In my lecture notes I have the following:
$f, g \in \mathbb{C}[x,y]$
$V(f)=V(g)$ if $f=p_1^{a_1} \cdot p_2^{a_2} \cdot \dots \cdot p_s^{a_s} , g=p_1^{b_1} \cdot p_2^{b_2} \cdot \dots \cdot p_s^{b_s}$, where $p_i(x,y)$ are the irreducible polynomials of $f, g$.
Definition: The curves that are defined by $p_i(x, y)$ are called irreducible components of the curve $f$ of multiplicity $a_i$.
It stands that $$V(f)=V(p_1) \cup V(p_2) \cup \dots \cup V(p_s)$$
What does it mean "the curves that are defined by $p_i(x, y)$" ?
Why does it stand that $V(f)=V(p_1) \cup V(p_2) \cup \dots \cup V(p_s)$ ? It is $V(f)=V(p_1^{a_1} \cdot p_2^{a_2} \cdot \dots \cdot p_s^{a_s}) \overset{ V(I \cdot J)=V(I) \cup V(J)}{ = }V(p_1^{a_1}) \cup V(p_2^{a_2}) \cup \dots \cup V(p_s^{a_s})$ or not?