In Hartshorne's Algebraic Geometry, Chapter 2, section 7, the trace of a linear system is defined as follows. Let $i:Y\hookrightarrow X$ be a closed immersion of nonsingular projective varieties over $k$. Let $\delta$ is a linear system on $X$ associated to an invertible sheaf $L$ and a subspace $V\subset H^0(X,L)$. Then $i^*L$ is an invertible sheaf on $Y$ and we have the morphism of vector spaces $H^0(X,L)\rightarrow H^0(Y,i^*L)$. Let $W$ be the image of $V$ under this morphism. Then the trace of $\delta$ denoted by $\delta|_Y$ is defined as the linear system on $Y$ associated to the line bundle $i^*L$ and the subspace $W$.
The following are my doubts.
1) There is a statement that - even if $\delta$ is a complete linear system, $\delta|_{Y}$ may not be complete. Why is this so? Is there some example?
2) If $L$ is very ample, then $i^*L$ on $Y$ is also very ample right? (If not, what is a counter example?)
3) By (2) if $L$ is very ample, $i^*L$ is also very ample. In this case won't the linear system $|L|$ restrict to $|i^*L|$?
Here is an example for 1). Take $C\subset\mathbb{P}^3$, embedding the projective line as a quartic. Then $V=H^0(\mathbb{P}^3,\mathcal{O}(1))$ is a complete linear system, but the restriction to $C$ is not, since $\dim V=4$, but $\dim H^0(C, \mathcal{O}_C(1))=5$. 2) is correct.The question in 3) is unclear. The linear system restricts, but completeness does not, so am not sure what you are asking.