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