Sharp Noether-type inequalities for $3$-folds of intermediate Kodaira dimensions

Abstract

For a smooth projective $3$-fold $V$ of intermediate Kodaira dimension, we establish Noether-type inequalities relating the Iitaka volume $\mathrm{Ivol}(V)$ and the geometric genus $p_g(V)$. Specifically, for a smooth projective $3$-fold $V$ of Kodaira dimension $2$, we show that $\mathrm{Ivol}(V)\ge \frac{3}{4}{p}_g(V)-\frac{5}{4};$ while if the Kodaira dimension of $V$ is $1$, then $\mathrm{Ivol}(V)\ge p_g(V)-1.$ Both inequalities are sharp. In the Kodaira dimension $2$ case, we prove that the equality holds only if the bicanonical system of $V$ induces the Iitaka fibration and the canonical system of $V$ is composed with a pencil of surfaces whose general member has Iitaka volume $\frac{1}{2}$ provided that $p_g(V)\ge 4$.

22 pages, comments are welcome