Peter Russell
Office: Burnside Hall, Room 1223
Office Hours: MWF 12:30-13:30 and by appointment
Office Phone: (514) 398-3835
Office FAX: (514) 398-3899


Research Interests

I am interested in the geometry of non-complete algebraic varieties, in particular affine spaces or varieties that closely resemble them. A central theme in my research over the past few years has been the linearizability of algebraic group actions on affine spaces, actions of tori in particular, which lead to quite fascinating geometric problems. Article (5) below gives an overview of the proof that C*-actions on C3 are linearizable, a result to which (3) and (4) make essential contributions. In continuation of the work in (4), Koras and I in (14) characterize C2/G, G a finite group, as a normal, affine, topologically contractible surface with negative Kodaira dimension and a unique quotient singular point. Together with recent work of Gurjar this implies that all two-dimensional quotients of affine n-space by a reductive group are of this form (21). Articles (6), (7), (12), (16) are interrelated and have as a common theme "affine rational surfaces with large automorphism groups (many additive actions)". (12) classifies normal affine surfaces with trivial Makar-Limanov invariant and torsion Picard group for the smooth locus. In (15) we classify certain birational endomorphisms of  C2, linking the problem to the description of  all open  subsets isomorphic to an affine plane in a ruled surface minus a section.  Another recurrent theme in my research has been the study of plane algebraic curves, rational ones in particular. (8) and (10) are in this area, and I am at present pursuing an old goal to understand all closed embeddings of C* into C2, with some progress in sight (17). I also have a longstanding interest in the special problems raised by positive characteristic, whether in the complete or non-complete context. In (20) and a follow up paper in preparation we study the relationship between the topology and the Kodaira dimension of a normal affine surface and the type of the singularities it can carry. This generalizes results of (14).

Recent Papers

  1. With S. M. Bhatwadekar. A note on geometric factoriality, Can. Math. Bull. 38 (4), 390-395, 1995.
  2. Sufficiently homogeneous embeddings of An-1 into An are linear, Can. J. Math. 48 (6), 1286-1295, 1996.
  3. With M. Koras. Contractible threefolds and C*- actions on C3, J. Alg. Geom. 6 671-695, 1997.
  4. With M. Koras. C3 / C* : the smooth locus is not of hyperbolic type, J. Alg. Geom. 8, 603-694, 1999.
  5. With S. Kaliman, M. Koras, L. Makar-Limanov, C*- actions on C3 are linear, Electronic Research Announcements of the AMS 3, 63-71, 1997.
  6. With D. Daigle. Affine rulings of normal rational surfaces, Osaka J. Math. 38 (2001), 37-100.
  7. With D. Daigle. On weighted projective planes and their affine rulings, Osaka J. Math. 38 (2001), 101-150.
  8. Normal crossing systems of plane curves with one place at infinity, Sitzungsberichte der Berliner Mathematischen Gesellschaft 1997-2000, 163-176, CICMA Reports 2000-01.
  9. Some formal aspects of the theorems of Mumford-Ramanujam, Proceedings of the International Colloquium in Algebra, Arithmetic and Geometry 2000, Tata Institute of Fundamental Research, Narosa Publishing, 2002, 557-584.
  10. Multiple planes ramified over one-place curves, CICMA reports 2000-10, Algebra, Arithmetic and Geometry with Applications, volume in honor of S.S. Abhyankar,s 70th birthday 673-685, Springer 2003, pp. 673-685.
  11.  With M. Koras. Linearization problems, in: Algebraic group actions and quotients, XXIII  Autumn School in Algebraic Geometry, Wykno, Poland, pp. 91-107, Hindawi Publishing 2004.
  12. With D. Daigle. On log Q-homology planes and weighted projective planes, Can. J. Math. 56 no. 6, 1145-1189 (2004).
  13. Book Review: Open Algebraic Surfaces, by M. Miyanishi, CMS Notes, v. 33, no. 8, Dec. 2001, 18-20.
  14. With M. Koras. Contractible surfaces with quotient singularities, Transformation Groups 12 no. 2 (2007), 293-340.
  15. With P. Cassou-Nogues. Birational endomorphisms of the affine plane and affine-ruled surfaces,  Proceedings of the Conference in Honor of M. Miyanishi, Kyoto University, T. Hibi, editor, Osaka U. Press 2007, 57-105.
  16. With R. Gurjar, K. Masuda and M. Miyanishi. Affine lines on affine surfaces and the Makar-Limanov invariant, Can. J. Math. 60 (2008), 109-139
  17. With P. Cassou-Nogues, M. Koras. Closed embeddings of C* in C^2, to appear in J. Algebra.
  18. Open Problems, Workshop on Group Actions on Rational Varieties, G. Freudenburg and P. Russell, editors, CICMA Reports 2002-04.
  19. Embedding problems in affine algebraic geometry, in Polynomial Automorphisms and related problems, Lecture notes of ICPA2006, Hanoi, Vietnam Publishing House for Science and Technology, 113-131.
  20. With R. Gurjar, M. Koras, M. Miyanishi, An affine Mumford theorem and related topics, submiited.
  21. With R. Gurjar, M. Koras, Two dimensional quotients of C^n by a reductive group, ERA-MS 15 (2008), 62-64.