View source for Polarized algebraic variety

A pair $(X,\xi)$, where $V$ is a complex smooth variety (cf.
[[Algebraic variety|Algebraic variety]]) over an algebraically closed
field $k$, $\def\Pic{{\mathop{\mathrm{Pic}}}} \xi\in\Pic V/\Pic^0 V$ is the class of some ample invertible sheaf (cf.
[[Ample sheaf|Ample sheaf]];
[[Invertible sheaf|Invertible sheaf]]) and $\Pic^0 V$ is the connected
component of the identity of the Abelian
[[Picard scheme|Picard scheme]] $\Pic V$. In the case when $V$ 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 $\def\cL{{\mathop{\mathcal{L}}}} \phi_\cL: V \to \Pic^0 V$ determined by a sheaf $\cL\in\xi$, namely 
$$\phi_\cL(x)=T_x^*\; \cL\otimes\cL^{-1} \in\Pic^0 V,$$
where $T_x$ is the morphism of translation by $x$, $x\in V$. 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 $f:X\to S$ be a
family of varieties with base $S$, that is, $f$ is a smooth projective
morphism from the scheme $X$ to the Noetherian scheme $S$, the fibres
of which are algebraic varieties. The pair $(X/S,\xi/S)$, where $X/S$ is the
family $f:X\to S$ with base $S$, while $\xi/S$ is the class of the
relatively-ample invertible sheaf $\cL_{X/S}$ in $\def\Hom{{\mathop{\mathrm{Hom}}}} \Hom(S,\Pic X/S)$ modulo $\Hom(S,\Pic^0 X/S)$, where $\Pic X/S$ 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|Moduli theory]]). For example, there is no moduli
space of all smooth algebraic curves of genus $g\ge 1$, while for polarized
curves there is such a space
[[#References|[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 $(V,\xi)$ is contained as a fibre in a polarized family $(X/S,\xi/S)$
with a connected base $S$ and relatively-ample sheaf $\cL{X/S}\in \xi/S$, then does
there exist a constant $c$ depending only on the
[[Hilbert polynomial|Hilbert polynomial]] $h(n)=\chi(V,\cL^n)$ such that for $n>c$ the
sheaves $\cL_S^n$ with Hilbert polynomial $h(n)$ and with $H^i(X_s,\cL^n_S)=0$ for $i>0$, are very
ample for all polarized algebraic varieties $(X_s,\xi_s)$, where $s\in S$? For smooth
polarized algebraic varieties over an algebraically closed field of
characteristic $0$ the answer to this question is affirmative
[[#References|[3]]], while in the case of surfaces of principal type
with the canonical polarization the constant $c$ is even independent
of the Hilbert polynomial (see
[[#References|[1]]],
[[#References|[2]]]).

====References====
<table>
<TR><TD valign="top">[1]</TD>
 <TD valign="top"> E. Bombieri, "Canonical models of surfaces of general type" ''Publ. Math. IHES'' , '''42''' (1973) pp. 171–220 {{MR|0318163}} {{ZBL|0259.14005}} </TD></TR>
<TR><TD valign="top">[2]</TD>
 <TD valign="top"> K. Kodaira, "Pluricanonical systems on algebraic surfaces of general type" ''J. Math. Soc. Japan'' , '''20''' : 1–2 (1968) pp. 170–192 {{MR|0224613}} {{ZBL|0157.27704}} </TD></TR>
<TR><TD valign="top">[3]</TD>
 <TD valign="top"> T. Matsusaka, D. Mumford, "Two fundamental theorems on deformations of polarized varieties" ''Amer. J. Math.'' , '''86''' : 3 (1964) pp. 668–684 {{MR|0171778}} {{ZBL|0128.15505}} </TD></TR>
<TR><TD valign="top">[4]</TD>
 <TD valign="top"> D. Mumford, "Geometric invariant theory" , Springer (1965) {{MR|0214602}} {{ZBL|0147.39304}} </TD></TR>
 <TR><TD valign="top">[a1]</TD><TD valign="top"> 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 {{MR|379494}} {{ZBL|}} </TD></TR>
</table>