I am reading by myself this book http://tinyurl.com/37z4bbt. But to be honest, I have several problems to fully understand some part of the text. Maybe because I have not yet solid knowledge or I still need to learn more about the subject, hence asked a little help
To be exact, is this part of the construction concerning to Algebraic Blow-Up that I do not understand like trying...
I quote part of the building in which I do not understand how to deal with some concepts
Consider the canonical map from $\mathbb{C}^2$ to the projective line $\mathbb{CP}^1$ that associates with each point $(x,y)$ different from the origin, the line {$(tx, ty) : t \in \mathbb{C}$} passing through this point. The graph of this map is a complex 2-dimensional surface in the complex 3-dimensional manifold (the Cartesian product) $\mathbb{C}^2$ × $\mathbb{CP}^1$, which is not closed. To obtain the closure, one has to add the exceptional curve $E={0} \times \mathbb{CP^{1}} \subset \mathbb{C^{2}} \times \mathbb{CP^{1}}$.
then, I do not understand is how to handle this graph. If, as a topological space or as complex manifold. Well, in the sense of complex manifold, say it is not closed, it means that the manifold is not compact without boundary.
I am really a little confused