Algebraic Transformation Groups and Algebraic Varieties: Proceedings of the conference Interesting Algebraic Varieties Arising in Algebraic Transformation Group Theory held at the Erwin Schrödinger Institute, Vienna, October 22–26, 2001

244 Pages·2004·1.72 MB·English

by  Ciro Ciliberto, Vincenzo Di Gennaro (auth.), Vladimir L. Popov (eds.)

Encyclopaedia of Mathematical Sciences
Volume 132
Invariant Theory and Algebraic Transformation Groups III
Subseries Editors:
R.V. Gamkrelidze V.L. Popov
Springer-Verlag Berlin Heidelberg GmbH
Vladimir L. Popov (Editor)
Algebraic
Transformation
Groups
and Algebraic
Varieties
Proceedings of the conference
Interesting Algebraic Varieties
Arising in Algebraic
Transformation Group Theory
held at the Erwin Schrödinger Institute,
Vienna, October 22-26, 2001
1 3
Vladimir L. Popov
Steklov Mathematical Institute
Russian Academy of Sciences
Gubkina 8
117966 Moscow
Russia
e-mail:
Preface
These are the proceedings of the conference Interesting Algebraic Varieties
Arising in Algebraic Transformation Groups Theory that was held at The
Erwin Schro¨dinger International Institute for Mathematical Physics, Vienna,
Austria, from October 22 through October 26, 2001.
The conference was made possible through interest and financial and orga-
nizational support of The Erwin Schro¨dinger International Institute for Ma-
thematical Physics, Vienna, Austria. On behalf of all participants I thank this
institution and especially P. W. Michor, one of its Directors, for this interest
and support.
It is an empirical fact that many interesting and important algebraic vari-
eties are intimately related to algebraic transformation groups. To name only
some, the examples are affine and projective spaces; quadrics; grassmanni-
ans, flag and, more generally, spherical (in particular toric) varieties; Schu-
bert varieties; nilpotent varieties; determinantal varieties, Severi, Scorza and,
more generally, highest vector (HV-) varieties; group varieties; generic tori
in algebraic groups; commuting varieties; categorical quotients of Geometric
Invariant Theory and the related moduli varieties of curves, vector bundles,
abelian varieties etc.; simple singularities realized as that of the corresponding
categorical quotients and nilpotent orbit closures. The idea of the conference
was to trace the new evidences of this relation.
For various reasons several talks given during the conference do not appear
in these proceedings. Below a complete listing of all talks given is presented for
the information about the conference. The talks which do appear are generally
expanded and/or modified versions of those given during the conference.
November 21, 2003 Vladimir L. Popov
List of Talks Given at the Conference
“Interesting Algebraic Varieties
Arising in Algebraic Transformation Groups Theory”,
ESI, Vienna, Austria, October 22–26, 2001
Monday, October 22, 2001
10.30–12.00
David J. Saltman (University of Texas at Austin, Austin, USA),
Invariants of Symplectic and Orthogonal Groups of Degree 8.
14.00–15.30
Dennis Snow (University of Notre Dame, Notre Dame, USA),
The Role of Exotic Affine Spaces in the Classification of Homogeneous Affine
Varieties.
15.45–17.15
Nikolai L. Gordeev (State Pedagogical University, St. Petersburg, Russia),
Branch Locus of Quotients of Finite Group Actions.
Tuesday, October 23, 2001
10.30–12.00
Alexei N. Parshin (Steklov Mathematical Institute, Moscow, Russia),
The Krichever Correspondence for Algebraic Varieties.
14.00–15.30
Claudio Procesi (Universita` di Roma “La Sapienza”, Roma, Italia),
Diagonal Harmonics.
15.45–17.15
Fyodor L. Zak (Central Economics Mathematical Institute, Moscow, Russia
and Independent University of Moscow, Moscow, Russia),
Orders and Classes of Projective Varieties.
VIII List of Talks
Wednesday, October 24, 2001
10.30–12.00
Corrado De Concini (Universita` di Roma “La Sapienza”, Roma, Italia),
On Semigroups Associated to Irreducible Representations of Algebraic Groups.
14.00–15.30
Joseph M. Landsberg (Georgia Institute of Technology, Atlanta, USA),
Deligne Dimension and Decomposition Formulas from a Geometric Perspec-
tive.
15.45–17.15
Vladimir L. Popov (Steklov Mathematical Institute, Moscow, Russia),
Self-dual Algebraic Varieties, Lie Algebras, and Symmetric Spaces.
Thursday, October 25, 2001
10.30–12.00
Laurent Manivel (Institut Fourier, Universit´e Grenoble, Saint Martin
d’H`eres cedex, France),
The Singularities of Schubert Varieties.
14.00–15.30
Shigeru Mukai (Research Institute of Mathematical Sciences, Kyoto Uni-
versity, Kyoto, Japan),
Minimal Counterexample to Hilbert’s Fourteenth Problem.
15.45–17.15
Ciro Ciliberto (Universita` di Roma Tor Vergata, Roma, Italia),
Varieties with One Apparent Double Point.
Friday, October 26, 2001
10.30–12.00
Jun–Muk Hwang (Korea Institute for Advanced Study, Seoul, Korea),
Automorphism Groups of the Spaces of Lines on Projective Manifolds with
Picard Number 1.
14.00–15.30
Ngaiming Mok (The University of Hong Kong, Hong Kong),
Holomorphic Vector Fields and Deformation Rigidity.
15.45–17.15
Evgueni A. Tevelev (Independent University of Moscow, Moscow, Russia),
Rank Stratification of the Tangent Space of G/P .
List of Contributors
Ciro Ciliberto Daniel Krashen
Universita` di Roma Tor Vergata Department of Mathematics
Dipartimento di Matematica University of California, Los Angeles
Via della Ricerca Scientifica Los Angeles, CA
00133 Roma, Italia 90095, USA
X List of Contributors
Vladimir L. Popov Evgueni A. Tevelev
Steklov Mathematical Institute Department of Mathematics
Russian Academy of Sciences The University of Texas
Gubkina 8 Austin, Texas 78712, USA
Moscow 117966, Russia
Contents
4
Factoriality Of Certain Hypersurfaces Of P
With Ordinary Double Points
Ciro Ciliberto, Vincenzo Di Gennaro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Boundedness For Low Codimensional Subvarieties
Ciro Ciliberto, Vincenzo Di Gennaro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Normality and Non Normality
Of Certain Semigroups and Orbit Closures
Corrado De Concini . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Deformation Rigidity Of the 20-dimensional F4-homogeneous
Space Associated To a Short Root
Jun–Muk Hwang, Ngaiming Mok . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Severi–Brauer Varieties and Symmetric Powers
Daniel Krashen, David J. Saltman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Representation Theory and Projective Geometry
Joseph M. Landsberg, Laurent Manivel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Geometric Realization Of T -shaped Root Systems
and Counterexamples To Hilbert’s Fourteenth Problem
Shigeru Mukai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Self-dual Projective Algebraic Varieties
Associated With Symmetric Spaces
Vladimir L. Popov, Evgueni A. Tevelev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
The Role Of Exotic Affine Spaces In the Classification
Of Homogeneous Affine Varieties
Dennis Snow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169