arXiv subjects

Zinovy Reichstein

Publications and source records attributed to Zinovy Reichstein.

76 records · Page 5Linked to original sources

arXiv1999

Parusiński's "Key Lemma" via algebraic geometry

The following ``Key Lemma'' plays an important role in Parusinski's work on the existence of Lipschitz stratifications in the class of semianalytic sets: For any positive integer n, there is a finite set of homogeneous symmetric polynomials W_1,...,W_N in Z[x_1,...,x_n] and a constant M >0 such that |dx_i/x_i| \le M \max_{j = 1,..., N} |dW_j/W_j| as densely defined functions on the tangent bundle of C^n. We give a new algebro-geometric proof of this result.

math.AG↗

arXiv1999

On a property of special groups

Let G be an algebraic group defined over an algebraically closed field k of characteristic zero. We give a simple proof of the following result: if H^1(L, G) = {1} for some finitely generated field extension L/k of transcendence degree \ge 3 then H^1(K, G) = {1} for every field extension K/k.

math.AG↗

arXiv1999

Splitting fields of G-varieties

Let $G$ be an algebraic group, $X$ a generically free $G$-variety, and $K=k(X)^G$. A field extension $L$ of $K$ is called a splitting field of $X$ if the image of the class of $X$ under the natural map $H^1(K, G) \mapsto H^1(L, G)$ is trivial. If $L/K$ is a (finite) Galois extension then $\Gal(L/K)$ is called a splitting group of $X$. We prove a lower bound on the size of a splitting field of $X$ in terms of fixed points of nontoral abelian subgroups of $G$. A similar result holds for splitting groups. We give a number of applications, including a new construction of noncrossed product division algebras.

math.AG↗

arXiv1999

Essential dimensions of algebraic groups and a resolution theorem for G-varieties

Let G be an algebraic group and let X be a generically free G-variety. We show that X can be transformed, by a sequence of blowups with smooth G-equivariant centers, into a G-variety X' with the following property: the stabilizer of every point of X' is isomorphic to a semidirect product of a unipotent group U and a diagonalizable group A. As an application of this and related results, we prove new lower bounds on essential dimensions of some algebraic groups. We also show that certain polynomials in one variable cannot be simplified by a Tschirnhaus transformation.

math.AG↗