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
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
- With S. M. Bhatwadekar. A
note on geometric factoriality, Can. Math. Bull. 38 (4), 390-395, 1995.
- Sufficiently homogeneous
embeddings of An-1 into An are linear,
Can. J. Math. 48 (6), 1286-1295, 1996.
- With M. Koras. Contractible
threefolds and C*- actions on C3, J. Alg. Geom. 6
- With M. Koras. C3
/ C* : the smooth locus is not of hyperbolic type, J. Alg. Geom. 8,
- With S. Kaliman, M. Koras, L.
Makar-Limanov, C*- actions on C3 are linear,
Electronic Research Announcements of the AMS 3, 63-71, 1997.
- With D. Daigle. Affine
rulings of normal rational surfaces, Osaka J. Math. 38 (2001), 37-100.
- With D. Daigle. On weighted
projective planes and their affine rulings, Osaka J. Math. 38 (2001),
- Normal crossing systems of
plane curves with one place at infinity, Sitzungsberichte der Berliner
Mathematischen Gesellschaft 1997-2000, 163-176, CICMA Reports
- 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.
- 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.
- 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.
- With D. Daigle. On log Q-homology
planes and weighted projective planes, Can. J. Math. 56 no. 6, 1145-1189
- Book Review: Open Algebraic
Surfaces, by M. Miyanishi, CMS Notes, v. 33, no. 8, Dec. 2001, 18-20.
- With M. Koras. Contractible surfaces
with quotient singularities, Transformation Groups 12 no. 2 (2007),
- 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
- 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
- With P. Cassou-Nogues, M.
Koras. Closed embeddings of C* in C^2, to appear in J. Algebra.
- Open Problems, Workshop on
Group Actions on Rational Varieties, G. Freudenburg and P. Russell,
editors, CICMA Reports 2002-04.
- Embedding problems in affine
algebraic geometry, in Polynomial Automorphisms and related problems,
Lecture notes of ICPA2006, Hanoi, Vietnam Publishing House for Science and
- With R. Gurjar, M. Koras, M.
Miyanishi, An affine Mumford theorem and related topics, submiited.
- With R. Gurjar, M. Koras, Two
dimensional quotients of C^n by a reductive group, ERA-MS 15 (2008),