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Let $X$ be a noetherian,integral,separated scheme which is nonsingular in codimension $1$. Let $Z_1, \ldots, Z_k$ be a fixed set of prime divisors. Now given a prime divisor $Y$, does there exist prime divisors $W_1, \ldots, W_m$, none of them equaling any of the $Z_j$s, such that $Y +\sum n_i W_i = 0$ in $\text{Cl}(X)$?

Further, when can we do it with $n_i \ge 0$? For example in $\mathbb P^n_k$ one can not.

I can easily do the first part for curves over algebraically closed fields by using a Chinese remainder type argument. But I don't see it in the general case or know if its true. Basically I am interested to find out how much I can modify the support of a Weil divisor. If true, my statement would inductively imply that we can always remove any chosen set of prime divisors from the support and still maintain the same linear equivalence.

Can someone please help me out?

Let $X$ be a noetherian,integral,separated scheme which is nonsingular in codimension $1$. Let $Z_1, \ldots, Z_k$ be a fixed set of prime divisors. Now given a prime divisor $Y$, does there exist prime divisors $W_1, \ldots, W_m$, none of them equaling any of the $Z_j$s, such that $Y +\sum n_i W_i = 0$ in $\text{Cl}(X)$?

Further, when can we do it with $n_i \ge 0$? For example in $\mathbb P^n_k$ one can not.

I can easily do the first part for curves over algebraically closed fields by using a Chinese remainder type argument. But I don't see it in the general case or know if its true. Basically I am interested to find out how much I can modify the support of a Weil divisor. If true, my statement would inductively imply that we can always remove any chosen set of prime divisors from the support and still maintain the same linear equivalence.

Let $X$ be a noetherian,integral,separated scheme which is nonsingular in codimension $1$. Let $Z_1, \ldots, Z_k$ be a fixed set of prime divisors. Now given a prime divisor $Y$, does there exist prime divisors $W_1, \ldots, W_m$, none of them equaling any of the $Z_j$s, such that $Y +\sum n_i W_i = 0$ in $\text{Cl}(X)$?

Further, when can we do it with $n_i \ge 0$? For example in $\mathbb P^n_k$ one can not.

I can easily do the first part for curves over algebraically closed fields by using a Chinese remainder type argument. But I don't see it in the general case or know if its true. Basically I am interested to find out how much I can modify the support of a Weil divisor. If true, my statement would inductively imply that we can always remove any chosen set of prime divisors from the support.

Can someone please help me out?

Let $X$ be a noetherian,integral,separated scheme which is nonsingular in codimension $1$. Let $Z_1, \ldots, Z_k$ be a fixed set of prime divisors. Now given a prime divisor $Y$, does there exist prime divisors $W_1, \ldots, W_m$, none of them equaling any of the $Z_j$s, such that $Y +\sum n_i W_i = 0$ in $\text{Cl}(X)$?

Further, when can we do it with $n_i \ge 0$? For example in $\mathbb P^n_k$ one can not.

I can easily do the first part for curves over algebraically closed fields by using a Chinese remainder type argument. But I don't see it in the general case or know if its true. Basically I am interested to find out how much I can modify the support of a Weil divisor. If true, my statement would inductively imply that we can always remove any chosen set of prime divisors from the support and still maintain the same linear equivalence.

Can someone please help me out?

Let $X$ be a noetherian,integral,separated scheme which is nonsingular in codimension $1$. Let $Z_1, \ldots, Z_k$ be a fixed set of prime divisors. Now given a prime divisor $Y$, does there exist prime divisors $W_1, \ldots, W_m$, none of them equaling any of the $Z_j$s, such that $Y +\sum n_i W_i = 0$ in $\text{Cl}(X)$?

Further, when can we do it with $n_i \ge 0$?

I can easily do the first part for curves over algebraically closed fields by using a Chinese remainder type argument. But I don't see it in the general case or know if its true. Basically I am interested to find out how much I can modify the support of a Weil divisor. If true, my statement would inductively imply that we can always remove any chosen set of prime divisors from the support.

Can someone please help me out?

Let $X$ be a noetherian,integral,separated scheme which is nonsingular in codimension $1$. Let $Z_1, \ldots, Z_k$ be a fixed set of prime divisors. Now given a prime divisor $Y$, does there exist prime divisors $W_1, \ldots, W_m$, none of them equaling any of the $Z_j$s, such that $Y +\sum n_i W_i = 0$ in $\text{Cl}(X)$?

Further, when can we do it with $n_i \ge 0$? For example in $\mathbb P^n_k$ one can not.

I can easily do the first part for curves over algebraically closed fields by using a Chinese remainder type argument. But I don't see it in the general case or know if its true. Basically I am interested to find out how much I can modify the support of a Weil divisor. If true, my statement would inductively imply that we can always remove any chosen set of prime divisors from the support.

Can someone please help me out?