Users Guide to Hartshorne Algebraic Geometry

Introduction
Hartshorne's introduction to algebraic geometry is a notoriously difficult book for beginners because of the technical mound one has to overcome in order to reach scheme theory. In addition, the lack of explanations for many of the pieces of machinery introduced creates an additional unneeded obstruction to learning. It is our intention to help ease these problems and help beginners overcome the hurdle of "big machinery". Our central goal is allow anyone to be able to appreciate the beauty and applications of scheme theory.

Overview
We will give a basic overview of the chapters and sections, giving intuition about what the reader should expect.

/Chapter 0/
This is a reference chapter that I am including to give the reader a sense of some of the commutative algebra and differential geometry required for scheme theory. This will include basic definitions, examples, and theorems, but will omit all proofs; Only computational technique will be discussed.

/Chapter 1/
Hartshorne starts his book with an overview of basic classical algebraic geometry. In the beginning mathematicians studied solutions of polynomials as subsets of $$\mathbb{Q}^n,\mathbb{R}^n,\mathbb{C}^n$$, or projective spaces constructed from these sets. This viewpoint is made rigorous through the classical theory of varieties, which is the one taken by many algebraists and geometers before Grothendieck. It is still useful to consider the state of algebraic geometry before scheme theory because the applications of algebra are much more transparent.

For technical simplicity he fixes an algebraically closed field $$k$$, so you should be thinking about the fields $$\mathbb{C},\overline{\mathbb{Q}},\overline{\mathbb{Q}}_p$$. For example, the variety
 * $$\mathbb{V}(x^2 +y^2 + 1) = \{(a,b) \in \mathbb{R}^2: a^2 + b^2 = -1 \}$$

is the empty set since both $$a^2,b^2\geq 0$$ for any $$(a,b) \in \mathbb{R}^2$$ but $$ -1 < 0 $$, hence
 * $$a^2 + b^2 > -1 \text{ for all } (a,b) \in \mathbb{R}^2$$

Chapter 2
This chapter is the core of the entire book. It introduces basic scheme theory and the various structures associated with a scheme. (Technical note: Hartshorne takes the locally ringed-space approach to scheme theory and does not mention the functor of points). The first and least complicated schemes introduced in this chapter are affine schemes which are denoted $$\text{Spec}(R)$$ for some commutative ring $$R$$. For example, the complex affine space $$\mathbb{A}^n_\mathbb{C} = \text{Spec}(\mathbb{C}[x_1,\ldots,x_n])$$. You can guess what the affine spaces for $$\mathbb{Q},\overline{\mathbb{Q}}, \overline{\mathbb{Q}}_p$$ should be defined as.

The core technical observations required for scheme theory are the following: \cong R^n$$ \cong \{p=(p_1,\ldots,p_n) \in R^n: f_1(p)=\cdots=f_k(p)=0\}$$
 * $$\text{Hom}_{\textbf{Aff}}(\text{Spec}(R),\text{Spec}(\mathbb{Z}[x_1,\ldots,x_n]))
 * $$\text{Hom}_{\textbf{Aff}}\left(\text{Spec}(R),\text{Spec}\left(\frac{\mathbb{Z}[x_1,\ldots,x_n]}{(f_1,\ldots,f_k)}\right) \right)
 * $$\text{Spec}(\mathbb{C}[x]/(x^2)) \not\cong \text{Spec}(\mathbb{C})$$

These first two observations allow one to consider universal spaces whose hom-sets give the vanishing locus of a polynomial in a given ring. Now, we are not forced to consider polynomials only as subsets for a fixed $$\mathbb{Q},\mathbb{R},\mathbb{C},$$ etc. The second observation is the other major benefit of scheme theory: one has a way to "remember" intersections of polynomials. Both schemes have the same underlying topological space as a point, but have different sheaves of commutative rings over them (you don't have to know the word "sheaf" yet, this is more for you to unpack once you read the book). For example, consider a family of intersections
 * $$X_t = \{(a,b)\in \mathbb{R}^2: b-a^2 = 0, b=t\}$$

if $$t=0$$, this is topologically just a point while if $$t > 0$$ then this is topologically two points. If you look at this algebraically, the two points are squashed down into a fat point which can be seen through the sheaf of rings on the scheme.

Once one understands the basics of what as scheme is, the next important feature of scheme theory are the morphisms in general. Typically morphisms are used for defining families of schemes. For example, consider the Weierstrass family of plane elliptic curves

\begin{matrix} \text{Spec}\left( \frac{\mathbb{C}[t,x,y]}{(y^2 - x(x-1)(x-t))} \right)\\ \downarrow\\ \text{Spec}(\mathbb{C}[t]) \end{matrix} $$

if you look at the fibers over the target of this morphism, you get the scheme $$\text{Spec}(\mathbb{C}[x,y]/(y^2 - x(x-1)(x-t)))$$ for some $$\alpha \in \mathbb{C}$$. This scheme can be endowed with the structure of an elliptic curve if $$t\neq 0,1$$. The notion of families form the basis of moduli theory where we should have a morphism $$\pi: \mathfrak{X} \to B$$ where $$B$$ is the moduli space or base space and $$\mathfrak{X}$$ the universal family. For example, consider the moduli space of conics in $$\mathbb{P}^2$$ which is given by the morphism



\begin{matrix} \text{Proj}_{\mathbb{P}_\mathbb{Z}^5}\left(\frac{\mathbb{Z}[a_1,\ldots,a_6][x,y,z]}{(a_1x^2 + a_2xy + a_3xy + a_4y^2 + a_5yz + a_6z^2)} \right)\\ \downarrow \\ \text{Proj}_\mathbb{Z}(\mathbb{Z}[a_1,\ldots,a_6]) = \mathbb{P}_\mathbb{Z}^5 \end{matrix} $$

Please note that this gives a toy example of a Hilbert scheme, something which is neither constructed nor discussed in this book.

Chapter 3
This chapter introduces the other half of basic scheme theory: sheaf cohomology of schemes and the cohomological structures used in scheme theory. Of them, one of the most import notions are flat morphisms. This is defined used derived tensor products, but has a simple geometric intuition: flat morphisms of schemes. The basic idea is that the cohomological properties of this morphism guarantee a certain amount of continuity of the family as the fibers change. For example, you will never see dimensions of the fibers change, as you would see with a blowup. For example, consider the blowup

\begin{matrix} \text{Proj}\left(\frac{\mathbb{C}[s,t][x,y]}{(sf(x,y) + tg(x,y))} \right) \\ \downarrow \\ \text{Spec}(\mathbb{C}[x,y]) \end{matrix} $$ The fibers over the vanishing locus $$\text{Spec}(\mathbb{C}[x,y]/(f,g))$$ are isomorphic to $$\mathbb{P}^1_\mathbb{C}$$ while over the rest of $$\mathbb{A}^2$$ they are a single point.

Chapter 4,5
These chapters introduces the classical subjects of algebraic curves and surfaces. These are fruitful subjects today and are the basis for much of modern research. In particular, the construction of the moduli space of curves depends on much of the material introduced in the fourth chapter.