On 3-dimensional Noether’s Inequality

The aim of this paper is to give effective inequalities in the form of K X ≥?pg(X) −?, where X is a minimal projective 3-fold of general type with at worst Q-factorial terminal singularities. Our method is quite effective in the sense that we have found the optimal form of Noether’s inequality: K X ≥ 4 3 pg(X) − a with a ∈ [ 10 3 , 14 3 ] among smooth canonical models. An interesting application of our inequalities is the so-called ”Boundedness Theorem” which has been expected by experts. Explicitly, suppose X is Gorenstein and Φ|KX | is of fiber type. Then X is canonically fibred by surfaces or curves with bounded invariants. Introduction Let S be a smooth minimal projective surface of general type. It is well known that M. Noether ([N]) gave the inequality K S ≥ 2pg − 4 whence K 2 S ≥ 2χ− 6 and that Bogomolov, Miyaoka and Yau ([M1], [Y1, Y2]) proved the inequality K S ≤ 9χ. These inequalities had played central roles to birational classification theory of surfaces during the last century. An effective higher dimensional version of these inequalities would be, by all means, much useful to classify higher dimensional varieties. This paper aims to give an effective Noether’s inequality for arbitrary minimal projective 3-folds of general type. Miles Reid put forward the following question in 1980s. Question. What is Noether’s inequality for smooth minimal 3-folds? As far as I know, there were a lot of partial results with regard to this problem. In 1981, J. Harris ([Hrs]) studied varieties with very ample canonical bundles and gave effective inequalities. In 1992, M. Kobayashi ([Kob]) systematically studied the above question and partially presented effective inequalities. An interesting point, due to Kobayashi, is that there is a smooth projective 3-fold Y of general type such that KY is ample and K Y = 1 3 [4pg(Y )− 10], (pg(Y ) = 7, 10, 13, · · · ). (0.1) This means that, in general, one can’t hope to obtain the expected inequality like K X ≥ 2pg(X) − 6. However, one may still ask what the authentic 3-dimensional Noether’s inequality is. Typeset by AMS-TEX 1