Poincare-Euler characteristic and sum of two divisors

Question

Let $X$ be a regular projective algebraic surface over a field $k$. Let $D$ be a divisor on $X$ and suppose that $Y$ is a prime irreducible divisor on $X$ (i.e. $Y$ is an irreducible curve in $X$).

Is there any relationship between $\chi_k(\mathcal O_X(D))$ and $\chi_k(\mathcal O_X(D-Y))$?

Here with $\chi_k$ I denote the Euler-Poincare characteristic of a sheaf.