I am trying to understand the paper On unique factorization domains by Pierre Samuel from 1960. Under the assumption that $C$ is a normal projective curve of order $2$ (hence genus $0$) in the projective plane over a field $k$, he states the following:
I have two questions:
(Q1) What does the multiplication sign between $C$ and $D$ mean, i.e., when is a divisor a "complete intersection" in this sense?
(Q2) How does the equivalence of $X$ being a "complete intersection" and its degree being even follow.
Unfortunately, he gives no reference for such an equivalence.
Thank you in advance for your help!
The order of a variety is classical terminology for what we now call its degree, so $C$ is a smooth plane conic (you probably understand this already, but I needed to look it up to be sure). The basic thing to understand is that if $D$ is a curve of degree $d$, then the intersection product $C\cdot D$ has cardinality $2d$ (this is Bezout's theorem, and as long as we understand the intersection product to be counted with multiplicities we don't need any assumptions of smoothness, irreducibility, or reducedness).
A complete intersection is a more general concept, but for divisors on a plane curve $C$ it simply means that $X$ is obtained by intersecting $C$ with some other curve $D$. To see that this is interesting, note that by the previous paragraph every complete intersection divisor on the conic $C$ is even, so a single reduced point $p \in C$ gives an example of a divisor $X = p$ that is not a complete intersection (more generally, take any divisor of any odd degree).
Now, it's clear that being a complete intersection implies even degree. To show that even degree implies complete intersection, note that we can simply group the multiset of points of $X$ (where a point of multiplicity $m$ occurs $m$ times in the set) into pairs, connect each pair with a line, and take $D$ to be the union of these lines; the business about multisets ensures that $D$ "cuts out" each point on $C$ with the correct multiplicity.
I hope this clears things up, but in case you are still wondering about the dot $\cdot$ formally, it is the unique interesting multiplication map $A^1 \times A^1 \to A^2$ in the Chow ring of $\mathbb P^2$; it can also be considered a map on some suitable cohomology $H^2 \times H^2 \to H^4$.