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Deformation of tropical Hirzebruch surfaces and enumerative geometry

We exhibit relations between enumerative invariants of the Hirzebruch surfaces $\Sigma_n$ and \Sigma_{n+2}, obtained by deforming the first surface to the latter. Our formula generalizes in particular a formula due to Abramovich and Bertram, extended later by Vakil.
We use methods from tropical geometry, involving a tropical counterpart of deformations of Hirzebruch surfaces, and tropical enumerative geometry on a tropical surface in three-space.