I'm studying intersection of curves with a fixed plane cubic, the first case I consider is of course lines, in particular lines intersecting the cubic at only one point. The problem is quite easy and I've solved it in terms of basic intersection theory on $\mathbb{P}^2$. Now I'm considering a new way of attacking the problem because it's easier to generalize to higher degrees but I'm a little bit unsure if I'm moving to the right direction.


I'm starting with $X=\mathbb{P}^2(\mathbb{C})$ and a cubic curve $B \subset X$ and a flex $P$ on $B$ such that for a hyperplane section $H$ I have $3dP \sim dH\vert\_B$ (where $d \in \mathbb{N}$). With these notations I'm considering the exact sequence of sheaves


$0 \rightarrow \mathcal{O}\_X(d-3) \rightarrow \mathcal{O}\_X(d) \rightarrow \mathcal{O}\_B(dH \vert\_B) \rightarrow 0$


In the case where $d=1$ (and of course when $d=2$ too), passing to the long exact sequence associated to it I get the isomorphism $H^0(X, O\_X(1)) \cong H^0(B,\mathcal{O}\_B(dH \vert\_B))$.
I have then two questions:


1. Is this correct? I mean, is the natural exact sequence obtained by the restriction map from $\mathcal{O}(1)$ to the rational functions on the cubic with at most a triple pole on $P$ and is the isomorphism I get from the long exact sequence precisely the restriction again?
2. Does it follow from this (case $d=1$ and $d=2$) that there is only one line (respectively a conic) meeting the cubic only at $P$? (Here I can choose $P$ such taht it has order 3 with respect to the addition in $B$ obviously)?


I'm thinking is it obvious regarding to the geometry but trying to writing down explicitly I'm getting a little bit of problems so I'm wondering if I'm misunderstanding something.


Thank you all



sequence of sheaves for studying intersection
1. is ok. For 2: I don't think you can prove the existence of flexes this way, but assuming P is a flex, then unicity of the line does follow from the iso. BTW, the unique "conic" that intersects B in 6P (P a flex) is the tangent line, doubled.


