Explicit equations for mirror families to log Calabi-Yau surfaces
Bull. Korean Math. Soc.
Published online December 11, 2019
Lawrence Jack Barrott
National Center for Theoretical Sciences
Abstract : Mirror symmetry for del Pezzo surfaces was studied by Auroux, Katzarkov and Orlov 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. 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 by Gross, Hacking and Keel 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 explain some of the motivation for their construction. This leads to general computational methods to construct these mirrors. In the end we will provide enumerative data supporting the claim that these are in fact the desired mirror families.
Keywords : Gross-Siebert program, Mirror Symmetry, Algebraic Geometry
MSC numbers : 14J33
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