Hartshorne exercise V.2.5: What is $|D+K-E|+E$? Hartshorne exercise V.2.5 is about possible values for $e$, the invariant of a ruled surface expressed as the relative proj of a rank-2 vector bundle $\mathcal{E}$ over a curve $C$ of genus $g>1$. Part (b) reads as follows:

(b) Let $e<0$, let $D$ be any divisor of degree $d=-e$, and let $\xi\in H^1(\mathcal{L}(-D))$ be a nonzero element defining an extension $$0\to\mathcal{O}_C\to\mathcal{E}\to\mathcal{L}(D)\to0.$$ Let $H\subset |D+K|$ be the sublinear system of codimension 1 defined by $\ker \xi$, where $\xi$ is considered as a linear functional on $H^0(\mathcal{L}(D+K))$. For any effective divisor $E$ of degree $d-1$, let $L_E\subset |D+K|$ be the sublinear system $|D+K-E|+E$. Show that $\mathcal{E}$ is normalized if and only if for each $E$ as above, $L_E\not\subset H$. Cf. proof of (2.15).

I am having trouble understanding what $L_E$ looks like inside $|D+K|$. I understand it's the divisors in $|D+K|$ which can be written as $E+F$ where $F\in |D+K-E|$, so this gives it the structure of a projective space. What I don't understand is that Hartshorne seems to imply via the phrase "sublinear system" that it's a linearly embedded projective space inside $|D+K|$, but I'm not sure how to see that.
 A: Let me belatedly answer my own question (a fellow student asked me something about this recently and I figured I had better clean up after myself).
Recall the definition of $|D|$: $|D|$ is the projectivization of the vector space of rational functions $f\in K(C)$ such that nonzero $f$ satisfy $(f)+D\geq 0$, i.e. $v_P(f)\geq -v_P(D)$ for all $P\in C$. We associate to each point in $|D|$ the divisor given by $(f)+D$.
If $E$ is effective, I claim that every rational function which satisfies $(f)+D-E\geq 0$ also satisfies $(f)+D\geq 0$ and these functions are closed under addition: if $v_P(f)+v_P(D)-v_P(E)\geq 0$ and $v_P(E)\geq 0$, then $v_P(f)+v_P(D)\geq 0$ as well; if $v_P(f)\geq -(v_P(D)-v_P(E))$ and $v_P(g)\geq -(v_P(D)-v_P(E))$, then as $v_P(f+g)\geq \min(v_P(f),v_P(g))$ by the properties of valuations, we have that $v_P(f+g)\geq -(v_P(D)-v_P(E))$.
Now consider what happens to the vector space of rational functions $f$ satisfying $(f)+D-E\geq 0$ under forming $|D-E|$ and $|D|$: the divisor associated to $f$ as a member of $|D-E|$ is $(f)+D-E$, while as a member of $|D|$ the divisor is $(f)+D$. So the natural inclusion of the vector space of rational functions $(f)$ satisfying $(f)+D-E\geq 0$ in to the vector space of rational functions $(f)$ satisfying $(f)+D\geq 0$ sends the linear system $|D-E|$ to $|D|$ by adding $E$. This shows that what Hartshorne writes down really does give $|D-E|$ the structure of a sublinear space of $|D|$.
