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{{for|complete intersection rings in commutative algebra, |complete intersection ring}}
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In mathematics, an [[algebraic variety]] ''V'' in [[projective space]] is a '''complete intersection''' if the ideal of V is generated by exactly ''n'' - ''m'' elements. That is, if ''V'' has [[dimension of an algebraic variety|dimension]] ''m'' and  lies in projective space ''P''<sup>''n''</sup>, there should exist ''n'' − ''m'' homogeneous polynomials


:''F''<sub>''i''</sub>(''X''<sub>0</sub>, ..., ''X''<sub>''n''</sub>), 1 ≤ ''i'' ≤ ''n'' − ''m'',
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in the [[homogeneous coordinates]] ''X''<sub>''j''</sub>, generate all other homogeneous polynomial which vanishes on V
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Geometrically, each ''F''<sub>''i''</sub> defines a [[hypersurface]]; the intersection of these hypersurfaces should be ''V''. The intersection of ''n''-''m'' hypersurfaces will always have dimension at least ''m'', assuming that the field of scalars is an [[algebraically closed field]] such as the [[complex number]]s. The question is essentially, can we get the dimension down to ''m'', with no  extra points in the intersection? This condition is fairly hard to check as soon as the codimension ''n'' − ''m''  ≥ 2. When ''n'' − ''m'' = 1 then ''V'' is automatically a hypersurface and there is nothing to prove.
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==Example of a space curve that is not a complete intersection==
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A classic example is the [[twisted cubic]] in <math>\mathbb{P}^3</math>: it is a set-theoretic complete intersection, i.e. as a set it can be expressed as the intersection of 2 hypersurfaces, but not an ideal-theoretic (or scheme-theoretic) complete intersection, i.e. its homogeneous ideal cannot be generated by 2 elements.  
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Its [[degree of an algebraic variety|degree]] is 3, so to be an ideal-theoretic complete intersection it would have to be the intersection of two surfaces of degrees 1 and 3, by the hypersurface [[Bézout theorem]]. In other words, it would have to be the intersection of a plane and a [[cubic surface]]. But by direct calculation, any four distinct points on the curve are not coplanar, so it cannot lie in a plane, ruling out the only possible case. The twisted cubic lies on many [[quadric]]s, but the intersection of any two of these quadrics will always contain the curve plus an extra line, since the intersection of two quadrics has degree <math>2\times 2 = 4,</math> and the twisted cubic has degree 3, so the only way to get degree 4 is to add a line.
</ul>
 
On the other hand, the twisted cubic, as a set, is the intersection of the quadric surface <math>x z - y^2 = 0</math> and the cubic surface <math> z (y w - z^2) - w (x w - y z)=0</math> in <math>\mathbb{P}^3</math>. Formally the degree of that intersection is 6, so in a more refined sense, the intersection is actually the twisted cubic counted with multiplicity two.
 
==Multidegree==
 
A complete intersection has a ''multidegree'', written as the [[tuple]] (properly though a [[multiset]]) of the degrees of defining hypersurfaces. For example taking quadrics in ''P''<sup>3</sup> again, (2,2) is the multidegree of the complete intersection of two of them, which when they are in [[general position]] is an [[elliptic curve]]. The [[Hodge number]]s of complex smooth complete intersections were worked out by [[Kunihiko Kodaira]].
 
==General position==
 
For more refined questions, the nature of the intersection has to be addressed more closely. The hypersurfaces may be required to satisfy a [[Transversality (mathematics)|transversality]] condition (like their [[tangent space]]s being in general position at intersection points). The intersection may be [[scheme (mathematics)|scheme-theoretic]], in other words here the [[homogeneous ideal]] generated by the ''F''<sub>''i''</sub>(''X''<sub>0</sub>, ..., ''X''<sub>''n''</sub>) may be required to be the defining ideal of ''V'', and not just have the correct [[radical of an ideal|radical]]. In [[commutative algebra]], the complete intersection condition is translated into [[regular sequence (algebra)|regular sequence]] terms, allowing the definition of [[local complete intersection]], or after some [[localization of a ring|localization]] an ideal has defining regular sequences.
 
==References==
*E. J. N. Looijenga (1984), ''Isolated Singular Points on Complete Intersections''
 
==External links==
*[http://www.map.mpim-bonn.mpg.de/Complete_intersections Complete intersections] at the Manifold Atlas
 
{{DEFAULTSORT:Complete Intersection}}
[[Category:Algebraic geometry]]
[[Category:Commutative algebra]]

Latest revision as of 21:20, 12 January 2015

Xing Canada Goose Coats

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