Gathmann's Plane Algebraic Curves
I do not understand the last statement of this corollary i.e. The irreducible components of F(but not their multiplicities) can be recovered from V(F). Does this mean that if I am given V(F) for some polynomial F(which is not known) then I can find its irreducible components/irreducible factorisation but without any information about multiplicities? If yes, then how does it follow from the corollary?
I have attached the relevant definitions and link to Gathmann's notes which I am referring. Thanks.
Question: "I do not understand the last statement of this corollary i.e. The irreducible components of F(but not their multiplicities) can be recovered from V(F). Does this mean that if I am given V(F) for some polynomial F(which is not known) then I can find its irreducible components/irreducible factorisation but without any information about multiplicities? If yes, then how does it follow from the corollary?"
Answer: If $f(x,y):=f_1(x,y)\cdots f_l(x,y)$ with $f_i \in k[x,y]$ an irreducible polynomial for all $i$ and $f_i \neq f_j$ for $i\neq j$, you may for any set of integers $a:=(a_1,..,a_l)$ with $a_i \geq 1$ define $F_a(x,y):=f_1^{a_1}\cdots f_l^{a_l}$ to get a new polynomial. The two polynomials $f,F_a$ have the same zero set:
$$V(f(x,y))=V(F_a(x,y))$$
but they are clearly different polynomials and different as "plane algebraic curves". If $a_i>1$ for some $i$ there is no element $\lambda\in k^*$ with $kf=F_a$. Hence you cannot recover the multiplicities $a_i$ from the zero set $V(F_a)$, since $V(F_a)=V(f)$ for any choice of $a$.
"..in fact this becomes a problem ,say,in C,the set of complex numbers"
Example: If $p:=(u,v)$ is a root/zero of $F_a(x,y)$, it follows
$$F_a(u,v)=\prod f_i^{a_i}(u,v)=0$$
is an equality in the field $k$, hence there is an $i$ with $f_i(u,v)=0$. It follows $f(u,v)=0$. Similarly if $f(u,v)=0$ it follows $F(u,v)=0$. This all follows from the fact that $k$ is a field, hence an integral domain: If a product of elements in $k$ is zero, one of the elements must be zero. Or: The zero ideal is a prime ideal.