A lower bound for the sectional genus of quasi-polarized surfaces (Q677647)

A pair \((X,L)\) of a smooth \(n\) dimensional projective variety \(X\) with a divisor \(L\) on it is called quasi polarized if \(L\) is nef and big. The sectional genus \(g(L)\) of \(L\) is then defined as in \[ 2g(L)-2 =(K_X+ (n-1)L) \cdot L^{n-1}. \] \(g(L)\) is an integer by looking at the coefficients of \(\chi (kL)\) viewed as a polynomial of \(k\). It was conjectured and proved in the case when \(L\) is ample and spanned that \(g(L)\geq q(X)= h^1({\mathcal O}_X)\). The paper under review examines the conjecture for quasi polarized surfaces. It verifies the conjecture for surfaces which are not of general type using the classification theory when \(\kappa (X)\leq 0\) and the canonical formula for elliptic fibrations when \(\kappa (X)=1\). For those surfaces whose Kodaira dimensions are bounded by 1, the extremal cases of \((X,L)\) with \(q(X)= g(L)\) are classified. The conjecture is also verified in some special case when \(\kappa (X)=2\).