mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Hodge index theorem for an

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 ...

''V'' determines the

signature A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...

of the intersection pairing on the

algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...

s ''C'' on ''V''. It says, roughly speaking, that the space spanned by such curves (up to

linear equivalence In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier (mathematician), Pierre Cartier ...

) has a one-dimensional subspace on which it is

positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * Positive-definite f ...

(not uniquely determined), and decomposes as a

direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...

of some such one-dimensional subspace, and a complementary subspace on which it is

negative definite In mathematics, negative definiteness is a property of any object to which a bilinear form may be naturally associated, which is negative-definite. See, in particular: * Negative-definite bilinear form * Negative-definite quadratic form * Nega ...

. In a more formal statement, specify that ''V'' is a

non-singular In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In cas ...

projective surface In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables ...

, and let ''H'' be the

divisor class In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mumfo ...

on ''V'' of a

hyperplane section In mathematics, a hyperplane section of a subset ''X'' of projective space P''n'' is the intersection of ''X'' with some hyperplane ''H''. In other words, we look at the subset ''X'H'' of those elements ''x'' of ''X'' that satisfy the single line ...

of ''V'' in a given projective embedding. Then the intersection :

H \cdot H = d\

where ''d'' is the

degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...

of ''V'' (in that embedding). Let ''D'' be the vector space of rational divisor classes on ''V'', up to

algebraic equivalence In algebraic geometry, a branch of mathematics, an adequate equivalence relation is an equivalence relation on algebraic cycles of smooth projective varieties used to obtain a well-working theory of such cycles, and in particular, well-defined inte ...

. 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 In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x = n_1x_1 + n_2x_2 + \cdots + n_sx_s for some integers n_1,\dots, n ...

, and the result is about its

tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...

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 In the theory of abelian groups, the torsion subgroup ''AT'' of an abelian group ''A'' is the subgroup of ''A'' consisting of all elements that have finite order (the torsion elements of ''A''). An abelian group ''A'' is called a torsion group (or ...

, 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 relation to the history of mathematics, the Italian school of algebraic geometry refers to mathematicians and their work in birational geometry, particularly on algebraic surfaces, centered around Rome roughly from 1885 to 1935. There were 30 ...

(in particular,

Francesco Severi Francesco Severi (13 April 1879 – 8 December 1961) was an Italian mathematician. He was the chair of the committee on Fields Medal on 1936, at the first delivery. Severi was born in Arezzo, Italy. He is famous for his contributions to algeb ...

, who in this case showed that ρ < ∞). Hodge's methods were the

topological In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...

ones brought in by

Lefschetz Solomon Lefschetz (russian: Соломо́н Ле́фшец; 3 September 1884 – 5 October 1972) was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear o ...

. The result holds over general (

algebraically closed In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...

) fields.

References

* {{Citation , last1=Hartshorne , first1=Robin , author1-link=Robin Hartshorne , title=

Algebraic Geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, publisher=

Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...

, location=Berlin, New York , isbn=978-0-387-90244-9 , oclc=13348052 , mr=0463157 , year=1977, see Ch. V.1 Algebraic surfaces Geometry of divisors Intersection theory Theorems in algebraic geometry