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.



What is the purpose of tangential base-points?
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.)


