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Deformation of tropical Hirzebruch surfaces and enumerative geometry. (English) Zbl 1396.14065

The “\(n\)th Hirzebruch surface”, denoted by \(\Sigma_n\), is the surface \[ \Sigma_n = \mathbb{P}(\mathcal{O}_{\mathbb{CP}^1(n)} \oplus \mathbb{C}) \] for \(n \geq 0\). The paper under review, establishes a formula relating genus zero enumerative invariants of the Hirzebruch surfaces \(\Sigma_n\) and \(\Sigma_{n+2}\) using tropical geometric techniques. This generalizes previous results of D. Abramovich and A. Bertram [Contemp. Math. 276, 83–88 (2001; Zbl 1044.14030)] and R. Vakil [Manuscr. Math. 102, No. 1, 53–84 (2000; Zbl 0967.14036)].
The method used is based on studying deformation theory of Hirzebruch surfaces using tools of tropical geometry. One of the main results establishes a correspondence between algebraic curves on \(\Sigma_n\), and tropical curves in their associated tropical models.

MSC:

14T05 Tropical geometry (MSC2010)
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References:

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