454 research outputs found
On automorphism groups of affine surfaces
This is a survey on the automorphism groups in various classes of affine
algebraic surfaces and the algebraic group actions on such surfaces. Being
infinite-dimensional, these automorphism groups share some important features
of algebraic groups. At the same time, they can be studied from the viewpoint
of the combinatorial group theory, so we put a special accent on
group-theoretical aspects (ind-groups, amalgams, etc.). We provide different
approaches to classification, prove certain new results, and attract attention
to several open problems.Comment: Proposition 2.10 from the previous version (published in Algebraic
Varieties and Automorphism Groups, ASPM 75) deleted. There is a mistake in
the proof kindly indicated by J.-P. Furter; the validity of the result
remains open. This does not affect the rest of the pape
Symplectic Lefschetz fibrations on S^1 x M^3
In this paper we classify symplectic Lefschetz fibrations (with empty base
locus) on a four-manifold which is the product of a three-manifold with a
circle. This result provides further evidence in support of the following
conjecture regarding symplectic structures on such a four-manifold: if the
product of a three-manifold with a circle admits a symplectic structure, then
the three-manifold must fiber over a circle, and up to a self-diffeomorphism of
the four-manifold, the symplectic structure is deformation equivalent to the
canonical symplectic structure determined by the fibration of the
three-manifold over the circle.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol4/paper18.abs.htm
Slices of the unitary spread
We prove that slices of the unitary spread of Q(+)(7, q), q equivalent to 2 (mod 3), can be partitioned into five disjoint classes. Slices belonging to different classes are non-equivalent under the action of the subgroup of P Gamma O+(8, q) fixing the unitary spread. When q is even, there is a connection between spreads of Q(+)(7, q) and symplectic 2-spreads of PG(5, q) (see Dillon, Ph.D. thesis, 1974 and Dye, Ann. Mat. Pura Appl. (4) 114, 173-194, 1977). As a consequence of the above result we determine all the possible non-equivalent symplectic 2-spreads arising from the unitary spread of Q(+)(7, q), q = 2(2h+1). Some of these already appeared in Kantor, SIAM J. Algebr. Discrete Methods 3(2), 151-165, 1982. When q = 3(h), we classify, up to the action of the stabilizer in P Gamma O(7, q) of the unitary spread of Q(6, q), those among its slices producing spreads of the elliptic quadric Q(-)(5, q)
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