Positivity of line bundles on general blow ups of Hirzebruch surfaces
Cyril J. Jacob, Bivas Khan
TL;DR
The paper addresses positivity of line bundles on general blow ups of Hirzebruch surfaces by deriving finite numerical criteria for ampleness, global generation, very ampleness, and k-very ampleness. It employs Nakai–Moishezon and Reider's criterion to prove global generation and very ampleness, and the Beltrametti–Francia–Sommese criterion for k-very ampleness, in conjunction with careful analysis of the blow-up geometry. It yields practical inequalities for line bundles on F_{e,r} and derives lower bounds for multi-point Seshadri constants. The results extend the line of work on blow-ups of P^2 to Hirzebruch surfaces and provide explicit tools for positivity questions in surface theory.
Abstract
We investigate various positivity properties of line bundles on general blow ups of Hirzebruch surfaces motivated by \cite{Han}, where the author has studied general blow ups of $\mathbb{P}^2$. For each of the properties: ampleness, global generation, very ampleness, and $k$-very ampleness, we provide several sufficient numerical conditions.