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Advances in Geometry

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Volume 18, Issue 3

Issues

On the quantum periods of del Pezzo surfaces with ⅓ (1, 1) singularities

Alessandro Oneto / Andrea Petracci
Published Online: 2018-04-06 | DOI: https://doi.org/10.1515/advgeom-2017-0048

Abstract

In earlier joint work with collaborators we gave a conjectural classification of a broad class of orbifold del Pezzo surfaces, using Mirror Symmetry. We proposed that del Pezzo surfaces X with isolated cyclic quotient singularities such that X admits a ℚ-Gorenstein toric degeneration correspond via Mirror Symmetry to maximally mutable Laurent polynomials f in two variables, and that the quantum period of such a surface X, which is a generating function for Gromov–Witten invariants of X, coincides with the classical period of its mirror partner f.

In this paper we give strong evidence for this conjecture. Contingent on conjectural generalisations of the Quantum Lefschetz theorem and the Abelian/non-Abelian correspondence, we compute many quantum periods for del Pezzo surfaces with 13(1, 1) singularities. Our computations also give strong evidence for the extension of these two principles to the orbifold setting.

Keywords: Gromov–Witten invariants; quantum cohomology; quantum period; del Pezzo surface

MSC 2010: 14J33; 14J45; 52B20; 14N35

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About the article


Received: 2016-02-12

Revised: 2016-06-01

Published Online: 2018-04-06

Published in Print: 2018-07-26


Citation Information: Advances in Geometry, Volume 18, Issue 3, Pages 303–336, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2017-0048.

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