9
$\begingroup$

The following theorem is often called the fundamental theorem of projective geometry:

Let $k$ be a field and let $n \geq 3$. Let $X$ be the partially ordered set of nonzero proper subspaces of $k^n$. Then every poset automorphism of $X$ is induced by a semi-linear automorphism of $k^n$, i.e. a set map $f:k^n\rightarrow k^n$ for which there exists a field automorphism $\tau:k \rightarrow k$ such that $$f(c_1 \vec{v}_1+c_2\vec{v}_2) = \tau(c_1) f(\vec{v}_1) + \tau(c_2) f(\vec{v}_2)$$ for all $c_1,c_2 \in k$ and $\vec{v}_1,\vec{v}_2 \in k^n$.

Question: who was the first person to prove this, and where does their proof appear? I know it has its origins in 19th century work of von Staudt, but I don't think that the above theorem appears in his work. On page 52 of Baer's book "Linear Algebra and Projective Geometry", he says that the first proof was due to Kamke, but he does not give a reference.

$\endgroup$
  • $\begingroup$ The reference I know is von Staudt's book "Beitrage zur Geometrie der Lage", Heft II, 1847. Did you read it? Von Staudt was interested in automorphisms of projective spaces, but the difference with your question is only minor. $\endgroup$ – Moishe Kohan Dec 30 '14 at 22:16
  • $\begingroup$ @studiosus : Yes, I did (attempt to) read it. The language is archaic enough that I am not entirely sure what is in there, but for instance I was not able to find any theorem in which a field automorphism showed up (and I have found references in other sources to errors in van Staudt's proof; there was apparently some kind of missing "continuity" assumption, which would explain why he did not see the field automorphisms). Of course, fields had not yet been defined at that point, so it could have been hidden somewhere. $\endgroup$ – Andrew Dec 30 '14 at 22:25
  • $\begingroup$ (eg: the paper arxiv.org/abs/0707.0229 discusses errors in von Staudt's work) $\endgroup$ – Andrew Dec 30 '14 at 22:36
  • $\begingroup$ I think you can find Kamke's paper here: eudml.org/doc/145757 $\endgroup$ – Carlo Beenakker Dec 30 '14 at 23:12
  • $\begingroup$ @CarloBeenakker: Thanks, I'll take a look at that paper. $\endgroup$ – Andrew Jan 1 '15 at 21:37
8
$\begingroup$

The version you state is definitely a 20th century development, only marginally related to Von Staudt's theorem. Here is a translation of the relevant section of Karzel & Kroll's Geschichte der Geometrie seit Hilbert, p. 51 (notation should be self-explanatory):

Examples of "non-linear" collineations were given by C. Segre [Seg 1890] for projective geometries over the complex numbers and by Veblen and Bussey [VB 06] for projective geometries over finite fields. Thus arose at the beginning of this century the problem whether these are all affinities resp. collineations. The following realization theorems state that this is the case.

  • (8.1) Let $(V,K)$ be a vector space with $\dim(V,K)\geqslant 2$ resp. $\dim(V,K)\geqslant 3$.

  • a) For every affinity $a$ of the corresponding affine space $A(V,K)$ there is exactly one semilinear permutation $\sigma$ of $(V,K)$ and one $\mathbf a\in V$ such that $a = \mathbf a^+\circ\sigma$.

  • b) For every collineation $\kappa$ of the corresponding projective space $\Pi(V,K)$ there is a semilinear permutation $\sigma$ of $(V,K)$ such that $\tilde\sigma = \kappa$; here $\sigma$ is unique up to a factor $\lambda\in K$, i.e. $\tilde\sigma = \tilde\sigma'$ for $\sigma' = \lambda_\ell\circ\sigma$, where $\lambda_\ell(\mathbf x) := \lambda\mathbf x$.

Part a) was first proved by E. Kamke [Kam 27]. A proof for the case where $K$ is a commutative field is found in the textbook [SS 35], §14 of Schreier and Sperner. Further proofs over arbitrary fields are found in the textbooks [Bae 52], [Le 65], [KSW 73]. In [KSW 73] and [Le 65], part a) is proved first and part b) then deduced as a consequence, which makes the proof simpler and more transparent. Veblen [Ve 07] already gives a proof for projective geometries over finite fields.

Theorem (8.1 b) is the generalization to [higher dimensional] space of Von Staudt's Theorem (9.1) on the group of projectivities.

One might add that Darboux (1880, p. 59) already states and attributes the theorem:

It is easy for example to recognize with v. Staudt (Geometrie der Lage § 121–122) that a projective or homographic correspondence in the plane or in space can be defined by the sole condition that aligned points in one figure correspond to aligned points in the other.

He only sketches a proof, and Schur (1881, p. 254) comments:

See v. Staudt, Geometrie der Lage, p. 60. Compare especially Möbius, der barycentrische Calcul, Chap. 6 and 7, where collineation in the plane and in space is first defined by the condition that straight lines correspond to straight lines.

Further references, giving various versions of the theorem but apparently never tracing it beyond Baer, are Dieudonné (1955, p. 72), Artin (1957, p. 88), Bourbaki (1970, Exerc. II.9.16), Jacobson (1974, p. 470), Samuel (1986, p. 32), Berger (1987, 5.4.8), Bennett (1995, p. 203), Jeffers (2000, p. 810).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.