An algebraic variety is a geometric object defined by the common zero set of a system of polynomial equations in several variables. As is obvious, it is extremely difficult to give explicit solutions to such systems of equations. Therefore, in algebraic geometry, we study the geometric structure of varieties and any nice properties that they possess.

We are particularly interested in a special type of algebraic variety, the Calabi-Yau manifold, due to its importance to theoretical physics, specially string theory. In this talk, I will give a brief introduction to the basic concepts of algebraic geometry and give an overview of the special properties of 3 dimensional Calabi-Yau manifolds. We will discuss how simple algebraic tools, namely the Tom and Jerry degenerations can be used to construct new families of Calabi-Yau 3-folds in codimension 4 by unprojection from lower codimensions.