Brief history of  Universality theorem for moduli spaces of line arrangements.


"For a long time, there was no detailed proof available for this theorem."
G. Zigler, "Lectures on polytopes"
1995, p. 182.


The problem of describing the moduli spaceces of line arrengements was posed to the autor by professor Vershik in connection with geometry of  optimality problems. Later we 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.
Tha latest  appearence, is probably, the problem of describing the thing 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.

The "universality theorem"  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. A typewritten copy was  available to everybody interested.
The main part of the proof was  a special   deformation of Horner computation scheme of a polynomial map.
This made the proof  looking complicate and almost computational.  

А brief 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;

At 1991 was announced а version of the 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 {\it Discrete and computational geometry (New Brunswick, NJ,
1989/1990)}, 237--243, Amer. Math. Soc., Providence,  RI, 1991

Simultaneously P. Shor has introduced a very important shortcut in the main trick of the proof

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.

The short complete proof,  together with very nice modernization of P. Shor's block was written by
Ju"rgen Richter-Gebert

Ju"rgen Richter-Gebert
"Mne"v's Universality Theorem Revisited"
Se'minaire Lotharingien de Combinatoire, B34h (1995), 15pp. [ps]


Almost simultaneously an  important combinatorial way in proof  was developed by
Harald Gu"nzel

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

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]
  
and L. Lafforgue

L. Lafforgue.
Chirurgie des grassmannienne
 IHES/M/02/31



By private communications,  recently R. Vakil  independently has proved the theorem in one night. Не 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

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.



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