What is the purpose of tangential base-points?

Question

Let $V$ be an affine complex variety. Let $x \in V$ be a closed point. Then a tangential base-point at $x$ is $x$ together with a regular function $t$ on $V$ that is zero exactly on $x$ (to degree $1$).

My familiarity with this notion is very meek, and I know that there are generalizations for other fields, and for normal crossing divisors (instead of one closed point; in this case they are called tangential morphisms), but I'm not sure I can recite a definition for these. In texts where I've seen this notion introduced, it said that the intuition should be that $t$ gives a direction: look at $t=\epsilon$ for $\epsilon$ going from $0$ to a small real positive number.

This hardly seems like motivation! Why would one want a base-point with a direction? What utility does it have? What problem does it solve? I've seen this in several papers, and I do wish to get over the hump and actually start understanding this.

Answer 1

Topologists' answer: often in topology it is useful to study the fundamental group of a surface (e.g.) with boundary with a basepoint sitting on a boundary component. In algebraic geometry, we don't really have surfaces with discs sliced out of them, only surfaces with punctures. But it turns out that a tangential basepoint at the puncture is a good substitute which makes sense algebraically. You have a circle's worth of choice of tangential basepoints at the puncture, which matches up exactly with the circle's worth of choice of basepoints on the boundary component (which is a circle.)

Number theorists' answer: Grothendieck's section conjecture asserts that, when X is a proper smooth variety over Spec k, for k a global field, the conjugacy classes of sections from Gal(k) (a.k.a. pi_1^et(Spec k)) up to pi_1^et(X) are in bijection with X(k). When X is not proper, this isn't true -- but (conjecturally) the sections are in bijection with the union of X(k) and the tangential basepoints.

Ultimate answer: if Deligne's monograph on the fundamental group of P^1 minus three points is not one of the papers where you've seen this, read that. (But this is a big, long paper and the tangential basepoints are near the end, so don't read unless you also want to learn about motives, periods, etale fundamental groups, Malcev completion, etc etc.)

Answer 2

I'm far from being an expert of the subject, but my naive opinion is that tangential basepoints makes many constructions more canonical and usually behave well with respect to the symmetries of the situation. Grothendieck said in his Esquisse:

Ceci est lié notamment au fait que les gens s’obstinent encore, en calculant avec des groupes fondamentaux, à fixer un seul point base, plutôt que d’en choisir astucieusement tout un paquet qui soit invariant par les symétries de la situation, lesquelles sont donc perdues en route.

On the fly translation:

This is because people working with fundamental groups are still choosing only one basepoint, instead of choosing a whole set of basepoints which is invariant under the symmetries of the situation, which are thus lost on the way.

Hence, we have to choose several basepoints (ie work with groupoids) and to choose them carefully.

If you take the fundamental example of $P^1(C)$ minus 0,1,$\infty$, there are two automorphisms $t\mapsto 1/t$ and $t\mapsto 1-t$ which induces an action of $S_3$ which is respected by the action of $G_{\mathbb{Q}}$. If you want to keep track of this action, you really want to take $0,1,\infty$ as a set of basepoints. Of course you can't, but you can take the set of tangential basepoints $01,10,1\infty,\infty 1, 0\infty,\infty 0$ which is stable under the action of $S_3$. Now if you decompose loops in appropriate pieces (moves between basepoints, small loops around basepoints) in such a way that the symmetries are again respected, then the action of $G_{\mathbb{Q}}$ can be written in a nice form.

Answer 3

One motivation comes from Grothendieck-Teichmüller theory. The idea, due originally to Grothendieck, is that one can study the absolute Galois group $G_{\mathbb{Q}}$ of $\mathbb{Q}$ by looking at its action on the fundamental groups of varieties defined over $\mathbb{Q}$. More specifically, Grothendieck suggested studying the Teichmüller tower, the collection of moduli spaces $M_{g, n}$ of genus $g$ curves with $n$ marked points together with any natural geometrically meaningful morphisms between them. Since all such morphisms should be defined over $\mathbb{Q}$, the action of $G_{\mathbb{Q}}$ on the fundamental groups $\widehat{\Gamma_{g, n}}$ will be compatible with the induced morphisms between them.

Tangential basepoints arise because most of the geometrically meaningful morphisms that one would like to define between moduli spaces take values not in the target moduli space itself but in the boundary of its compactification. For example, taking a smooth genus 0 curve with $m + 1$ marked points and a smooth genus 0 curve with $n + 1$ marked points and gluing them together along a specified pair of marked point gives a nodal genus 0 curve with $m + n$ marked points: This procedure defines a map $M_{0, m + 1} \times M_{0, n + 1} \to \partial\overline{M_{0, m + n}}$. In order for this to induce a map $\widehat{\Gamma_{0, m + 1}} \times \widehat{\Gamma_{0, n + 1}} \to \widehat{\Gamma_{0, m + n}}$, we are forced to consider tangential basepoints. (Note that the fundamental groups of the compactifications $\overline{M_{g, n}}$ are not interesting, so it does not do us much good to study the action of $G_{\mathbb{Q}}$ on these.)