WHEN IS THE AUTOMORPHISM GROUP OF AN AFFINE VARIETY NESTED?

The automorphism group of an affine quadric

We determine the automorphism group for a large class of affine quadrics over a field, viewed as affine algebraic varieties. The proof uses a fundamental theorem of Karpenko’s in the theory of quadratic forms [13], along with some useful arguments of birational geometry. In particular, we find that the automorphism group of the n-sphere {x0 + · · · + x 2 n = 1} over the real numbers is just the...

THE AUTOMORPHISM GROUP OF FINITE GRAPHS

Let G = (V,E) be a simple graph with exactly n vertices and m edges. The aim of this paper is a new method for investigating nontriviality of the automorphism group of graphs. To do this, we prove that if |E| >=[(n - 1)2/2] then |Aut(G)|>1 and |Aut(G)| is even number.

Finite flag-transitive affine planes with a solvable automorphism group

In this paper, we consider finite flag-transitive affine planes with a solvable automorphism group. Under a mild number-theoretic condition involving the order and dimension of the plane, the translation complement must contain a linear cyclic subgroup that either is transitive or has two equal-sized orbits on the line at infinity. We develop a new approach to the study of such planes by associ...

The affine flag variety

When is a polynomial ideal binomial after an ambient automorphism?

Can an ideal I in a polynomial ring k[x] over a field be moved by a change of coordinates into a position where it is generated by binomials x − λx with λ ∈ k, or by unital binomials (i.e., with λ = 0 or 1)? Can a variety be moved into a position where it is toric? By fibering the G-translates of I over an algebraic group G acting on affine space, these problems are special cases of questions a...