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?
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$\begingroup$ What do you mean when you say "how does it look"? Does this mean you want a description of the intersection as a subscheme of the curve or something like that? $\endgroup$– Joel DodgeCommented May 24, 2011 at 15:44
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$\begingroup$ @Joel: I mean can we locally say the elliptic curve is the graph of some function like $z\to z^k$ for $k\in \mathbb{Z}$ $\endgroup$– user13559Commented May 24, 2011 at 15:53
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1$\begingroup$ Please make your question more precise. Right now it's basically meaningless, and I'm inclined to close it. You haven't even said whether the other curve is nonsingular at the point of intersection, and it's unclear whether that is even relevant given the information you've provided. Are you fixing a dimension of the projective space in question? $\endgroup$– S. Carnahan ♦Commented May 24, 2011 at 18:30
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2 Answers
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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).
answered May 24, 2011 at 15:48
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$\begingroup$ Dear Charles, thanks for your nice comment. Lets think we are in a surface and locally C is the curve which is one of the axis in the point of intersection with the elliptic curve. Can we write the elliptic curve is the graph of some function like $z\to z^k$ for both case of (a) this intersection point is an inflection point of the elliptic curve. (b) this point is not the inflection point? $\endgroup$ Commented May 24, 2011 at 16:08
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$\begingroup$ I'm not quite sure in what terms to answer. "Writing as the graph of a function" for a relation is one way to pose the implicit function; which doesn't really belong in algebraic geometry as such. Perhaps you are looking for something like the Weierstrass preparation theorem, which gives a local picture for holomorphic functions. $\endgroup$ Commented May 24, 2011 at 18:06
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Let $E$, $C$ be your curves and suppose they intersect at a point $P$. I think asking for a description of $E$ "locally" at $P$ must mean that you want to understand the closed subscheme of $\mathcal{O}_{C, P}$ which is defined by an equation for $E$ in an affine neighborhood of $P$, say $f(\underline{x}) = 0$. Of course, if your curve is regular, then $\mathcal{O}_{C, P}$ is a discrete valuation ring and the ideal of $\mathcal{O}_{C, P}$ that $f$ generates will just be a power of the maximal ideal, say $(f) = \mathfrak{m}^k$. Then I would say that $E$ locally looks like $z\mapsto z^k$. Further, this business of thinking inside of the local ring is necessary because you can't make algebraic sense of a curve locally looking like $\mathbb{A}_1$ until you go all the way to the local ring.
answered May 24, 2011 at 16:33
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$\begingroup$ Of course, this has nothing to do with $E$ being an elliptic curve. This is just a very naive discussion about intersections of curves. $\endgroup$ Commented May 24, 2011 at 16:34