Polarized algebraic variety
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
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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) |
Comments
References
[a1] | D. Mumford, "Matsusaka's big theorem" R. Hartshorne (ed.) , Algebraic geometry (Arcata, 1974) , Proc. Symp. Pure Math. , 29 , Amer. Math. Soc. (1975) pp. 513–530 |
Polarized algebraic variety. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Polarized_algebraic_variety&oldid=19594