# Polarized algebraic variety

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A pair , where is a complex smooth variety (cf. Algebraic variety) over an algebraically closed field , is the class of some ample invertible sheaf (cf. Ample sheaf; Invertible sheaf) and is the connected component of the identity of the Abelian Picard scheme . In the case when is an Abelian variety, the concept of the degree of polarization of a polarized algebraic variety is also defined: It coincides with the degree of the isogeny determined by a sheaf , namely

where is the morphism of translation by , . A polarization of degree one is called a principal polarization.

The concept of a polarized algebraic variety is closely connected with the concept of a polarized family of algebraic varieties. Let be a family of varieties with base , that is, is a smooth projective morphism from the scheme to the Noetherian scheme , the fibres of which are algebraic varieties. The pair , where is the family with base , while is the class of the relatively-ample invertible sheaf in modulo , where is the relative Picard scheme, is called a polarized family.

The introduction of the concept of a polarized family and a polarized algebraic variety is required for the construction of moduli spaces of algebraic varieties (see Moduli theory). For example, there is no moduli space of all smooth algebraic curves of genus , while for polarized curves there is such a space [4]. One of the first questions connected with the concept of polarization of varieties is the question of simultaneous immersion in a projective space of polarized varieties with numerical invariants. If is contained as a fibre in a polarized family with a connected base and relatively-ample sheaf , then does there exist a constant depending only on the Hilbert polynomial such that for the sheaves with Hilbert polynomial and with for , are very ample for all polarized algebraic varieties , where ? For smooth polarized algebraic varieties over an algebraically closed field of characteristic the answer to this question is affirmative [3], while in the case of surfaces of principal type with the canonical polarization the constant is even independent of the Hilbert polynomial (see [1], [2]).

#### References

 [1] E. Bombieri, "Canonical models of surfaces of general type" Publ. Math. IHES , 42 (1973) pp. 171–220 [2] K. Kodaira, "Pluricanonical systems on algebraic surfaces of general type" J. Math. Soc. Japan , 20 : 1–2 (1968) pp. 170–192 [3] T. Matsusaka, D. Mumford, "Two fundamental theorems on deformations of polarized varieties" Amer. J. Math. , 86 : 3 (1964) pp. 668–684 [4] D. Mumford, "Geometric invariant theory" , Springer (1965)