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Gary Kennedy

Publications and source records attributed to Gary Kennedy.

14 recordsLinked to original sources
arXiv2025

The small growth invariants of Goursat distributions

This is the second of a pair of papers devoted to the local invariants of Goursat distributions. The study of these distributions naturally leads to a tower of spaces over an arbitrary surface, called the monster tower, and thence to connections with the topic of singularities of curves on surfaces. In the prior paper we studied those invariants of Goursat distributions akin to those of curves on surfaces, which we call structural invariants. In this paper we study invariants arising from the small growth sequence of a Goursat distribution, and relate them to the the structural invariants.

math.DG↗
arXiv2023

The structural invariants of Goursat distributions

This is the first of a pair of papers devoted to the local invariants of Goursat distributions. The study of these distributions naturally leads to a tower of spaces over an arbitrary surface, called the monster tower, and thence to connections with the topic of singularities of curves on surfaces. Here we study those invariants of Goursat distributions akin to those of curves on surfaces, which we call structural invariants. In the subsequent paper we will relate these structural invariants to the small growth invariants.

math.DG↗
arXiv2019

On the Milnor fiber boundary of a quasi-ordinary surface

We give a recursive formula, expressed in terms of the characteristic tuples, for the Betti numbers of the boundary of the Milnor fiber of an irreducible quasi-ordinary surface. The singular locus of the surface consists of two components, and for each component we introduce a sequence of increasingly simpler surfaces. Our recursion depends on a detailed comparison of these two sequences. We indicate how we expect pieces of these associated surfaces to glue together to reconstruct the Milnor fiber and its boundary.

math.AG↗
arXiv2016

A coarse stratification of the monster tower

The monster tower is a tower of spaces over a specified base; each space in the tower is a parameter space for curvilinear data up to a specified order. We describe and analyze a natural stratification of these spaces.

math.AG↗
arXiv2013

Computing Severi Degrees with Long-edge Graphs

We study a class of graphs with finitely many edges in order to understand the nature of the formal logarithm of the generating series for Severi degrees in elementary combinatorial terms. These graphs are related to floor diagrams associated to plane tropical curves originally developed by Brugalle and Mikhalkin, and used by Block, Fomin, and Mikhalkin to calculate Severi degrees of the projective plane and node polynomials of plane curves.

math.AG↗
arXiv2011

Zariski decomposition: a new (old) chapter of linear algebra

In a 1962 paper, Zariski introduced the decomposition theory that now bears his name. Although it arose in the context of algebraic geometry and deals with the configuration of curves on an algebraic surface, we have recently observed that the essential concept is purely within the realm of linear algebra. In this paper, we formulate Zariski decomposition as a theorem in linear algebra and present a linear algebraic proof. We also sketch the geometric context in which Zariski first introduced his decomposition.

math.AG↗
arXiv2009

Monodromy of plane curves and quasi-ordinary surfaces

We develop recursive formulas for the horizontal and vertical monodromies of a quasi-ordinary surface. These are monodromies associated to the Milnor fiber of a slice transverse to a component of the singular locus. In the course of working out these recursions, we have discovered what appears to be a new way to express the monodromy associated to the Milnor fibration of a singular plane curve.

math.AG↗
arXiv2000

Contact formulas for rational plane curves via stable maps

We use stable maps, and their stable lifts to the Semple bundle variety of second-order curvilinear data, to calculate certain characteristic numbers for rational plane curves. These characteristic numbers involve first-order (tangency) and second-order (inflectional) conditions. Although they may be virtual, they may be used as inputs in an enumeratively significant formula for the number of rational curves having a triple contact with a specified plane curve and passing through 3d-3 general points.

math.AG↗
arXiv1998

Contact Cohomology of the Projective Plane

We construct an associative ring which is a deformation of the quantum cohomology ring of the projective plane. Just as the quantum cohomology encodes the incidence characteristic numbers of rational plane curves, the contact cohomology encodes the tangency characteristic numbers.

alg-geom↗
arXiv1993

The enumeration of simultaneous higher-order contacts between plane curves

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.

alg-geom↗