Polarized algebraic variety

A pair $(X,\xi)$, where $V$ is a complex smooth variety (cf. 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; Invertible sheaf) and $\Pic^0 V$ is the connected component of the identity of the Abelian 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). 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 [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 $h(n)=\chi(V,\cL^n)$ 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 [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 [1], [2]).

References

[1]	E. Bombieri, "Canonical models of surfaces of general type" Publ. Math. IHES , 42 (1973) pp. 171–220 MR0318163 Zbl 0259.14005
[2]	K. Kodaira, "Pluricanonical systems on algebraic surfaces of general type" J. Math. Soc. Japan , 20 : 1–2 (1968) pp. 170–192 MR0224613 Zbl 0157.27704
[3]	T. Matsusaka, D. Mumford, "Two fundamental theorems on deformations of polarized varieties" Amer. J. Math. , 86 : 3 (1964) pp. 668–684 MR0171778 Zbl 0128.15505
[4]	D. Mumford, "Geometric invariant theory" , Springer (1965) MR0214602 Zbl 0147.39304
[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 MR379494