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In mathematics, a '''Fermat quintic threefold''' is a special [[quintic threefold]], in other words a degree 5 dimension 3 [[hypersurface]] in 4-dimensional projective space, given by
 
:<math>V^5+W^5+X^5+Y^5+Z^5=0. \, </math>
 
It is a [[Calabi-Yau manifold]].
 
The [[Hodge diamond]] of a non-singular quintic 3-fold is
{| style="text-align:center"
|-
|width="20"|
|width="20"|
|width="20"|
|width="20"|1
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|width="20"|
|width="20"|
|-
|
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|0
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|0
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|-
|
|0
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|1
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|0
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|-
|1
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|101
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|101
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==Rational curves==
{{harvtxt|Clemens|1984}} conjectured that the number of rational curves of a given degree on a generic quintic threefold is finite. The Fermat quintic threefold is not generic in this sense, and {{harvtxt|Albano|Katz|1991}} showed that its lines are contained in 50 1-dimensional families of the form (''x''&nbsp;:&nbsp;&minus;''&zeta;x''&nbsp;:&nbsp;''ay''&nbsp;:&nbsp;''by''&nbsp;:&nbsp;''cy'') for ''&zeta;''<sup>&nbsp;5</sup>&nbsp;=&nbsp;1 and ''a''<sup>5</sup>&nbsp;+&nbsp;''b''<sup>5</sup>&nbsp;+&nbsp;''c''<sup>5</sup>&nbsp;=&nbsp;0. There 375 lines in more than 1 family, of the form (''x'' : &minus;''&zeta;x''&nbsp;:&nbsp;''y''&nbsp;:&nbsp;&minus;''&eta;y''&nbsp;:&nbsp;0) for 5th roots of&nbsp;1 ''&zeta;''&nbsp;and&nbsp;''&eta;''.
 
==References==
*{{Citation | last1=Albano | first1=Alberto | last2=Katz | first2=Sheldon | title=Lines on the Fermat quintic threefold and the infinitesimal generalized Hodge conjecture | jstor=2001512 | mr=1024767  | year=1991 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=324 | issue=1 | pages=353–368 | doi=10.2307/2001512}}
*{{Citation | last1=Clemens | first1=Herbert | title=Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982) | publisher=[[Princeton University Press]] | series=Ann. of Math. Stud. | mr=756858  | year=1984 | volume=106 | chapter=Some results about Abel-Jacobi mappings | pages=289–304}}
*{{Citation | last1=Cox | first1=David A. | last2=Katz | first2=Sheldon | title=Mirror symmetry and algebraic geometry | url=http://www.ams.org/bookstore-getitem/item=surv-68.s | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Mathematical Surveys and Monographs | isbn=978-0-8218-1059-0 | mr=1677117  | year=1999 | volume=68}}
 
[[Category:Threefolds]]
[[Category:Complex manifolds]]

Revision as of 17:45, 23 December 2013

In mathematics, a Fermat quintic threefold is a special quintic threefold, in other words a degree 5 dimension 3 hypersurface in 4-dimensional projective space, given by

V5+W5+X5+Y5+Z5=0.

It is a Calabi-Yau manifold.

The Hodge diamond of a non-singular quintic 3-fold is

1
0 0
0 1 0
1 101 101 1
0 1 0
0 0
1

Rational curves

Template:Harvtxt conjectured that the number of rational curves of a given degree on a generic quintic threefold is finite. The Fermat quintic threefold is not generic in this sense, and Template:Harvtxt showed that its lines are contained in 50 1-dimensional families of the form (x : −ζx : ay : by : cy) for ζ 5 = 1 and a5 + b5 + c5 = 0. There 375 lines in more than 1 family, of the form (x : −ζx : y : −ηy : 0) for 5th roots of 1 ζ and η.

References

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  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

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    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

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  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

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    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

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