TITLE: Cartier divisors on singular curves
QUESTION [7 upvotes]: (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.

REPLY [7 votes]: Here is a part of an email I sent you because I couldn't log in MO. 
The answer to (3) is yes: there is a Cartier divisor $D$ on $X$ such that the associated Weil divisor ($0$-cycle for a  curve) is $[P]$. This idea is just to consider  the normalization $f : Y\to X$. Let $Q\in Y$ be a point lying over $P$. Then $[Q]\sim Z$ (rational equivalence) for some $0$-cycle $Z$ on $Y$ with support disjoint from $f^{-1}(P)$ (use e.g. Riemann-Roch). By pushforward, $[P]=f_*[Q] \sim f_*Z$. By definition of rational equivalence, this means that $[P]=f_*Z+[\mathrm{div}(h)]$ for some rational function $h$ on $X$. As the support of $f_*Z$ is disjoint from $P$, this means that in an open neighborhood $V$ of $P$, $[P]$ coincides with $[\mathrm{div}(h)]$. Define a Cartier divisor $D$ by ${D|}\_{V}=\mathrm{div}(h)$ and $D|_{X\setminus { P}}=1$. Then $[P]$ equals to  (the $0$-cycle associated to) $D$.  The idea of taking normalization can be found, say, in a paper of Colliot-Thélène (Inv. Math. 2005, p. 599).
This works over any algebraically closed field. If the base field is not necessarily algebraically closed, let $Q_1, \cdots, Q_r$ be the points of $f^{-1}(P)$, let $n$ be the gcd of the $[k(Q_i): k(P)]$. Then the above argument shows that $n[P]$ is associated to a Cartier divisor on $X$. And $n$ is the smallest possible.
In higher dimension, as you already known, this conclusion is false. However the following weaker result holds. Let $X$ be a noetherian scheme, let $E$ be a prime cycle. Then for some positive integr $n$, $nE$ is the cycle associated to a Cartier divisor (on an integral closed subscheme of $X$ containing $E$) in an open neighborhood of the generic point $P$ of $E$. The integer $n$ can be controlled by some invariants related to the singularity at $P$ (can take $n=1$ is $P$ is a regular point of $X$, but this is not necessary as we saw in the case of curves). See Theorem 4.5, §5, and Theorem 6.5 in this paper.