Puiseux series expansion for space curves?

2012-02-18T23:42:37Z

This result is apparently well known and used by many people. I am, however, quite frustrated that I cannot seem to find a proof that I can understand. For plane algebraic curves, this is not too hard. For an irreducible polynomial $F(x,y) \in \mathbb{C}[x,y]$ nonconstant in both $x$ and $y$ with $F(0,0) = 0$, one can formally expand $y$ as a Puiseux series in $x$ $$ y(x) = \sum_{i=0}^\infty y_i x^{\alpha_i} $$ The Puiseux theorem actually states that such a series converges (in some sense) near $x=0$. Alternatively, we can construct Riemann surface over the point $(0,0)\in \mathbb{C}^2$ of the algebraic function $y(x)$ which has the normal representation as holomorphic element $$ \begin{eqnarray} x &=& t^m \cr y &=& \sum_{i=1}^\infty y_i t^i \end{eqnarray} $$ With some abuse of notation, I think we can even say the two are the same. I can find many sources, among which I like Walker's representation the best.

Now how about space algebraic curves (of one complex dimension) in $\mathbb{C}^n$? For a polynomial system $F(x_0,\ldots,x_n) = (f_1(x_0,\ldots,x_n),\ldots,f_n(x_0,\ldots,x_n))$ where each $f_i \in \mathbb{C}[x_0,\ldots,x_n]$ with $F(0,\ldots,0) = (0,\ldots,0)$, and lies only on a one (complex) dimensional irreducible component of $V(F)$, if we fix a place of the this algebraic curve centred at $(0,\ldots,0)$, we should be able to find a parametrization $$ \begin{eqnarray} x_0 &=& t^m \cr x_k &=& \sum_{i=1}^\infty c_{k,i} t^i \end{eqnarray} $$ with convergent power series. Or equivalently, we could express $x_1,\ldots,x_n$ as Puiseux series in $x_0$ that converge in certain sense.

I could only find "proofs" that reference Hironaka's resolution of singularity, which I don't think I can understand any time soon. I'm hoping to find a proof using only complex geometry or basic complex algebraic geometry. In particular, I was thinking maybe I can repeatedly apply Weierstrass preparation theorem together with Puiseux theorem, however, I'm not quite sure how to continue after the first step.

Answer by Samuele for Puiseux series expansion for space curves?

2012-03-16T23:06:00Z

If $(C,0)$ is a germ of complex curve in $\mathbb{C}^n$, then you can find coordinates $(z_1, z')$ and a polydisc $V=V_1\times V'$ centered in $0$ such that the canonical projection $V\ni (z_1,z')\mapsto \pi(z_1,z')=z_1\in V_1$ is a ramified covering when restricted to $C$ with $p$ sheets. Let $S$ be the ramification locus, then $S$ is an analytic set in $V_1$, that is, a discrete set of points. Therefore, upon taking a smaller $V_1$, we can think that $S={0}$ or it is empty. Let us suppose $S$ is not empty, otherwise the projection is a local biholomorphism with the unit disc and this gives the thesis with $z_1=t$, $z_k=f_k(t)$, $f_k\in\mathcal{O}(D )$. Now, $C^*=C\cap \pi^{-1}(V_1\setminus S)$ is a $p-$sheeted covering of $V_1\setminus S=D\setminus{0}$, therefore it is isomorphic to the standard $p-$sheeted covering: $$D\setminus{0}\ni t\mapsto t^p\in D\setminus{0}.$$ This means that there exists a map $g:D\setminus{0}\to C^*$ such that $\pi\circ g (t)=t^p$; this map extends clearly to a holomorphic bijection by setting $g(0)=0$ (it is continuous in $0$ and holomorphic outside). Therefore, we have $$z_1=t^p$$ $$z_k=g_k(t)=\sum_{j=0}^\infty a_{kj}t^j.$$

The existence of such a system of coordinates is a standard result in local complex geometry, sometimes called local parametrization theorem.

Puiseux series expansion for space curves? - MathOverflow

Puiseux series expansion for space curves?

Answer by Samuele for Puiseux series expansion for space curves?