Synthetic projective lines

Question

The classical synthetic notion of projective plane consists of a set of points, a set of lines, and a relation of incidence between the two, such that any two distinct points lie on a unique line and any two distinct lines intersect in a unique point (plus some nondegeneracy assumptions). There are similar notions of projective 3-space, $n$-space, and so on — but 1-dimensional projective space seems harder to capture synthetically, since there is no "room", dimensionally, for subspaces in between the points and the entire space.

Has anyone attempted to define a synthetic notion of "projective line"? Ideally such a definition would have properties like the following:

• The space $P^1(k)$ is naturally a projective line for any division ring $k$, and from any projective line $L$ satisfying enough axioms we can construct a skew field $c(L)$ such that $c(P^1(k)) \cong k$ and $P^1(c(L))\cong L$ (unnaturally). The corresponding facts for Desarguesian projective planes are classical.

• Any line in a projective plane is a projective line, and any projective line satisfying enough axioms can be embedded as a line in some projective plane. This would be analogous to how any plane in a projective 3-space is a Desarguesian projective plane, while any Desarguesian projective plane can be embedded in a projective 3-space.

I have an idea for how one might do this, by axiomatizing the "quadrangular hexad" relation on a line in a projective plane; but before I try very hard, I'm looking for references where something like it has been tried before.

Answer 1

Building on previous work by Paul Libois, and related to work by Libois' student Jean van Buggenhaut from 1969, Francis Buekenhout considered and solved this question in "Foundations of one Dimensional Projective Geometry based on Perspectivities" Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 43 (1975) 21-29. Note that his approach also adapts to lines in Moufang planes. See also F. Buekenhout and A. Cohen Diagram geometry Springer 2013, Section 6.2 Projective lines.

Answer 2

Following up on Matthias Wendt's comment, the language of Moufang sets is indeed a suitable axiomatic approach to (generalizations of) projective lines.

Formally speaking, a Moufang set is a set $X$ together with a collection of groups $U_x \leq \operatorname{Sym}(X)$ (one group for each $x \in X$), such that:

• each $U_x$ fixes $x$ and acts sharply transitively on $X \setminus \{ x \}$;
• the group $G := \langle U_y \mid y \in X \rangle$ permutes the $U_x$'s by conjugation.

The groups $U_x$ are called the root groups of the Moufang set, and the group $G$ is called the little projective group.

It is not hard to see that $X = \mathbb{P}^1(k)$ and $G = \mathrm{PSL}_2(k)$ has the structure of a Moufang set, but there are many more interesting examples, most notably those arising from semisimple linear algebraic groups of $k$-rank one.

It is possible to single out the genuine projective lines over fields among the Moufang sets. For fiels $k$ with $\operatorname{char}(k) \neq 2$, this was done by Richard Weiss and me (see section 6 of our paper "Moufang sets and Jordan division algebras", Math. Ann. 335 (2006), no. 2, 415–433); this result has been generalized to arbitrary fields (including the case $\operatorname{char}(k) = 2$) by Matthias Grüninger, where the result is somewhat more subtle ("Special Moufang sets with abelian Hua subgroup", J. Algebra 323 (2010), no. 6, 1797–1801).

At the risk of giving too much advertisement for my own papers, I can recommend the course notes "A course on Moufang sets", Innov. Incidence Geom. 9 (2009), 79–122 that I wrote together with Yoav Segev, for an introduction to the subject.

Edit: As requested in the comments below, I am adding some more details, in particular about the example with $X = \mathbb{P}^1(k)$ and $G = \mathrm{PSL}_2(k)$.

First of all, notice that the little projective group $G$ is generated by any two of the root groups, $G = \langle U_x, U_y \rangle$ for all $x,y \in X$ with $x \neq y$. So to give an explicit description of an example, it suffices to describe two of these root groups; all others are then obtained by conjugation inside $G$.

We now take $X = \mathbb{P}^1(k) = k \cup \{ \infty \}$, acted upon by $G=PSL_2(k)$, the elements of which I will denote with matrices with square brackets (determined up to a non-zero scalar), so $$ \begin{bmatrix} a & b \\ c & d \end{bmatrix} .x = \frac{ax+b}{cx+d} \quad \text{for all } x \in X.$$ Notice that $\operatorname{Stab}_G(\infty) = \left\{ \begin{bmatrix} a & b \\ 0 & a^{-1} \end{bmatrix}\right\}$ and $\operatorname{Stab}_G(0) = \left\{ \begin{bmatrix} a & 0 \\ c & a^{-1} \end{bmatrix}\right\}$. We now define the root groups $U_\infty$ and $U_0$ to be the group of unipotent elements of these point stabilizers, i.e. $$ U_\infty =\left\{ \begin{bmatrix} 1 & b \\ 0 & 1 \end{bmatrix} \mid b \in k \right\}, \quad U_\infty =\left\{ \begin{bmatrix} 1 & 0 \\ c & 1 \end{bmatrix} \mid c \in k \right\} .$$ The point is that it is now possible to forget about the matrix representation and even about the original ambient group $G$ all together, and only retain the corresponding permutations of $X$. We then get $$ U_\infty = \{ x \mapsto x + b \mid b \in k \}, \quad U_0 = \{ x \mapsto (x^{-1}+c)^{-1} \mid c \in k \} . $$ It is now not so hard to imagine that this description makes sense for more general algebraic structures than commutative fields only. And indeed, this works equally well for skew fields, octonion division algebras, and even more generally for Jordan division algebras. To make examples with non-abelian root groups, similar ideas make sense by replacing the multiplicative inverse by more complicated maps that "behave like a multiplicative inverse".

Another relevant comment, related to your first "ideal property": in the case of skew fields, for instance, it is only possible to recover the skew field up to opposition, i.e., in the case of $\mathrm{PSL}_2(D)$, we can recover the pair $(D, D^{\mathrm{op}})$ from the Moufang set, but not $D$ itself. (Here, $D^{\mathrm{op}}$ is the skew field with same underlying additive group as $D$, and with multiplication given by $x*y := yx$.)

Answer 3

One defining feature of $\mathbb P^1(k)$ is that it provides a sharply 3-transitive permutation representation for $\operatorname{PGL}_2(k)$. I believe that the abstraction of projective line to "sharply 3-transitive permutation group" is the most studied one.

The characterization of sharply 3-transitive groups as groups of projectivities over KT-fields came up in an answer to Jacob Lurie's question Action of PGL(2) on Projective Space. That answer mentions that every sharply 3-transitive group is the "group of projectivities" of a KT-field $F$, but note that the correspondence goes both ways, and one can construct $F$ out of the permutation representation.

Judging by how there is an equivalence of categories of near fields and of sharply 2-transitive groups, I wouldn't be surprised if one can say something similar for KT-fields and sharply 3-transitive groups. A reference for this is

"Kerby, W., Wefelscheid, H. "Uber eine scharf 3-fach transitiven Gruppen zugeordnete algebraische Struktur." Abh. Math. Sem. Univ. Hamburg 37 (1972), 225–235.

Answer 4

You might benefit from reading Section 5.3 of John Faulkner's book The role of nonassociative algebra in projective geometry AMS 2014. The results there may have been what you had in mind by 'axiomatizing the "quadrangular hexad" relation'. These are (almost) all very old theorems, appearing in (and mostly predating) Pickert's Projektive Ebenen book from 1955, one going back to Veblen and Young from 1910 and another to von Staudt from 1860. The basic underlying idea of using quadrangular sets goes back to Desargues in 1639. (It was Desargues who introduced the term involution into mathematics, in this context, often referred to as an involution of six points.) But Desargues' work is an elaboration and extension of results of Pappus from c.340.