrices of multihomogeneous multilinear forms. The subschemes now lie in products of projective spaces and the Veronese embeddings are replaced with Segre embeddings.     

The Space of Unordered Real Line Arrangements

The set of all unordered real line arrangements of given degree in the real projective plane is known to have a natural semialgebraic structure. The nonreduced arrangements are singular points of this structure. We show that the set of all unordered real line arrangements of given degree also has a natural structure of a smooth compact connected affine real algebraic variety. In fact, as such, it is isomorphic to a real projective space. As a consequence, we get a projectively linear structure on the set of all real line arrangements of given degree. We also show that the universal family of unordered real line arrangements of given degree is not algebraic.     

A numerical approach to proving projectivity

Summary We present a nonconstructive method which uses intersection numbers and linear space theory for proving the existence of projective embeddings of suitable algebraic schemes, and we apply it to establish Chevalley's conjecture that a complete nonsingular variety such that any finite number of points is contained in an open affine subset is projective. In memory of Guido Castelnuovo in the recurrence of the first centenary of his birth.     

On the intersection problem for linear sets in the projective line

The aim of this paper is to investigate the intersection problem between two linear sets in the projective line over a finite field. In particular, we analyze the intersection between two clubs with possibly different maximum fields of linearity. We also consider the intersection between a certain linear set of maximum rank and any other linear set of the same rank. The strategy relies on the study of certain algebraic curves whose rational points describe the intersection of the two linear sets. Among other geometric and algebraic tools, function field theory and the Hasse–Weil bound play a crucial role. As an application, we give asymptotic results on semifields of BEL-rank two.     

The algebraic degree of semidefinite programming

Given a generic semidefinite program, specified by matrices with rational entries, each coordinate of its optimal solution is an algebraic number. We study the degree of the minimal polynomials of these algebraic numbers. Geometrically, this degree counts the critical points attained by a linear functional on a fixed rank locus in a linear space of symmetric matrices. We determine this degree using methods from complex algebraic geometry, such as projective duality, determinantal varieties, and their Chern classes.     

Vanishing Ideals of Projective Spaces over Finite Fields and a Projective Footprint Bound

We consider the vanishing ideal of a projective space over a finite field. An explicit set of generators for this ideal has been given by Mercier and Rolland. We show that these generators form a universal Gr¨obner basis of the ideal. Further we give a projective analogue for the so-called footprint bound, and a version of it that is suitable for estimating the number of rational points of projective algebraic varieties over finite fields. An application to Serre's inequality for the number of points of projective hypersurfaces over finite fields is included.     

On the representations of projective geometries in algebraic combinatorial geometries

In this paper, we show that the full algebraic combinatorial geometry is not a projective geometry, it is only semimodular, but the p-polynomial points give a projective subgeometry. Also, we show that the subgeometry can be coordinatized by a skew field, which is quotient ring of an Ore domain. As a corollary, we prove the existence of algebraic representations over fields of prime characteristic of the non-Pappus matroid and its dual matroid. Regarding the existence of algebraic representations of the non-Pappus matroid, this result was earlier proved by Lindström [7] for finite fields.     

Unique range sets and uniqueness polynomials for algebraic curves

We study unique range sets and uniqueness polynomials for algebraic functions on a smooth projective algebraic curve over an algebraically closed field of characteristic zero.

     

Real plane algebraic curves

We study real algebraic plane curves, at an elementary level, using as little algebra as possible. Both cases, affine and projective, are addressed. A real curve is infinite, finite or empty according to the fact that a minimal polynomial for the curve is indefinite, semi-definite nondefinite or definite. We present a discussion about isolated points. By means of the P operator, these points can be easily identified for curves defined by minimal polynomials of order bigger than one. We also discuss the conditions that a curve must satisfy in order to have a minimal polynomial. Finally, we list the most relevant topological properties of affine and projective, complex and real plane algebraic curves.     

On the complexity of deciding connectedness and computing Betti numbers of a complex algebraic variety

We extend the lower bounds on the complexity of computing Betti numbers proved in [P. Bürgisser, F. Cucker, Counting complexity classes for numeric computations II: algebraic and semialgebraic sets, J. Complexity 22 (2006) 147–191] to complex algebraic varieties. More precisely, we first prove that the problem of deciding connectedness of a complex affine or projective variety given as the zero set of integer polynomials is PSPACE-hard. Then we prove PSPACE-hardness for the more general problem of deciding whether the Betti number of fixed order of a complex affine or projective variety is at most some given integer.     

Poincaré series of divisorial filtration corresponding to a curve with one place at infinity

The exceptional divisor component of the projective plane modified by a sequence of blow-ups determines filtration on the ring of polynomials in two variables. The set of such components determines the multi-index filtration on this ring. The Poincaré series of this filtration is calculated for some sets of components provided that the modification under study is the minimal resolution of a plane algebraic curve with one place at infinity.     

New examples (and counterexamples) of complete finite-rank differential varieties

Differential algebraic geometry seeks to extend the results of its algebraic counterpart to objects defined by differential equations. Many notions, such as that of a projective algebraic variety, have close differential analogues but their behavior can vary in interesting ways. Workers in both differential algebra and model theory have investigated the property of completeness of differential varieties. After reviewing their results, we extend that work by proving several versions of a “differential valuative criterion" and using them to give new examples of complete differential varieties. We conclude by analyzing the first examples of incomplete, finite-rank projective differential varieties, demonstrating a clear difference from projective algebraic varieties.     

The isostrophisms of planar ternary rings part I

It is well known that a projective plane nay be coordinatized by picking four arbitrary, ordered points such that no three of them are collinear. This gives rise to an algebraic structure called the planar ternary ring T with respect to the base points A,B,C,D. The existence of certain collineations in the plane will be reflected by algebraic properties in T, and conversely. In this paper the interrelationships between certain algebraic properties in T, and the linearity of those ternary rings which are obtained from T and various permutations of A,B,C,D are considered. Many well known types of algebraic structures, such as quasifields and alternative division rings will occur if the ternary rings generated by suitable permutations of A,B,C,D are linear.This research is included in the author's doctoral dissertation submitted to Temple University. The author would like to express his appreciation to Professor R. Artzy for his assistance in completing this work.     

On the Geometry of Point-Plane Constraints on Rigid-Body Displacements

In this paper the rigid-body displacements that transform a point in such a way that it remains on a particular plane are studied. These sets of rigid displacements are referred to as point-plane constraints and are given by the intersection of the Study quadric of all rigid displacements with another quadric in 7-dimensional projective space. The set of all possible point-plane constraints comprise a Segre variety. Two different classes of problems are investigated. First instantaneous kinematics, for a given rigid motion there are points in space which, at some instant, have no torsion or have no curvature to some order. The dimension and degrees of these varieties are found by very simple computations. The corresponding problems for point-sphere constraints are also found. The second class of problems concern the intersections of several given constraints. Again the characteristics of these varieties for different numbers of constraints are found using very simple techniques.     

Recognizing ℙn in Classical and Modern Setting

In classical projective algebraic geometry, ?ⁿ was seen mainly as a linear subspace. The modern setting has produced in the last 40 years several remarkable abstract characterizations of projective space. We survey some interaction between these two points of view.     

Arbitrary Pole Assignability by Static Output Feedback under Structural Contraints

Using a few elementary concepts from algebraic geometry such as multidimensional projective space, quadric hypersurface and its tangential variety, the known problem of arbitrary pole placement is transformed into a system of well‐structured (partly non‐linear) algebraic equations. Necessary and sufficient solvability conditions are derived. Finally, it is outlined how to calculate admissible output feedback matrices which ensure the desired pole assignment.     

Algebraic Topology for Minimal Cantor Sets

It will be shown that every minimal Cantor set can be obtained as a projective limit of directed graphs. This allows to study minimal Cantor sets by algebraic topological means. In particular, homology, homotopy and cohomology are related to the dynamics of minimal Cantor sets. These techniques allow to explicitly illustrate the variety of dynamical behavior possible in minimal Cantor sets. submitted 20/07/05, accepted 18/10/05     

Any algebraic variety in positive characteristic admits a projective model with an inseparable Gauss map

We determine the values attained by the rank of the Gauss map of a projective model for a fixed algebraic variety in positive characteristic p. In particular, it is shown that any variety in p>0 has a projective model such that the differential of the Gauss map is identically zero. On the other hand, we prove that there exists a product of two or more projective spaces admitting an embedding into a projective space such that the differential of the Gauss map is identically zero if and only if p=2.     

Difference Sets and Hyperovals

We construct three infinite families of cyclic difference sets, using monomial hyperovals in a desarguesian projective plane of even order. These difference sets give rise to cyclic Hadamard designs, which have the same parameters as the designs of points and hyperplanes of a projective geometry over the field with two elements. Moreover, they are substructures of the Hadamard design that one can associate with a hyperoval in a projective plane of even order.     

Projective planes over rings of stable rank 2

This paper gives an axiomatic characterization of projective planes over rings of stable rank 2. These rings, which are known from algebraic K-theory, beautifully reflect simple geometric properties. The basic relations in the plane are incidence and the neighbor relation. The axioms consist of a number of axioms expressing elementary relations between points and lines such as, e. g., the existence of a unique line joining any two non-neighboring points, and a couple of axioms ensuring the existence of transvections and dilatations.