Smoothing toric Fano surfaces using the Gross–Siebert algorithm. Issue 3 (12th June 2018)

• Record Type: Journal Article
• Authors: Prince, Thomas
• Is Part Of: Proceedings of the London Mathematical Society. Volume 117:Issue 3(2018)
• Page Start: 617
• Page End: 660
• Publication Date: 2018-06-12
• Subjects: 14J33 -- 14J45 (primary) -- 14D06 (secondary); Mathematics -- Periodicals; Mathematics; Periodicals; 510
• DOI: 10.1112/plms.12153
• Languages: English
• ISSNs: 0024-6115

Abstract

Abstract: A toric del Pezzo surface X P with cyclic quotient singularities determines and is determined by a Fano polygon P . We construct an affine manifold with singularities that partially smooths the boundary of P ; this is a tropical version of a Q ‐Gorenstein partial smoothing of X P . We implement a mild generalization of the Gross–Siebert reconstruction algorithm — allowing singularities that are not locally rigid — and thereby construct (a formal version of) this partial smoothing directly from the affine manifold. This has implications for mirror symmetry: roughly speaking, it implements half of the expected mirror correspondence between del Pezzo surfaces with cyclic quotient singularities and Laurent polynomials in two variables.