A Noether type inequality (Q2430827)

Let \(L\) be a Cartier divisor on a normal projective variety \(X\), and let \(\phi\) be the rational map defined by the linear system \(|L|\). If dim\(\phi(X)=k\), then the author proves that \[ h^0(2L)\geq (k+1)h^0(L)-\frac{1}{2}k(k+1). \] Particularly, if \(X\) is a minimal Gorenstein 3-fold of general type and the canonical map of \(X\) is generically finite, then \[ K^3_X\geq 2p_g +6 h^2(O_X)-6h^1(O_X)-6. \]