In Hartshorne's Algebraic geometry, page 158 gives the definition of the trace of $\mathfrak d$ on $Y$, where $i: \, Y \to X$ is a closed immersion of nonsingular projective varieties over an algebraically closed field $k$ and $\mathfrak d$ is a linear system on $X$. Let $\mathfrak d $ corresponds to an invertible sheaf $\mathscr L$ on $X$ and a sub-vector space $V \subset \Gamma(X, \mathscr L)$ over $k$. Then the trace $\mathfrak d \big |_Y$ of $\mathfrak d$ on $Y$ is defined as the linear system corresponding to $i^* \mathscr L$ and the image of $V$ under the natural map $\Gamma(X, \mathscr L) \to \Gamma (Y, i^* \mathscr L)$ which comes from the natural map $\mathscr L \to i_* i^* \mathscr L$.
I have some difficulty to understand its geometrical interpretation of $\mathfrak d \big |_Y$. The book says $\mathfrak d \big |_Y$ consists of all divisor $D\cdot Y$ where $D \in \mathfrak d$ is a divisor whose support does not contain $Y$. I do not understand why it is true. I really appreciate if someone could show me some details on it. One more question is: if the support of a divisor corresponding to $s \in \mathfrak d$ does contain $Y$, I think the image of $s$ under the map $\Gamma(X, \mathscr L) \to \Gamma (Y, i^* \mathscr L)$ should be $0$. I do not know if it is true or not.
Thank you very much.