Kodaira-Itaka dimension

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


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


then tr.degR(X,L)=κ(L)+1.

When the line bundle we have is the canonical bundle KX of X, then its Kodaira-Itaka dimension is called Kodaira dimension of X.

In paticular, if for some m we have dimφ|mL|(X)=dimX then κ(L)=dimX and L is called big.

If κ(X)=κ(KX)=dimX, then X is said to be of general typePlanetmathPlanetmath.

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