Linear equivalence of Weil divisors ($D\sim D+\mathrm{div}(q)$)

Two Weil divisors are linearly equivalent when their difference is a principal Weil divisor. This is the equivalence relation defining the divisor class group, and applies to normal varieties of any dimension, not just curves.

If a Weil divisor $D$ on a normal variety is not Cartier at $x$, every linearly equivalent divisor has $x$ in its support of a Weil divisor. Otherwise that representative would vanish near $x$, expressing $D$ as a principal Weil divisor there. On $V(ac-b^2)$ the prime divisor $V(a,b)$ is not Cartier at the origin: its ideal has two independent generators modulo the maximal ideal times that ideal. This gives an unavoidable support point even for representatives with negative coefficients.