Explicit equations for mirror families to log Calabi-Yau surfaces

Abstract

Mirror symmetry for del Pezzo surfaces was studied by Auroux, Katzarkov and Orlov who suggested that the mirror should take the form of a Landau-Ginzburg model with a particular type of elliptic fibration. 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 of $S$ a Fano surface they proved 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. We also provide an implementation of the Kontsevich Soibelman lemma in Sage.