algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, given an

ample line bundle In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of ...

''L'' on a compact complex manifold ''X'', Matsusaka's big theorem gives an integer ''m'', depending only on the

Hilbert polynomial In commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a field are three strongly related notions which measure the growth of the dimension of the homog ...

of ''L'', such that the tensor power ''L''^''n'' is very ample for ''n'' ≥ ''m''. The theorem was proved by

Teruhisa Matsusaka (1926–2006) was a Japanese-born American mathematician, who specialized in algebraic geometry. Matsusaka received his Ph.D. in 1954 at Kyoto University; he was a member of the Brandeis Mathematics Department from 1961 until his retirement in 1 ...

in 1972 and named by Lieberman and Mumford in 1975. The theorem has an application to the theory of

Hilbert scheme In algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space (or a more general projective scheme), refining the Chow variety. The Hilbert scheme is a ...

Notes

Theorems in algebraic geometry {{algebraic-geometry-stub