On the Dimension of the Adjoint Linear System for Quadric Fibrations

Abstract

Let L^ be a very ample line bundle on a smooth, n-dimensional, projective manifold X^ , i.e. assume that ${L^ \wedge } \approx {i^*}{O_{pn}}$ (1) for some embedding $i:{X^ \wedge } \to {\mathbb{P}^N}$. In [S1] it is shown that for such pairs, (X^ , L^), the Kodaira dimension of ${K_{{X^ \wedge }}} \otimes {L^{ \wedge n - 2}}\;is \geqslant 0$, i.e. there exists some positive integer, t, such that ${h^0}\left( {{{\left( {{K_{{X^ \wedge }}}{L^{ \wedge n - 2}}} \right)}^t}} \right) \geqslant 1$, except for a short list of degenerate examples. It is moreover shown that except for this short list there is a morphism $r:{X^ \wedge } \to X$ expressing X^ as the blow-up of a projective manifold X at a finite set B, and such that:

• 1. ${K_{{X^ \wedge }}} \otimes {L^{ \wedge n - 1}} \approx {r^*}\left( {{K_X} \otimes {L^{n - 1}}} \right)$ where $L: = {\left( {{r_*}{L^ \wedge }} \right)^{**}}$ is an ample line bundle and ${K_X} \otimes {L^{n - 1}}$

• 2. ${K_X} \otimes {L^{n - 2}}$ is nef, i.e. $\left( {{K_X} \otimes {L^{n - 2}}} \right) \cdot \geqslant 0$ for every effective curve C ⊂ X.