(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.

(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.

(1) Let $X$ be a projective 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.

(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.