In algebraic geometry, Max Noether's theorem on curves is a theorem about curves lying on
algebraic surface
In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of di ...
s, which are
hypersurfaces in ''P''
3, or more generally
complete intersection
In mathematics, an algebraic variety ''V'' in projective space is a complete intersection if the ideal of ''V'' is generated by exactly ''codim V'' elements. That is, if ''V'' has dimension ''m'' and lies in projective space ''P'n'', there shou ...
s. It states that, for degree at least four for hypersurfaces, the ''
generic
Generic or generics may refer to:
In business
* Generic term, a common name used for a range or class of similar things not protected by trademark
* Generic brand, a brand for a product that does not have an associated brand or trademark, other ...
'' such surface has no curve on it apart from the
hyperplane section. In more modern language, the
Picard group is
infinite cyclic, other than for a short list of degrees. This is now often called the Noether-Lefschetz theorem.
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Algebraic geometry