"For a long time, there was no detailed proof available for this theorem."

G. Zigler, "Lectures on polytopes" 1995, p. 182.

Known to me surways are:

*Jürgen Richter-Gebert "The Universality Theorems for Oriented Matroids and Polytopes"*

for the ordered geometric part of story and

*Seok Hyeong Lee and Ravi Vakil **"Mnev-Sturmfels universality for schemes*"

for the scheme-level part of story. * *

The problem of describing the moduli spaceces of line arrengements was posed to the autor by professor Vershik in 1976 at a student seminar. Much later (after observing the universality) it was discovered that the problem has a long history in very different subjects. The earliast appearence is, probobably,

Ringel's isotopy problem in

G. Ringel, Teilung der Ebene durch Geraden oder topologische Geraden, Math. Zeitschr. 64 (1956), 79 -102.

The latest appearence, is probably, the problem of describing the thin Schubert cells posed in

I.M. Gelfand, M. Goresky, R.D. MacPherson, and V.V. Serganova,

*Combinatorial geometries*, *convex polyhedra*, *and Schubert cells*, Adv. Math. 63 (1987) 301–316.

What is now called by universality theorem was discoverd in December 1984, the first examples computed and it was announced in

*N. E. Mn\"ev, Varieties of combinatorial types of projective configurations and convex ** polyhedra, Dokl. Akad. Nauk SSSR {\bf 283} (1985), no.~6, 1312--1314*

Detailed proof together with an application to Smale's problem on classification of Pareto-critical points of a smooth vector-map was written

in

*Н.Е. Мнeв, Топология многообразий комбинаторных типов проективных конфигураций ** и выпуклых многогранников, кандидатская диссертация, 116 стр., Ленинград, 1986 ** (N. E. Mn\"ev, The topology of configuration varieties and convex polytopes varieties, ** PHD thesis, 116 pp., Leningrad, 1986) (Russian, scanned typewritten manuscript 4,7MB)*

but it was not published in any way. In the USSR PHD dissertations got state approval and were considered as a sort of book publication. Copies were officialy available by request from VINITI. But surely it was not easily available text, not easy to read even in Russian.

The main part of the proof was a special deformation of Horner computation scheme of a polynomial map to make it "monotone" -- order preserving on all the steps of computation.

The deformation was simple-minded and straightforward - introducing extra variables and linear coordinate changes at any step of computaions. The resulting new deformed computation scheme was discribed and analysed and the proof that it became monotone required analasys of 45 simple algebraic inequalities. In fact it is pages and pages of trivilal manipulatios with brackets. In this form it was painfully difficult to autor to present it completely in talks or some short clear writings for nice journals. The deformation is a sort of homotopy in appropriate category and in its various forms is the only nontrivial point of the theorems.

A brief and not that good description of the proof from PhD was published in

*N. E. Mn\"ev, **The universality theorems on the classification problem of configuration varieties and convex polytopes varieties**, in {\it Topology and geometry---Rohlin Seminar}, 527--543, Lecture Notes in Math., 1346, Springer, Berlin-New York, Berlin, 1988.*

An interest to the theorem was complete surprise for the author. It is truly "elementary", it use basicaly tools of Greek mathematicians. It is useless, in a sense that it state that a class on nice looking problems is universal and therefor useless. However, it happens that it has viral property to show up in remote corners of some natural geometric moduli problems and make it useless too, understandably provoking some uneasy feelings in the correspondent community.

At 1991 was announced a version of the universality theorem for oriented matroid stratification of Grassmanian.

*N. Mn\"ev, "The universality theorem on the oriented matroid stratification of the space **of real matrices", in Discrete and computational geometry (New Brunswick, NJ, ** 1989/1990)}, 237--243, Amer. Math. Soc., Providence, RI, 1991*

Simultaneously P. Shor, basing only on the brief description of the proof from Rohlin Seminar paper, has introduced a key important shortcut in the main trick of the proof - the surgery of a computation scheme of a polynomial map. Perhaps as an educated object-oriented computer scientist Peter figured out that in this context polynomial system one should consider as a result of gluing elementary quadratic and linear expressions and then deform elementary parts and gluing rules. This observation of locality and "homotopy" pasting really opens eyes.

*Shor, Peter W. ** Stretchability of pseudolines is NP-hard. Applied geometry and discrete** mathematics, 531--554,** DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 4, Amer. Math. Soc., Providence, RI, 1991.*

Then the surgery was modernised to truly projective geometry surgery manipulating by projectivites and geometric identities by Harald Gu"nzel:

*Harald Gu"nzel,** "The Universal Partition Theorem for Oriented Matroids", ** Discrete & Computational Geometry 15, 121-145, 1996.*

and Ju"rgen Richter-Gebert assembled a short clear proof in

*Ju"rgen Richter-Gebert** "Mne"v's Universality Theorem Revisited", 1995*

This develoment made clear that the theorem actually should be attributed to Karl Georg Christian von Staurt, or even to Pappus of Alexandria.

Later the scheme-language versions were introduced by

M. Kapovich and J. J. Millson

*Kapovich, Michael; Millson, John J. On representation varieties of Artin groups, projective arrangements and the fundamental groups of smooth complex algebraic varieties. Inst. Hautes Études Sci. Publ. Math. No. 88 (1998), 5--95 (1999)* [ps],

*Prakash Belkale and Patrick Brosnan*

*"Matroids, motives and conjecture of Kontsevich"*

and L. Lafforgue

*L. Lafforgue, "Chirurgie des grassmanniennes", CRM Monograph Series, Volume: 19; 2003*

In 2004 R. Vakil independently has proved the scheme-level theorem in one night. He has presented exciting applications in

*R. Vakil, ** Murphy's Law in algebraic geometry: Badly-behaved deformation spaces ** Inventiones Mathematicae Volume 164, Number 3 / June, 2006 p 569-590. ** Arxiv math.AG/0411469, 2004*

The updated proof separetly presented in

*Seok Hyeong Lee and Ravi Vakil*

*Mnev-Sturmfels universality for schemes*

Author strongly believes that the theorem (and its derivatives) is not in its final form.

The results in

*H. Lombardi, N. Mnev and M.-F. Roy, The Positivstellensatz and small deduction rules for ** systems of inequalities, Math. Nachr. {\bf 181} (1996), 245--259 [ps]*

were designed as a first step of the next step

but the project was frozen, due to lack of interested people exept myself.