On the mirrors of low-degree del Pezzo surfaces

We compare different constructions of mirrors of del Pezzo surfaces, focusing on degree $d \leq 3$. In particular, we extract Lefschetz fibrations, with associated exceptional collections, from the mirrors obtained via the Hori-Vafa and Fanosearch program constructions, which we relate to one another. We show with geometric methods that the Lefschetz fibrations define categorical mirrors. With a more explicit approach, we give a sequence of (numerical) mutations relating the exceptional collections considered by Auroux, Katzarkov, and Orlov with those arising in this paper. This uses the theory of surface-like pseudolattices, and extends some of the string junction results of Grassi, Halverson and Shaneson. Our argument lifts directly to an equivalence of certain Fukaya-Seidel categories arising from our fibrations and those of Auroux, Katzarkov, and Orlov.