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http://dx.doi.org/10.4134/BKMS.b190128

EXPLICIT EQUATIONS FOR MIRROR FAMILIES TO LOG CALABI-YAU SURFACES  

Barrott, Lawrence Jack (National Center for Theoretical Sciences, National Taiwan University)
Publication Information
Bulletin of the Korean Mathematical Society / v.57, no.1, 2020 , pp. 139-165 More about this Journal
Abstract
Mirror symmetry for del Pezzo surfaces was studied in [3] where they suggested that the mirror should take the form of a Landau-Ginzburg model with a particular type of elliptic fibration. This argument came from symplectic considerations of the derived categories involved. This problem was then considered again but from an algebro-geometric perspective by Gross, Hacking and Keel in [8]. Their construction allows one to construct a formal mirror family to a pair (S, D) where S is a smooth rational projective surface and D a certain type of Weil divisor supporting an ample or anti-ample class. In the case where the self intersection matrix for D is not negative semi-definite it was shown in [8] that this family may be lifted to an algebraic family over an affine base. In this paper we perform this construction for all smooth del Pezzo surfaces of degree at least two and obtain explicit equations for the mirror families and present the mirror to dP2 as a double cover of ℙ2.
Keywords
Mirror symmetry; Gross-Siebert; Fano surfaces; scattering diagrams;
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