<< Chapter < Page Chapter >> Page >
To understand the behavior of polynomials at isolated singularity, we use the process of blowing up. This process yields an important characteristic number called the log canonical threshold (LCT). Our goal is to explore some uses for the LCT and understand methods for finding this value.

A Theorem on the Log Canonical Threshold Mike Clendenen, Ira Jamshidi, Lauren Kirton, David Lax Dr. Hassett, Dr. Varilly S. Li, Z. Li, B. Waters NSF grant 0739420 July 16, 2009

Introduction

In the study of solutions to polynomial equations, there are points of interest called singularities. A singularity on a plane curve is a point that does not look locally like the graph of a function (see Figures 1). There is a number associated with every singularity called the log canonical threshold (LCT) that carries some information about the singularity. In a sense, this number is a measure of how bad a singularity is. A higher LCT corresponds to a “nicer” singularity. For example, the node below has LCT 1 and the cusp has LCT 5 6 . This corresponds to the notion that the node seems to be a nicer singularity.

In this paper we examine the LCT of polynomials of the form y q - x p . Such polynomials have a singularity at the origin when p , q 2 . This is an important case to understand since many singularities can be related to a singularity of this form. We give an elementary proof of the following theorem:

Theorem 1.1 L C T ( y q - x p , 0 ) = 1 p + 1 q .

This is a special case of J. Igusa's theorem from 1977. In his notation for his theorem, this is the case where the Puiseux series of a plane curve is of the form: y ( x ) = x p q [link] Section 1. Before giving the proof of Theorem [link] we will first define the algebro-geometric terms necessary to understand the theorem and then review concepts from continued fractions which are useful in the proof of the theorem.

Definitions

In this section we give precise definitions for some of the algebro-geometric concepts mentioned in "Introduction" ; we establish some notation along the way. Throughout, we work over the field of complex numbers.

Definition 2.1 A plane curve is the set of all points ( x , y ) in the affine plane such that a given function f ( x , y ) x , y ] is zero.

Definition 2.2 A singularity is a point on a plane curve where all of the partial derivatives vanish, that is, a point ( x 0 , y 0 ) such that f ( x 0 , y 0 ) = f x ( x 0 , y 0 ) = f y ( x 0 , y 0 ) = 0 . A singulariy is called an isolated singularity if in some small neighborhood of the point, it is the only singularity of the curve.

Example 2.3 Let f ( x , y ) = y 2 - x 3 . The partial derivatives of f are given by

f x = 3 x 2 , f y = 2 x .

Thus, at the origin, f = f x = f y = 0 , and ( 0 , 0 ) is a singularity of f ( x , y ) = y 2 - x 3 . One can check that this is an isolated singularity.

Given a singularity, we can “resolve” it into slopes by implementing a series of substitutions of a particular kind and graphing the resulting curves. For example, a curve in x and y can be resolved by the substitutions y = t x or x = s y , where t and s represent slopes of lines through the singularity. This process is called blowing-up , and each stage at which a new substitution is introduced is known as a blow-up . The axes we use in the blow-up diagrams are called exceptional divisors , and can be thought of as “slope axes” at the point of blowing-up. We denote the exceptional divisor of the i t h blow-up as E i . We continue the process of blowing-up until we achieve a normal crossings , which is a result of blowing-up where the graph has no singularities and meets the exceptional divisors at unique intersections, without being tangent to an exceptional divisor at any point.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'The art of the pfug' conversation and receive update notifications?

Ask