Prerequisite of Projective Geometry for Algebraic Geometry

I studied Euclidean Geometry in high-school, and I have not studied anything relates to geometry since I started studying in university. I am now intending to study Algebraic Geometry, however, I lacked in geometric intuition, especially in projective aspect.

Could you please suggest me some text book about Projective Geometry(or Geometry in general) with a view toward Algebraic Geometry, or have promising intuition to Algebraic Geometry?

I am sorry if my poor English lead to any misunderstanding.

Thank for reading.

Solution 1:

One of my favorite sets of notes on projective geometry is from a course by Enrique Arrorndo found here. Since you are probably posting for more that just the book, here's a small tour of things you should know while learning about projective geometry.

1. Don't forget your linear algebra! You can use this knowledge in a bunch of different areas. The basic example is just changing coordinates, so $$ \mathbb{C}[x,y,z] \cong \mathbb{C}[x,x+y,x+z] $$ describe the same ring, hence give the same projective spaces. Another place this pops up is with vector bundles and the like.

2. Become familiar with the notation of using schemes. I'm not saying you should learn scheme theory on your first introduction to algebraic geometry, but this will save you some headaches in the future. What I mean by this is best explained in an example. If you want to consider a complex projective variety, you're considering the vanishing locus of a finite set of homogeneous polynomials $f_1,\ldots,f_k$. One interesting scheme that pops up is the cubic surface $$ \textbf{Proj}\left( \frac{\mathbb{C}[x,y,z,w]}{(x^3 + y^3 + z^3 + w^3)} \right) $$ this will be important when you learn about intersection theory and enumerative geometry because a generic cubic polynomial will have 27 lines on it.

3. Become familiar with the usage of the line bundle $\mathcal{O}(a)$ over projective space $$ \mathbb{P}^n_\mathbb{C} = \textbf{Proj}(\mathbb{C}[x_0,\ldots,x_n]) $$ when taking global sections $\Gamma(\mathbb{P}^n_\mathbb{C},\mathcal{O}(a))$, this will give you all homogeneous polynomials in $\mathbb{C}[x_0,\ldots,x_n]$ of degree $a$. For example, $$ \Gamma(\mathbb{P}^1_\mathbb{C},\mathcal{O}(2)) = \text{Span}_\mathbb{C}(x^2,xy,y^2) $$

4. When you do learn schemes, make sure to spend a lot of time thinking about fiber products, and relative morphisms of schemes. Understanding the following diagram will give you a lot $$ \begin{matrix} \textbf{Proj}\left( \frac{\mathbb{C}[x,y,z]}{x^n + y^n - z^n}\right) & \to & \textbf{Proj}\left( \frac{\mathbb{C}[s,t][x,y,z]}{x^n + y^n - z^n + sx^{n-2}y^2 + tx^n}\right) \\ \downarrow & & \downarrow \\ \textbf{Spec}(\mathbb{C}) & \xrightarrow{(0,0)} & \textbf{Spec}(\mathbb{C}[s,t]) \end{matrix} $$ and will help you on your journey for appreciating the functor of points.

5. Don't forget the golden rule: Compute Everything You Possibly Can and then go on math stackexchange and get help/answer questions.