In David Mumford's book Algebraic Geometry I, Complex Projective Varieties the proof of Bezout's Theorem (5.16 page 80) contains terminology I not understand. Mumford speaks at pages 83 & 84 about 'differentials of equation system (that defines a variety)'. I know what differentials of regular functions are, but I don't know what kind of thing is a 'differential of an equation system ' in classical complex algebraic geometry. We assume:
Let $X^r,Y^s \subset \mathbb{P}^n$ complex algebraic varieties of dimensions $r, s$ with $r+s=n$. Let $L \subset \mathbb{P}^n$ a linear subspace with $L \cap X = \emptyset$. After a reasonable choice of cordinates we can assume $L= V(X_0,...,X_r)$ where $X_i \in \mathbb{C}[X_0,...,X_n]$.
Then the rational map
$$ \sigma:(\mathbb{P}^n - L) \times \mathbb{C} \to \mathbb{P}^n - L$$
def by
$$ \sigma(x_0,...,x_n, t) = (x_0,...,x_r,t x_{r+1},..., t x_n)$$
restricts to regular map $\text{res } \sigma: X \times \mathbb{C} \to \mathbb{P}^n$. Define the algebraic set
$$ \mathcal{X} = \{(x,y,t) \ \vert \ \sigma_t x=y \} \subset X \times Y \times \mathbb{C} $$
and consider the restriction of the $3$rd projection $\pi: \mathcal{X} \to \mathbb{C} $. Locally, $\mathcal{X}$ is defined by $n$ equations:
$$ \tag{EQ} X_i (\sigma_t x) = X_i (y), \ 1 \ge i \ge n $$
$$ X_1,...,X_n \text{ affine coord in } \mathbb{P}^n $$
Fix a $(x,y, t) \in \mathcal{X}$.
My question: Mumford uses terminology in this context that I don't understand. On pages 83 & 84 he talked about 'differentials of the equations (EQ) defining $\mathcal{X}$ restricted to tangent space $T_{(x,y,t),X \times Y, \{t\}} $, then ...'
I'm not familar with this terminology. What is a 'differential of an equation system' like (EQ) in this example?
Note, that we can regard $X_i$ as functions $\mathbb{A}^n \to \mathbb{C}$ from affine parts of $\mathbb{P}^n$, therefore compositions like $X_i \circ \sigma_t$ can be also regarded as locally defined functions $X \to \mathbb{C}$ and we can associate to differentaible functions a differential in usual way.
But here Mumford associates a differential to equations, instead to functions and I not understand it's meaning and construction. Is there a standard method in classical algebraic geometry to calculate a 'differential' of a equation system? In simple words: what is it?
When talking about algebraic varieties it is common to speak interchangeably between "equations" that define a variety and the corresponding regular functions which vanish on the variety. So, when referring to the differential of an equation $f(x_1,\dots,x_n)=g(x_1,\dots,x_n)$, that would usually just mean the differential of $f(x_1,\dots,x_n)-g(x_1,\dots,x_n)$, the function whose vanishing is equivalent to that equation.