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. KeywordsLine BundleAmple Line BundleAmple DivisorProjective ManifoldKodaira DimensionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.