Explicit equations for mirror families to log Calabi-Yau surfaces
Bull. Korean Math. Soc. 2020 Vol. 57, No. 1, 139-165
Published online January 31, 2020
Lawrence Jack Barrott
No. 1 Sec. 4 Roosevelt Rd., National Taiwan University
Abstract : Mirror symmetry for del Pezzo surfaces was studied in~\cite{MirrorSymmetryForDelPezzoSurfacesVanishingCyclesAndCoherentSheaves} 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~\cite{MirrorSymmetryforLogCY1}. 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~\cite{MirrorSymmetryforLogCY1} 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 $dP_2$ as a double cover of $\PP^2$.
Keywords : Mirror symmetry, Gross-Siebert, Fano surfaces, scattering diagrams
MSC numbers : 14J33, 14N10, 14T05
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