Editing Hodge index theorem

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{{More footnotes needed|date=December 2022}}
In [[mathematics]], the '''Hodge index theorem''' for an [[algebraic surface]] ''V'' determines the [[signature (quadratic form)|signature]] of the [[intersection theory|intersection pairing]] on the [[algebraic curve]]s ''C'' on ''V''. It says, roughly speaking, that the space spanned by such curves (up to [[linear equivalence]]) has a one-dimensional subspace on which it is [[positive definite]] (not uniquely determined), and decomposes as a [[direct sum of modules|direct sum]] of some such one-dimensional subspace, and a complementary subspace on which it is [[negative definite]].

In a more formal statement, specify that ''V'' is a [[Algebraic curve#Singularities|non-singular]] [[projective surface]], and let ''H'' be the [[divisor class]] on ''V'' of a [[hyperplane section]] of ''V'' in a given [[projective embedding]]. Then the intersection

:<math>H \cdot H = d\ </math>

where ''d'' is the [[degree of an algebraic variety|degree]] of ''V'' (in that embedding). Let ''D'' be the vector space of rational divisor classes on ''V'', up to [[algebraic equivalence]].  The dimension of ''D'' is finite and is usually denoted by ρ(''V''). The Hodge index theorem says that the subspace spanned by ''H'' in ''D'' has a complementary subspace on which the intersection pairing is negative definite. Therefore, the signature (often also called ''index'') is (1,ρ(''V'')-1).

The abelian group of divisor classes up to algebraic equivalence is now called the [[Néron-Severi group]]; it is known to be a [[finitely-generated abelian group]], and the result is about its [[tensor product]] with the rational number field. Therefore, ρ(''V'') is equally the rank of the Néron-Severi group (which can have a non-trivial [[torsion subgroup]], on occasion).

This result was proved in the 1930s by [[W. V. D. Hodge]], for varieties over the complex numbers, after it had been a conjecture for some time of the [[Italian school of algebraic geometry]] (in particular, [[Francesco Severi]], who in this case showed that ρ < ∞). Hodge's methods were the [[topological]] ones brought in by [[Lefschetz]]. The result holds over general ([[algebraically closed]]) fields.

==References==
* {{Citation | last1=Hartshorne | first1=Robin | author1-link=Robin Hartshorne | title=[[Algebraic Geometry (book)|Algebraic Geometry]] | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90244-9 | oclc=13348052 |mr=0463157 | year=1977}}, see Ch. V.1

[[Category:Algebraic surfaces]]
[[Category:Geometry of divisors]]
[[Category:Intersection theory]]
[[Category:Theorems in algebraic geometry]]