<< Chapter < Page Chapter >> Page >

Example 2.4 Consider f ( x , y ) = y 2 - x 3 . As shown in example [link] , f = 0 has a singularity at the origin.

First Blow-up

L e t y = s x .
T h e n s 2 x 2 - x 3 = 0 .
x 2 ( s 2 - x ) = 0 .

In this equation, x 2 is the exceptional divisor while s 2 - x is what is called the proper transform . Note that the proper transform is tangent to the exceptional divisor.

Second Blow-up

L e t x = t s .
T h e n t 2 s 2 ( s 2 - t s ) = 0 .
t 2 s 3 ( s - t ) = 0 .

In this equation, t 2 and s 3 are exceptional divisors while s - t is the proper transform. Note that the proper transform intersects two exceptional divisors at one point.

Third Blow-up

L e t t = r s .
T h e n r 2 s 5 ( s - r s ) = 0 .
r 2 s 6 ( 1 - r ) = 0 .

In this equation, r 2 and s 6 are exceptional divisors while 1 - r is the proper transform. Note that the proper transform is now resolved.

Definition 2.5 The multiplicity of the exceptional divisor is the highest degree of the factored variables of the equation provided after a blow-up substitution. We denote the multiplicity of the i t h exceptional divisor as C i .

Example 2.6 Using the blow-up sequence in Example [link] , we can see that C 1 = 2 , C 2 = 3 , and C 3 = 6 .

Definition 2.7 When blowing-up, we track the result of the series of substitutions leading up to and including that blow-up on the original differential form (in our examples d x d y ). Note that the change of variables formula will always contribute an additional power to the higher degree variable of the differential form. The multiplicity of the canonical divisor is the highest degree of the factored variables of the differential form. We denote the multiplicity of the i t h canonical divisor as K i .

Example 2.8 Using the blow-up sequence in Example [link] , we start with d x d y . After the first blow-up, the change of variables gives us the differential form x d s d x . After the second blow-up, the differential form is t s 2 d t d s . After the third blow-up, the differential form is r s 4 d r d s . We can see that K 1 = 1 , K 2 = 2 , and K 3 = 4 .

Definition 2.9 Given a normal crossings blow-up sequence with multiplicities of the canonical divisor and the exceptional divisor known, we calculate α i = K i + 1 C i . The log canonical threshold of the function is the infimum of the set of α i for all i . We denote the log canonical threshold of a function f at a singularity p as L C T ( f , p ) .

Example 2.10 Using the blow-up sequence in Example [link] , we can see that α 1 = 1 , α 2 = 1 , and α 3 = 5 6 . Thus, L C T ( f , 0 ) = 5 6 .

Continued fractions

The proof of the Theorem [link] relies heavily on continued fractions. In this section we recall some basic facts about continued fractions and prove some Lemmas about them relevant to the theorem. Consider the rational number q p . First, using the Euclidean algorithm, we write:

q = r 1 p + n 1 n 1 < p
p = r 2 n 1 + n 2 n 2 < n 1
n 1 = r 3 n 2 + n 3 n 3 < n 2
n m - 2 = r m n m - 1 + 0 .

The continued fraction of q p is

q p = r 1 + 1 r 2 + 1 r 3 + 1 r 4 +

This can be abbreviated with the standard notation for continued fractions: q p = [ r 1 , r 2 , r 3 , r 4 , . . . ] . Since q p is a rational number, we know that the continued fraction expression will eventually terminate, meaning that there exists a smallest positive integer δ such that n δ = 0 .

We call [ r 1 , r 2 , . . . , r v ] the v t h convergent of q p . Note that the δ t h convergent equals q p . We will then rearrange the variables of the continued fraction convergents so that they are in the form of a simple fraction. For example, we will rearrange the second convergent of q p to be r 1 r 2 + 1 r 2 ; the third convergent will become r 1 + r 3 r 2 r 3 + 1 = r 1 r 2 r 3 + r 2 + r 3 r 2 r 3 + 1 ; etc. Once the v t h convergent is in this final form, we will denote the numerator of this expression h v and the denominator of this expression k v . Denote h v k v by A v . Notice that q p = A δ , since the δ t h convergent is the entire continued fraction.

Questions & Answers

what is mutation
Janga Reply
what is a cell
Sifune Reply
how is urine form
Sifune
what is antagonism?
mahase Reply
classification of plants, gymnosperm features.
Linsy Reply
what is the features of gymnosperm
Linsy
how many types of solid did we have
Samuel Reply
what is an ionic bond
Samuel
What is Atoms
Daprince Reply
what is fallopian tube
Merolyn
what is bladder
Merolyn
what's bulbourethral gland
Eduek Reply
urine is formed in the nephron of the renal medulla in the kidney. It starts from filtration, then selective reabsorption and finally secretion
onuoha Reply
State the evolution relation and relevance between endoplasmic reticulum and cytoskeleton as it relates to cell.
Jeremiah
what is heart
Konadu Reply
how is urine formed in human
Konadu
how is urine formed in human
Rahma
what is the diference between a cavity and a canal
Pelagie Reply
what is the causative agent of malaria
Diamond
malaria is caused by an insect called mosquito.
Naomi
Malaria is cause by female anopheles mosquito
Isaac
Malaria is caused by plasmodium Female anopheles mosquitoe is d carrier
Olalekan
a canal is more needed in a root but a cavity is a bad effect
Commander
what are pathogens
Don Reply
In biology, a pathogen (Greek: πάθος pathos "suffering", "passion" and -γενής -genēs "producer of") in the oldest and broadest sense, is anything that can produce disease. A pathogen may also be referred to as an infectious agent, or simply a germ. The term pathogen came into use in the 1880s.[1][2
Zainab
A virus
Commander
Definition of respiration
Muhsin Reply
respiration is the process in which we breath in oxygen and breath out carbon dioxide
Achor
how are lungs work
Commander
where does digestion begins
Achiri Reply
in the mouth
EZEKIEL
what are the functions of follicle stimulating harmones?
Rashima Reply
stimulates the follicle to release the mature ovum into the oviduct
Davonte
what are the functions of Endocrine and pituitary gland
Chinaza
endocrine secrete hormone and regulate body process
Achor
while pituitary gland is an example of endocrine system and it's found in the Brain
Achor
what's biology?
Egbodo Reply
Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
Lisah
biology is the study of life.
Alfreda
Biology is the study of how living organisms live and survive in a specific environment
Sifune
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




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