Constructing new ample divisors out of old ones

We prove a gluing theorem which allows to construct an ample divisor on a rational surface from two given ample divisors on simpler surfaces. This theorem combined with the Cremona action on the ample cone gives rise to an algorithm for constructing new ample divisors. We then propose a conjecture relating continued fractions approximations and Seshadri-like constants of line bundles over rational surfaces. By applying our algorithm recursively we verify our conjecture in many cases and obtain new asymptotic estimates on these constants. Finally, we explain the intuition behind the gluing theorem in terms of symplectic geometry and propose generalizations.