Brian Pike

Brian Pike
University of Toronto | U of T · Department of Computer and Mathematical Sciences at Scarborough

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9
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Publications

Publications (9)
Article
A prehomogeneous vector space is a rational representation $\rho:G\to GL(V)$ of a connected complex linear algebraic group $G$ that has a Zariski open orbit $\Omega\subset V$. M. Sato showed that the hypersurface components of $D:=V\setminus \Omega$ are related to the characters $H\to GL(\mathbb{C})$ of $H$, an algebraic abelian quotient of $G$. Mi...
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Let $\varphi :X\to S$ be a morphism between smooth complex analytic spaces and let $f=0$ define a free divisor on $S$. We prove that if the deformation space $T^1_{X/S}$ of $\varphi $ is a Cohen–Macaulay $\mathcal {O}_X$-module of codimension 2, and all of the logarithmic vector fields for $f=0$ lift via $\varphi $, then $f\circ \varphi =0$ defines...
Article
Associated to the germ of an analytic set $(X,p)$ in $\C^n$ is the $\mathscr{O}_{\C^n,p}$--module $\Der(-\log X)$ of logarithmic vector fields, the ambient germs of holomorphic vector fields tangent to the smooth locus of $X$. For a submodule $L\subseteq \Der(-\log X)$, we seek to use Fitting ideals associated to $L$ and $\Der(-\log X)$ as submodul...
Article
We apply previous results on the representations of solvable linear algebraic groups to construct a new class of free divisors whose complements are K(π,1)ʼs. These free divisors arise as the exceptional orbit varieties for a special class of “block representations” and have the structure of determinantal arrangements.Among these are the free divis...
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This paper is the first part of a two part paper which introduces a method for determining the vanishing topology of nonisolated matrix singularities. A foundation for this is the introduction of a method for obtaining new classes of free divisors from representations $V$ of connected solvable linear algebraic groups $G$. For equidimensional repres...
Article
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In this paper we use the results from the first part to compute the vanishing topology for matrix singularities based on certain spaces of matrices. We place the variety of singular matrices in a geometric configuration of free divisors which are the "exceptional orbit varieties" for repesentations of solvable groups. Because there are towers of re...
Article
Survival rates of juvenile reptiles are critical population parameters but are difficult to obtain through mark-recapture programs because these small, secretive animals are rarely caught. This scarcity has encouraged speculation that survival rates of juveniles are very low, and we test this prediction by estimating juvenile survival rates indirec...

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