A polarization on a projective variety \(X\) is the class in \(\NS(X)\) of an ample line bundle. A polarized variety is a pair \((X, \mcl)\) with \(\mcl\) ample, considered up to isomorphisms preserving the class of \(\mcl\) in \(\NS(X)\). The degree of a polarization on \(X\) of dimension \(n\) is the intersection number \(\mcl^n\).
\((\PP^n, \OO(1))\) is polarized of degree \(1\), and a hypersurface of degree \(d\) with \(\OO(1)\) is polarized of degree \(d\). A K3 surface with an ample class \(h\) has a polarization of degree \(h^2 = 2g - 2\), and a quartic surface in \(\PP^3\) is a K3 surface of degree \(4\). For an abelian variety \(A\), a polarization is equivalently an isogeny \(A \to \dualof{A}\) of the form \(x \mapsto t_x^* \mcl \otimes \mcl^{-1}\); it is principal when this is an isomorphism, as for the theta divisor on the Jacobian of a curve.
Dependencies
D-DIVAMPLEAmple and very ample divisorsusesD-DIVEQUIVLinear, algebraic, and numerical equivalenceuses