Lawrence Brenton

On singular complex surfaces with negative canonical bundle, with applications to singular compactifications ofC2 and to 3-dimensional rational singularities

IfX is a compact complex manifold of complex dimension two, denote by K x the canonical line bundle (the second exterior power of the cotangent bundle) of X. It is immediate from the classification of surfaces [15, 5] that if K x is negative then X is either the complex projective plane IP 2, or the product tP ~ x tP 1 of two projective lines ( t h e non-singular quadric surface Q2C IP3), or is derived from one of these by blowing up points. Here a vector bundle E on a complex space X (singularities allowed) is negative if the zero section X o is exceptional in E. Now if (X, (gx) is an analytic surface with singular points, then if each singularity is Cohen-Macaulay (the homological codimension of each stalk (gx, x equals the dimension of X = 2; this is equivalent to normal (integrally closed) in this dimension) then at least the canonical sheaf ~ x is defined and Serre-Grothendieck duality holds. And if each point is Gorenstein ((gx, x has finite injective dimension) then J{x is locally trivial and so it is the sheaf of sections of a holomorphic line bundle Kx, the canonical bundle of the singular space X. The purpose of this note is to expose all compact Gorenstein surfaces (compact two-dimensional complex spaces with only Gorenstein singularities) for which K x is negative.

Math Corps Summer Camp: An Inner City Intervention Program

The underrepresentation and underachievement of African American students in mathematics is a national tragedy. One response to these problems is the design and implementation of mathematics intervention programs aimed at primarily African American student populations living in the inner city, many of whom are placed at risk. This article describes such a program and presents a philosophical framework grounded in empirical studies of effective schools. The authors suggest that key aspects of the program design help students avoid the trap of disidentification with the educational system. They also suggest that a large number of factors that have been demonstrated to be associated with higher student achievement (acting in combination) contribute to the effectiveness of this intervention program.

On Recursive Solutions of a Unit Fraction Equation

We consider the Egyptian fraction equation and discuss techniques for generating solutions. By examining a quadratic recurrence relation modulo a family of primes we have found some 500 new infinite sequences of solutions. We also initiate an investigation of the randomness of the distribution of solutions, and show that there are infinitely many solutions not generated by the aforementioned technique.

Some examples of singular compact analytic surfaces which are homotopy equivalent to the complex projective plane

IN THEIR fundamental work[S] on analytic structures on the complex projective spaces Hirzebruch and Kodaira show in particular the following result in dimension two: If X is a compact complex Klhler manifold homeomorphic to the complex projective plane p = p(C) and with negative canonical line bundle, then X is analytically isomorphic to p. Indeed, it is easy to check that more is true (Proposition 4, Corollary 5 below): If X is a 2-(complex)-dimensional compact complex manifold with H,(X, Z) = 0 and second Betti number b*(X) = 1, then X is projective algebraic and II*(X, Z) is generated (over Z) by the Chern class of a positive (ample) line bundle L. If L admits a non-trivial section (which will be the case if the canonical bundle KX is negatiue) then X is biholomotphic to p. The purpose of this paper is to show by example that nevertheless there exist singular compact 2-dimensional complex spaces X which are homotopy equivalent with p, with H*(X, Z) = H*(p, Z) generated by the Chern class of the bundle of a positively embedded divisor P, and even with negative “canonical” bundle; and secondly, to give a uniqueness result in the form of an algorithm for constructing such spaces. In particular we show (Theorem 6 below): Let X be a 2-dimensional compact complex analytic space each of whose singular points is an isolated rational double point. Suppose that II*(X, Z) is isomorphic to Z[t]/(t3) and is generated by the Chem class of the line bundle of a holomorphic divisor T. Then X is biholomorphic either to p or to a singular rational projective algebraic surface obtained from p by the successive application of precisely 8 monoidal transformations followed by the collapsing of a curve with precisely 8 analytic components, each a non-singular rational curve with self-intersection 2, to 1 or more singular points. Every such space X is homotopy equivalent to p, and the generator T may always be taken to be the divisor of a non-singular elliptic curve contained in the regular points of X.

Sets of Sets: A Cognitive Obstacle

Lawrence Brenton (brenton turing.math.wayne.edu) was educated at the University of Pennsylvania (BA, 1968), and the University of Washington (PhD, 1974). He held posts as :>+.Xr, na post-doctoral research fellow at the University of Bonn, Germany and as a visiting professor at Tulane University before joining the faculty of Wayne State University, where he has served for twenty-six years. His research interests are several complex variables and algebraic geometry, especially the theory of singularities of analytic varieties. His current passion is directing the WSU Undergraduate Research Program.

Remainder Wheels and Group Theory

Lawrence Brenton (brenton@math.wayne.edu) was educated at the University of Pennsylvania, the University of Washington in Seattle, and the University of Bonn, Germany. He has served on the faculty of Wayne State University for 33 years, specializing in singularity theory in complex algebraic and analytic varieties. His teaching activities include directing the undergraduate research program for many years as well as mentoring graduate students in mathematics and mathematics education.

Graph theoretic techniques in algebraic geometry II: construction of singular complex surfaces of the rational cohomology type of CP2

Methods of graph theory are used to obtain rational projective surfaces with only rational double points as singularities and with rational cohomology rings isomorphic to that of the complex projective plane. Uniqueness results for such cohomologyCP2s and for rational and integral homologyCP2s are given in terms of the typesAk,Dk, orEk of singularities allowed by the construction.

Perfectly Weighted Graphs

Abstract. We present a classification of those graphs which arise as dual intersection graphs in the resolutions of complex surface singularities with perfect local fundamental group. Using computation intensive methods, we examine the possibilities for weights on the vertices corresponding to self-intersections of the exceptional curves.

Homologically simple singularities

We discuss a family of topological models for spacetimes whose spacelike hypersurfaces are exotic cohomology 3-spheres, and examine the structure of their initial singularities.