Matsusaka's big theorem

In algebraic geometry, given an ample line bundle L on a compact complex manifold X, Matsusaka's big theorem gives an integer m, depending only on the Hilbert polynomial of L, such that the tensor power Lⁿ is very ample for n ≥ m.

The theorem was proved by Teruhisa Matsusaka in 1972 and named by Lieberman and Mumford in 1975.^[1][2][3]

The theorem has an application to the theory of Hilbert schemes.

Notes

1. Matsusaka, T. (1972). "Polarized Varieties with a Given Hilbert Polynomial". American Journal of Mathematics. 94 (4): 1027–1077. doi:10.2307/2373563. JSTOR 2373563.
2. Lieberman, D.; Mumford, D. (1975). "Matsusaka's big theorem". Algebraic Geometry. Providence, RI: American Mathematical Society. pp. 513–530.
3. Kollár, János (August 2006). "Teruhisa Matsusaka (1926–2006)" (PDF). Notices of the American Mathematical Society. 53 (7): 766–768.