(1) Let $X$ be a projective (integral) curve over $\mathbb{C}$ and let $P$ be a singular point of $X$. Is there always a Cartier divisor whose support is exactly $P$ (set-theoretically)?
The following may be two related questions:
(2) Let $f: Y \to X$ be the normalisation of the curve $X$. Then we have a pushforward functor for Weil divisors from $Y$ to $X$. Is there some analogous contruction for Cartier divisors?
(3) Is every Weil divisor on a singular curve $\mathbb{Q}$-Cartier (i.e. a mutiple of the divisor is Cartier)? A 2-dimensional analog seems to have been discussed in the following MO question:
Is every Weil divisor on an arithmetic surface Q-Cartier
I still wonder what is known in the 1-dimensional situation.
Best Answer
Ok, the first thing you have to specify is what does it mean for to talk about a Cartier divisor (or Weil divisors). There are a number of options here. The one I like best for Cartier divisors would be invertible subsheaves of $K(X)$, the fraction field of $X$. This coincides with the one in Fulton's intersection theory if I recall correctly.
Now, what do you mean by Weil divisors? One option is formal sums of points. This is ok, but there are problems (mentioned below). Another option is $O_X$-module subsheaves of $K(X)$ in general (if you weren't dealing with curves maybe we should require these to be S2). In higher dimension sometimes people require them to be Cartier in codimension 1. But this just gives us Cartier divisors...
Why is the naive notion of Weil divisor problematic? The map from Cartier divisors to Weil divisors is not necessarily injective for non-normal varieties. Let me give an example. Consider the curve singularity $k[x,y]/(xy + x^3 + y^3)$ and consider the Cartier divisors corresponding to the principal ideals $(x)$ and $(y)$ respectively. These are different subsheaves of $K(X)$ (they are different ideals). However, they both give Weil divisors $3P$ (I assume you compute the map from Cartier to Weil divisors by computing lengths at points).
This implies that there isn't a well defined notion of $O_X(nP)$. For example, if there are two Cartier divisors giving the same $nP$, which sheaf do you choose to represent $O_X(nP)$?
The subsheaf notion is also problematic since not all subsheaves are invertible. So, I think the best behaved notion of Weil is the one that requires divisors to be Cartier in codimension 1. Unfortunately, for curves, this is not interesting.
Now, let me answer your questions.
Yes, depending on what you mean by Cartier divisor. Embed your curve in projective space. Take a hyperplane $H$ (not containing $X$) passing through $P$ and some other smooth points on $X$. $O_{\mathbb{P}^n}(H) \cdot O_X$ is certainly invertible. Let $D$ denote the effective Cartier divisor on $X$ such that $O_X(D)$ agrees with $O_{\mathbb{P}^n}(H) \cdot O_X$ away from $P$ and agrees with $O_X$ at $P$. Now simply consider $O_{\mathbb{P}^n}(H) \cdot O_X(-D)$. This agrees with $O_X$ away from $P$, and is not trivial at $P$, so I would say it is a Cartier divisor supported at $P$.
I don't think so. This doesn't work in higher dimensions either. I assume you want some properties? Like preserving linear equivalence?
Since $O_X(nP)$ is not well defined, then I don't think you can define the notion of $\mathbb{Q}$-Cartier. I guess you could say that there is some Cartier divisor that maps to $nP$ in the natural map from Cartier divisors to Weil divisors?