Singular points of algebraic varieties and parametrization by Puiseux series

Question

Let $V\subset \mathbb{R}^n$ (or $\mathbb{C}^n$ if that makes anything easier) be an algebraic variety and $p\in V$ a possibly singular point. Let $U\subset V$ be a sufficiently small neighborhood of $p$ and $q\in U$ a nearby point. Can every such point $q$ be reached by a curve $\gamma\colon [0,1] \to U\subset \mathbb{R}^n$, where $\gamma(0) = p$, $\gamma(1) = q$ and $\gamma(t) = \gamma_1 t + \gamma_2 t^2 + \cdots$ is a convergent Taylor series with $\gamma_1 \ne 0$?

I have a feeling that the answer might be No, because some $q$ may only be reachable if fractional powers of $t$ are included along with integer powers. In that case, for a given $q$ (as above) does there always exist an integer $r>0$ and a convergent Puiseux series $\gamma(t) = \sum_{k=r}^\infty \gamma_k t^{k/r}$ with $\gamma_r \ne 0$ such that $\gamma(0) = p$, $\gamma(1) = q$ and $\gamma([0,1]) \subset U$? If so, is it possible to express the integer $r$ (for a given $q\in U$ or the maximal such integer over all $q\in U$) in terms of algebraic invariants associated to the variety $V$ or the tangent cone of $V$ and $p$? Also, with point $q$ and leading coefficient $\gamma_r$ fixed, are the higher order coefficients $\gamma_k$ ($k>r$) allowed to vary arbitrarily or perhaps must they belong to some special subsets of $\mathbb{R}^n$? If so, how to characterize these subsets?

Please note that my background is not in algebraic geometry, so apologies if this question is elementary. However, I've not been able to find it, at least not posed in this form, after scanning through some textbooks. My interest in it comes from the same context as this question.

Answer 1

Cut $V$ with a linear subspace of appropriate dimension passing through $p$ and $q$ and reduce to the case of curves. Through $p$ there are only finitely many branches of the curve (and one of them contains $q$). For each of these branches there is a local parameter $t$ and the branch can be expressed as the image of a map given by power series in $t$ with $t=0$ mapping to $p$. Now, it may not start with $t$ but take the lowest power of $t$ occurring in the parametrization, say $t^r$ and replace $t$ by $t^{1/r}$. Now you end up with a Puiseux series in the new $t$ the way you want it. Standard books on curves (e.g. Fulton) will have proofs of the statements about branches and local parameters to fill in the details of this argument. There is also the issue that things might depend on $q$, I don't think that's a big deal.

Answer 2

(This is essentially a very long comment...)

What is r?

If your original $V$ is a curve, or alternatively after (as per Filipe) cutting your space down to a curve by taking hyperplane sections, then one has recourse to the following basic fact of algebraic geometry

normalization of curves: given any (possibly singular) curve algebraic curve $C \subset \mathbb{C}^n$, there is a unique map $\nu: C^\nu \to C$ such that $C^\nu$ is smooth, $\nu$ is surjective, and is a bijection away from the singularities of $C$.

So locally at some $p^\nu \in C^\nu$ mapping to your point of interest $p \in C$ you can choose a coordinate $z$ and write $$\nu(z) = (\nu_1(z), \cdots, \nu_n(z)) = (z^{a_1} + \ldots, z^{a_2} + \ldots, z^{a_3} + \ldots)$$

Filipe was explaining above that if you wanted something with a leading linear term, you have no choice but to write $z^{\mathrm{min}(a_i)} = w$. Note that this minimum doesn't depend on the coordinate $z$ you chose.

But what is this minimum?

If $V$ is a unibranch* curve, then $r = \mathrm{mult}_p(V)$, the multiplicity of $V$ at $p$.

The multiplicity of a variety at a point has various characterizations. If $V$ were cut out by one equation and $p=0$, then it's just the degree of the minimal degree monomial in that equation. In general, it's

• The number of intersections, near $p$, of $V$ with a general hyperplane passing near $p$
• The same for the tangent cone of $V$ at $p$
• The number of copies of the exceptional divisor when you blowup $V$ at $p$
• Take $\mathcal{O}_p, \mathfrak{m}_p$ to be the local (or complete local) ring of functions of $V$ at $p$, and its maximal ideal. Then the quantity $HS(n) = \mathrm{dim}_\mathbb{C} \mathcal{O}_p / \mathfrak{m}_p^{n+1}$ is called the Hilbert-Samuel function; it's eventually a polynomial in $n$ with leading term $n^{\mathrm{dim}(V)}/\mathrm{dim}(V)!$ times the multiplicity.

We can use that last characterization to see $\mathrm{mult}_p(C) = \mathrm{min}(a_i)$: the point is that the completed ring of functions $\mathcal{O}_p$ is the subring of $\mathbb{C}[[z]]$ generated by the $\nu_i(z)$. One sees from the fact that the normalization is generically 1-1 that $\mathrm{gcd}(a_i) = 1$, from which it follows that sufficiently high powers $\mathfrak{m}_p^N$ will just be $z^{N \cdot \mathrm{min}(a_i)} \mathbb{C}[[z]]$.

What is this word unibranch?

Well, when you normalize a curve, the preimage of the point $p$ might be several points $p^\nu_i$, the neighborhood of each of which maps to a different analytically locally irreducible component of the (reducible) analytic local curve $C$. One would like to say: just treat each local irreducible component separately. However one could for example have a connected rational curve with a unique singularity $p$ such that $C$ at $p$ is analytically locally of the form $x(y^2- x^3) = 0$. This has two branches, one smooth ($x = 0$), and the other singular ($y^2 - x^3 = 0$). For any point on the curve, including points arbitrarily close to $p$ along the singular branch, one can get to them by traveling only along the smooth branch. Thus for this curve, the minimal $r$ is $1$, despite the presence of the singularity.

You demanded your paths stay in a small open set $U$, which would fix the problem if $V$ is a curve. But I don't see why it couldn't reappear for higher dimensional $V$ with the property that for any neighborhood $U$ of $p$, there exists some slice of $V$ through $p$ (depending on the neighborhood) such that this slice is a curve with an analytically locally reducible singularity at $p$ while however the branches meet at a smooth point inside $U$. I suspect this means you've asked slightly the wrong question, and should demand something like that your paths `never travel too far away from the shortest path from $p$ to $q$', but I'm not exactly sure what's the best way to make that precise.

What if I just want to bound $r$?

Then Filipe's answer tells you the following: $r$ is less than the maximal multiplicity of any curve obtained by cutting the variety by an appropriate-dimensional linear slice. (I see no a-priori reason to believe that linear slices give better bounds than some other kind of slices, but also as mentioned above I don't know how to correctly formalize the question so that `what is $r$' actually makes sense.)

I do not know the name of this number -- i.e., the maximal multiplicity of a linear-slice-down-to-a-curve -- in algebraic geometry.

A related thing which does have a name is the answer to this question for a general $q$ near $p$, i.e. the multiplicity of a general linear-slice-down-to-a-curve. Such a curve is called a local polar curve and its multiplicity is called a local polar multiplicity; people study these things.

I do not think either of these things can be computed from the tangent cone.