Hartshorne Chapter V Proposition 1.5 This proposition states as follows,
What confuses me is the last sentence. Notice that the Lemma 1.3 states as follows.

We also have the fact that,

*

*The pairing  Div $X \times \operatorname{Div} X \rightarrow \mathbf{Z},$ only depends on the linearly equivalence class.


*If $C$ and $D$ are nonsingular curves meeting transversally, then $C . D=$ $\#(C \cap D),$ the number of points of $C \cap D.$
So my qusetion is
(1) In the case of Adjunction formula, how to guarantee  there exists   $H\in|C+K|$ such that $H$ meets $C$ transversally?
(2) As mentioned in the statement of self-intersection, for a nonsingular curve $C$ on the smooth projective variety, we also have $C^2=\text{deg}\mathcal O_C(C)$.
So, it seems that we always have $C . D=\text{deg}\mathcal O_C(D)$ for any nonsingular irreducible curve $C$? Is it right? Why?
Any help would be appreciated. Thanks a lot!
 A: Let me answer (2) in my way (and afterwards how Hartshorne possibly wanted to do it):
My way:
In Proposition 1.4 Harthorne shows that we have $C \cdot D = h^0(\mathscr{O}_{C \cap D})$ for all curves $C$ and $D$ not having common irreducible components. Therefore it suffices to show $$\text{deg}(\mathscr{O}_C(D)) = h^0(\mathscr{O}_{C \cap D}).$$ For this we tensor the short exact sequence $$0 \rightarrow \mathscr{O}_C(-D) \rightarrow \mathscr{O}_C \rightarrow \mathscr{O}_{C \cap D} \rightarrow 0$$
(here we assume that no irreducible component of $C$ is contained in $D$ to get exactness on the left side) with the line bundle $\mathscr{O}_X(D)$ (which is an exact functor) and get the short exact sequence $$0 \rightarrow \mathscr{O}_C \rightarrow \mathscr{O}_C(D) \rightarrow \mathscr{O}_{C \cap D} \rightarrow 0.$$ Now we have $h^0(\mathscr{O}_{C \cap D}) = \chi(\mathscr{O}_{C \cap D}) = \chi(\mathscr{O}_C(D)) - \chi(\mathscr{O}_C) = \text{deg}(\mathscr{O}_C(D)),$ where the second equality comes from the additivity of the Euler characterstic and the third equality comes from Riemann-Roch for curves.
Harthorne's way (I guess):
I guess Hartshorne wants to combine 1.3 and 1.4 in the following way. Due to 1.3 we have that the degree $\text{deg}(\mathscr{O}_C(D))$ counts the intersection points of $C$ and $D$ and by 1.4 we can compute $C \cdot D$ in terms of the local intersection multiplicities. These local intersection multiplicities at some point $p$ are $\geq 1$ if and only if the point $p$ lies in the intersection. Therefore one has the inequality $$\#(C \cap D) \leq \sum_p (C \cdot D)_p,$$ where $p$ runs over all points in the intersection $C \cap D$. If we now can assure that we never have local intersection multiplicity $\geq 2$ by some assumptions on $C$ and $D$, we have the wanted equality. This is the case if we have transversal intersections, which is the assumption Hartshorne states.
