Hartshorne Chapter V Proposition 1.5

Question

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,

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

2. 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!

Answer 1

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.