Nonlocal Lagrangian: Difference between revisions

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In [[geometry]], the '''Fermat cubic''', named after [[Pierre de Fermat]], is a surface defined by
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:<math> x^3 + y^3 + z^3 = 1. \ </math>
 
Methods of [[algebraic geometry]] provide the following parametrization of Fermat's cubic:
:<math> x(s,t) = {3 t - {1\over 3} (s^2 + s t + t^2)^2 \over t (s^2 + s t + t^2) - 3} </math>
 
:<math> y(s,t) = {3 s + 3 t + {1\over 3} (s^2 + s t + t^2)^2 \over t (s^2 + s t + t^2) - 3} </math>
 
:<math> z(s,t) = {-3 - (s^2 + s t + t^2) (s + t) \over t (s^2 + s t + t^2) - 3}. </math>
 
In projective space the Fermat cubic is given by
:<math>w^3+x^3+y^3+z^3=0.</math>
The 27 lines lying on the Fermat cubic are easy to describe explicitly: they are the 9 lines of the form (''w'' : ''aw'' : ''y'' : ''by'') where ''a'' and ''b'' are fixed numbers with cube &minus;1,  and their 18 conjugates under permutations of coordinates.
 
[[Image:FermatCubicSurface.PNG]]
::::''Real points of Fermat cubic surface.''
 
==References==
*{{Citation | last1=Ness | first1=Linda | title=Curvature on the Fermat cubic | url=http://projecteuclid.org/getRecord?id=euclid.dmj/1077313099 | mr=518106  | year=1978 | journal=[[Duke Mathematical Journal]] | issn=0012-7094 | volume=45 | issue=4 | pages=797–807}}
*{{cite web|url=http://www.math.harvard.edu/~elkies/4cubes.html|title=Complete cubic parametrization of the Fermat cubic surface|first=Noam|last=Elkies}}
 
[[Category:Algebraic surfaces]]

Latest revision as of 21:22, 6 March 2014

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