Orbifold Del Pezzo surfaces: Constructions, toric degenrations and links to mirror symmetry

This project can be divided into three parts. The first part of this project is to construct a large (possibly infinite) number of families of orbifold Del Pezzo surfaces such that their equations can be defined in terms of equations of the image of Segre embedding of P1 P1 P1 into P7 and another set of equations known as Rolling factors format. One of the important questions in classification problems in algebraic geometry is to realize the geometric objects ( algebraic varieties) by relatively simple sets of polynomial equations. Instead of studying orbifold Del Pezzo surfaces abstractly, we write down explicit equations which gives this project a special flavor and power in applications. In the 2nd part we aim to study the mirror symmetry conjecture for these constructed surfaces which basically mean to compute their toric degenerations explicitly and compute the mirror pairs of the surfaces constructed in part 1. In the last part of the project, we aim to calculate the cascade of unprojection of these orbifold Del Pezzo surfaces.