Assume that $X$ is a scheme over an algebraic closed field $k$. $Y$ is a irreducible curve on $X$ with generic point $y$ and $Y$ is proper.
$Y_{red}$ is its reduced scheme. Then for any invertible sheaf $\mathcal{L}$ on $X$, can we prove that $(\mathcal{L}\cdot O_Y)=length(O_{Y,y})(\mathcal{L}\cdot O_{Y_{red}})$.
$(\mathcal{L}\cdot \mathcal{F})$ is the intersection number between $\mathcal{L}$ and $\mathcal{F}$ where the support of $\mathcal{F}$ is 1-dimensional. And it’s defined by the leading coefficient of $n$ in the euler characteristic of $\mathcal{L}^n\otimes\mathcal{F}$.
After clearing the notation, the answer is positive. This is a basic result about how to compute the intersection numbers on irreducible and reduced components, see Proposition 1.2.5iii) in these lecture notes.