Questions about algebraic curve definition The algebraic curve definition states as following
$S_{2}^m$ denotes homogeneous polynomial of degree $m$ in $x$ and $y$
$f_{m}(x, y) = \sum_{j, k \ge 0, j+k=m} C_{j,k}x^{j}y^{k}$
The elements of $f \in S_{2}^m$ have the property of $\alpha^{m}f(x,y) =  f(\alpha x, \alpha y )$ 
I'm just wondering why the $f_{m}(x, y)$ does't include some terms: 
e.g. $ax \text{ or } by$
In other word, $S_{2}^{m}$ does't include the term of the sum of the power of x and y is not equal to m
 A: $\newcommand{\Proj}{\mathbf{P}}$Let $k$ be a positive integer. A monomial $f$ of degree $k$ in two variables satisfies the homogeneity conditon $f(ax, ay) = a^{k} f(x, y)$ (for all non-zero $a$) by elementary algebra. It's not difficult to see that a sum of monomials of different degrees is not homogeneous at all. Consequently if a polynomial $f$ in two variables is homogeneous of degree $m$, then $f$ is a sum of monomials of degree (precisely) $m$.
Added in edit: The preceding paragraph is mathematically correct, but at least partially misses the point of the question (i.e., doesn't fully address the issues mentioned in the comments).
Points in the projective plane $\Proj^{2}$ (i.e., lines through the origin in three-dimensional space) are described by triples (not pairs) of homogeneous coordinates $[X : Y : Z]$. (If $a \neq 0$, then $[X : Y : Z] = [aX : aY : aZ]$ as points in the projective plane. Consequently, the homogeneous coordinates $X$, $Y$, and $Z$ aren't "coordinates" (scalar-valued functions) in $\Proj^{2}$ in the sense of elementary geometry; only their ratios $x = X/Z$, $y = Y/Z$, etc. are well-defined numerical functions.)
If $F$ is a homogeneous polynomial of degree $m > 0$ (in three variables), in the sense that
$$
F(aX, aY, aZ) = a^{m} F(X, Y, Z)\quad\text{for all $a$,}
$$
then $F$ is not a well-defined scalar-valued function on $\Proj^{2}$. Despite this, the condition $F(X, Y, Z) = 0$ is well-defined on $\Proj^{2}$. (Why?) The locus
$$
\bigl\{[X : Y : Z] \in \Proj^{2} : F(X, Y, Z) = 0\bigr\}
$$
is the plane curve cut out by $F$. (Exercise: If $F$ is a non-homogeneous polynomial in three variables, the condition $F(X, Y, Z) = 0$ is not well-defined on $\Proj^{2}$.)
To see the relationship with the definition from secondary school, write the general homogeneous polynomial of degree $m$ as
$$
F(X, Y, Z) = \sum_{\substack{j, k, \ell \geq 0 \\ j + k + \ell = m}} C_{jk\ell}\; X^{j} Y^{k} Z^{\ell}.
$$
In $\bigl\{[X : Y : Z] : Z \neq 0\bigr\}$ (a well-defined subset of $\Proj^{2}$), introduce the "affine coordinates" $x = X/Z$ and $y = Y/Z$. Dividing $F(X, Y, Z)$ by $Z^{m}$ gives the non-homogeneous polynomial
$$
f(x, y) = \sum_{\substack{j, k, \ell \geq 0 \\ j + k + \ell = m}} C_{jk\ell}\; x^{j} y^{k}.
$$
Formally, set $Z = 1$, replace $X$ with $x$, and replace $Y$ with $y$. Because $x$ and $y$ are well-defined numerical coordinates on the affine chart $\{Z \neq 0\}$, the polynomial $f$ has a well-defined zero locus, the "finite part" of the projective curve $\{F = 0\}$ with respect to the affine chart $\{Z \neq 0\}$. (In this chart, $\{Z = 0\}$ is the line at infinity.)
This process can be inverted. For example, the general inhomogeneous quadratic
$$
f(x, y) = ax^{2} + bxy + cy^{2} + dx + ey + f
$$
is "homogenized" by replacing $x$ with $X$ and $y$ with $Y$, then multiplying each monomial of degree $2 - \ell$ by $Z^{\ell}$:
$$
F(X, Y, Z) = aX^{2} + bXY + cY^{2} + dXZ + eYZ + fZ^{2}.
$$
The diagram (a frame from an animation of a rotating cone) shows the circle $x^{2} + y^{2} - 1 = 0$ in the plane $\{Z = 1\}$ (the red grid) represented as the projective conic cut out by the homogeneous equation
$$
(X\cos\theta - Z\sin\theta)^{2} + Y^{2} - (Z\cos\theta + X\sin\theta)^{2} = 0,
$$
i.e., by $X^{2} + Y^{2} - Z^{2} = 0$ when $\theta = 0$.

