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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