Lets assume an elliptic curve intersect a curve inside a projective space. How does the graph of an elliptic curve (complex curve) locally look like at the point of intersection. For example how does it look like inside a line bundle of $\mathbb{P}\_2$ at the point of intersection?



graph of elliptic curve inside projective space
I'm not quite sure about the formulation of the question: but there is something worth saying anyway, since it isn't often emphasised in basic texts. The points of order 3 can be identified with the inflection points of a plane cubic E (or, more accurately, taking one inflection point as origin on E for the group law, the nine 3-torsion points are the nine inflection point of a smooth plane cubic). There is a generalisation for the embedding of E in projective space of dimension n - 1 (essentially unique if E spans the space): the n-torsion is picked out by inflection of order n (higher tangency of a hyperplane). 


