Minimal threefolds with small slope and the Noether inequality for canonically polarized threefolds

Meng Chen

paper

Minimal threefolds with small slope and the Noether inequality for canonically polarized threefolds

arXiv:math/0312179

Abstract

1) We give a 3-dimensional analogue of M. Noether's inequality for canonically polarized threefolds: $K^{3} \geq 2 (2 p_{g} - 5) /3$ . This inequality is sharp by known examples of M. Kobayashi. 2) Given a minimal 3-fold $X$ of general type with canonical singularities, if $K^{3} < (3 p_{g} - 5) /2$ , we show that $X$ must be birationally fibred by curves of genus 2.

20 pages, modified version, to appear in Math. Res. Lett