ProfessorThe Department of Mathematics and Statistics
Burnside Hall, Room 1223
805 Sherbrooke W.
H3A 0B9, Canada
Phone: (514) 398-3835
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 (25) below gives an overview of the proof that $\C^*$-actions on $\C^3$ are linearizable, a result to which (23) and (24) make essential contributions. In (43) we extend this result to non-algebraically closed fields in the novel context of studying separable forms of $\G_m$-actions on $\A^3$. In (44) we study linearizability in a relative context, in particular for reductive actions on $\A^3$ that admit a semi-invariant variable, making use of a very general generic equivalence theorem for for morphisms with isomorphic closed fibers. Specific results, modest first steps in the study of reductive actions with general orbit of co-dimension 2, are the linearizability of finite group actions on $\C^3$ with a semi-invariant variable (48) and $\C^*$-actions on $\C^4$ with an invariant variable (49).
In continuation of the work in (24), Koras and I in (33) characterize $\C^2/G$, $G$ 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 (37). Articles (26), (27), (31), (36) are interrelated and have as a common theme "affine rational surfaces with large automorphism groups (many additive actions)". (31) classifies normal affine surfaces with trivial Makar-Limanov invariant and torsion Picard group for the smooth locus. In (39) and (40) 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 (33). We have recently begun similar investigations in dimension 3 (42). In (35) we classify certain birational endomorphisms of $\C^2$, 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. (28) and (30) are in this area, and I am pursuing an old goal to understand all closed embeddings of $\C^*$ into $\C^2$, with some progress in sight (38) (41) (47). I also have a longstanding interest in the special problems raised by positive characteristic, whether in the complete or non-complete context. In collaboration with R. Ganong and A. Sathaye I have recently taken up again the problem of understanding lines in the plane in positive characteristic.
23. With M. Koras. Contractible threefolds and $\C^*$-actions on $\C^3$ , J. Alg. Geom. 6, 671-695, 1997.
24. With M. Koras. $\C^3/\C^*$: the smooth locus is not of hyperbolic type, J. Alg. Geom. 8, 603-694, 1999.
25. With S. Kaliman, M. Koras, L. Makar-Limanov. $\C^*$-actions on $\C^3$ are linear, Electronic Research Announcements of the AMS 3, 63-71, 1997.
26. With D. Daigle. Affine rulings of normal rational surfaces, Osaka J. Math., 38 (1), 2001 pp. 37-100.
27. With. D. Daigle. On weighted projective planes and their affine rulings, Osaka J. Math., 38 (1), 2001, pp. 101-150.
28. Normal crossing systems of plane curves with one place at infinity, Sitzungsberichte der Berliner Mathematischen Gesellschaft, 1997-2000, pp. 163-176.
29. Some formal aspects of the theorems of Mumford-Ramanujan, Proceedings of the International Colloquium in Algebra, Arithmetic and Geometry 2000, Tata Institute of Fundamental Research, Narosa Publishing 2002, pp. 557-584.
30. Multiple planes ramified over one-place curves, Algebra, Arithmetic and Geometrey, papers from S.S. Abhyankar's 70th birthday conference, Springer, 2003, pp. 673-685.
31. With D. Daigle, On log Q-homology planes and weighted projective planes, Canadian Journal of Mathematics 56 no.6, 2004, pp. 1145-1189.
32. 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.
33. With M. Koras, Contractible surfaces with a quotient singularity,Transformation Groups, 12(2007), no. 2, 293-340.
34. Embedding problems in affine algebraic geometry, in Polynomial Automorphisms and related problems, ICPA 2006, 113-135, Publishing House for Science and Technology, 2007. Hanoi, Vietnam.
35. With P. Cassou-Nogues. Birational endomorphisms of the affine plane and affine-ruled surfaces, Proceedings of the conference in honour of M. Miyanishi, Kyoto, 2003, Affine algebraic geometry, 57-105, Osaka Univ. Press, Osaka, 2007
36. With R. Gurjar, K. Masuda, M. Miyanishi, Affine lines on affine surfaces and the Makar-Limanov Invariant, Canadian Journal of Mathematics 60, 2008, 109-139.
37. With R. Gurjar, M. Koras, Two dimensional quotients of $\C^n$ by a reductive group, ERA-MS, 15 (2008), 62-64.
38. With P. Cassou-Nogues, M. Koras, Closed Embeddings of $\C^*$ in$\C^2$, J. Algebra 322 (2009), 2950-3002.
39. With R. Gurjar, M. Koras, M. Miyanishi, Affine normal surfaces with simply connected smooth locus, Math. Ann. 353(2012) no. 1, 127-144.
40. With R. Gurjar, M. Koras, M. Miyanishi, A homology plane of general type can have at most a cyclic quotient singularity, J. Alg. Geom. 23(2014), 1-62, posted electronically May 2013.
41. With M. Koras, Some properties of $\C^*$ in $\C^2$, in Affine Algebraic Geometry, Proceedings of a Conference in Honour of M. Miyanishi's 70th birthday, World Scientific Publishing, 2013.
42. With R. Gurjar, M. Koras, K. Masuda, M. Miiyanishi, $\A^1_*$-fibrations on affine threefolds, in Affine Algebraic Geometry, Proceedings of a Conference in Honour of M. Miyanishi's 70th birthday, World Scientific Publishing, 2013.
43. With M. Koras, Separable forms of $\G_m$-actions on $\A^3$, Transformation Groups 18 (2013), no. 4, 1155-1103.
44. With H. Kraft, Families of group actions, generic isotriviality and linearization, Transformation Groups 19 (2014) no.3, 779-792.
45. Cancellation, in: Automorphisms in Birational and Affine Geometry, Proceedings of the Levico Terme conference GABAG2012 (2014), 495-518.
46. With A. Sathaye, 40 Years of the Epimorphism Theorem, Feature Article, Newsletter of the European Mathematical Society, December 2013.
47. With M. Koras, K. Palka, The geometry of sporadic embeddings of $C^*$ in $C^2$, submitted for publication.
48. With H.Kraft, G. Schwarz, Finite group actions on $\C^3$ with a semi-invariant variable, in preparation.
49. With R. Gurjar, M. Koras, M. Miyanishi, Affine 3-folds admitting $\G_a$-actions, in preparation.