Home Algebraic Geometry - Hartshorne 1.1

Algebraic Geometry - Hartshorne 1.1

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This is a series on studying Algebraic Geometry based on the textbook “Algebraic Geometry” by Hartshorne. We start off with the so called “classical algebraic geometry” where the unifying concept of schemes has yet to be introduced. We will define the geometric objects we shall be studying known as algebraic varieties and associate with them algebraic objects where by geometric properties can be extracted via algebraic means.

A Little Motivation

It wasn’t until a little while after learning a little geometry and algebra past undergraduate level that it finally hits me: “$x^2 + y^2 = 1$” is the circle well… with lots mathematical preamble suppressed. But that’s what the definitions in this section is for., not just a mere representation or tool to manipulate the set of points on the circle. Not only that, the way – the algebraic way – it is presented lends itself to various obvious generalisations that are “correct” read: plays well with intuitions and other mathematical theories. For instance, definitions higher dimensional analogs is immediate with “$x_1^2 + x_2^2 + \dots + x_n^2 = 1$” and, with perhaps more historical significant, the study of conic section can be unified under the study of general degree 2 polynomials

\[ax^2 + bxy + cy^2 + dx + ey + f = 0.\]

Similarly, most students of mathematics need no convincing when it comes to the power of talking about points, lines, planes and hyperplanes and so on in the language of linear algebra.

We shall now define the rather general setting of algebraic varieties to study such association between algebra and geometry.

Definition of Affine Varieties

Let’s fixed a field $k$. Unless otherwise specified, we shall assume that $k$ is algebraically closedwe shall see why when discuss the celebrated Hilbert Nullstellenstaz. But, with apology, we will be drawing pictures as though $k = \R$..

Let $n \in \N$. We shall call the set of all $n$-tuples of elements in the field $k$ the $n$-dimensional affine space or the affine $n$-space and denote it as $$ \A_k^n := k^n = \set{(a_1, a_2, \dots, a_n) \wh a_i \in k \text{ for all } i} $$ We call any element $P = (a_1, \dots, a_n) \in \A_k^n$ of the space a point with the $a_i$'s being its coordinates .

This should remind us of the familiar case of cartesian coordinates $(x, y) \in \R^2$ used above to defined the circle $x^2 + y^2 = 1$. We are simply generalising this with arbitrary field $k$meaning we would like to retain the ability to add, multiple, divide and have access to $0$ and $1$. and to arbitrary dimensions. This is the algebraic stage where our geometric objects shall abide.

Let $k[x_1, \dots, x_n]$ denote the ring of polynomials in $n$ variables over $k$. We naturally interpret each polynomial $f \in k[x_1, \dots, x_n]$ as a function $A^n_k \to k$ $$ P = (a_1, \dots, a_n) \mapsto f(P) = f(a_1, \dots, a_n) $$ The preimage of zero of such a polynomial $f$ is called the zero set or the vanishing set hence the notation $\V$ for "vanishing" of the function $$ \V(f) = \set{(a_1, \dots, a_n) \in \A_k^n \st f(a_1, \dots, a_n) = 0}. $$ We can also talk about the _common vanishing set_ of a set of functions $F \subset k[x_1, \dots, x_n]$ $$ \V(F) = \bigcap_{f \in F}\V(f) = \set{P \in \A^n_k \st f(P) = 0 \text{ for all } f \in F} $$ The set of points in $\A^n_k$ that corresponds to precisely such vanishing sets are called algebraic sets $$ \begin{align*} X \subset \A^n_k &\text{ is an algebraic set } \\ &\iff \\ X = \V(F) &\text{ for some } F \subset k[x_1, \dots, x_n]. \end{align*} $$

Now, given such a set of polynomials $F \subset k[x_1, \dots, x_n]$, we can consider $\abrac{F}$, the ideal generated by $F$

\[I = \abrac{F} = \set{\sum_{i = 1}^k a_i(x)f_i(x) \st k \in \N, f_i \in F, a_i \in k[x_1, \dots, x_n]}\]

Observe that if a point $P \in A^n_k$ is a zero for all polynomials in $F$, then it is a zero for all polynomials in $I$ too. Conversely, since $F \subset I$, a common zero of polynomials in $I$ is a common zeros for those in $F$. We conclude,

\[\V(F) = \V(\abrac{F}) = \V(I).\]

We shall henceforth always consider the ideal $\abrac{F}$ instead of just $F$ itself. Moreover, by Hilbert Basis Theorem, $k[x_1, \dots, x_n]$ is a noetherian ring a ring if noetherian if all its ideals are finitely generated., even if the initial family of polynomial $F$ is infinite, there exist a finite set $\set{g_1, \dots, g_r}$ of generators of $\abrac{F}$ so that we can express the algebraic set as

\[\V(F) = \V(\abrac{F}) = \V(g_1, \dots, g_r).\]
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