# Kodaira-Itaka dimension

Given a projective algebraic variety $X$ and a line bundle $L\to X$,
the *Kodaira-Itaka dimension* of $L$
is defined to be the supremum of the dimensions of the image of $X$
by the map ${\phi}_{|mL|}$ associated to the linear system $|mL|$,
when $m$ is a positive integer, namely

$$\kappa (L)=\underset{m\in \mathbb{N}}{sup}\{dim{\phi}_{|mL|}(X)\}.$$ |

It is a standard fact that if we consider the graded ring

$$R(X,L)=\underset{m\in \mathbb{N}}{\oplus}{H}^{0}(X,mL),$$ |

then $\text{tr.deg}R(X,L)=\kappa (L)+1$.

When the line bundle we have is the canonical bundle ${K}_{X}$ of $X$,
then its Kodaira-Itaka dimension is called *Kodaira dimension* of $X$.

In paticular, if for some $m$ we have $dim{\phi}_{|mL|}(X)=dimX$
then $\kappa (L)=dimX$ and $L$ is called *big*.

If $\kappa (X)=\kappa ({K}_{X})=dimX$,
then $X$ is said to be of *general type ^{}*.

Title | Kodaira-Itaka dimension |
---|---|

Canonical name | KodairaItakaDimension |

Date of creation | 2013-03-22 16:12:43 |

Last modified on | 2013-03-22 16:12:43 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 17 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 14E05 |

Defines | Kodaira dimension |

Defines | bigness |

Defines | general type |