# Zinovy Reichstein

#### Relevant Degree Programs

## Graduate Student Supervision

##### Doctoral Student Supervision (Jan 2008 - Nov 2019)

The overarching theme of this thesis is to assign, and sometimes find, numerical values which reflect complexity of algebraic objects. The main objects of interest are field extensions of finite degree, and more generally, etale algebras of finite degree over a ring.Of particular interest to us is the invariant known as essential dimension. The essential dimension of separable field extensions was introduced by J. Buhler and Z. Reichstein in their landmark paper. A major (still) open problem arising from that work is to determine the essential dimension of a general separable field extension of degree n (or equivalently, the essential dimension of the symmetric group). Loosening the separability assumption we arrive at the case of inseparable field extensions. In the first part of this thesis we study the problem of determining the essential dimension of inseparable field extensions. In the second part of this thesis, we study the essential dimension of the double covers of symmetric groups and alternating groups, respectively. These groups were first studied by I. Schur and their representations are closely related to projective representations of symmetric and alternating groups. In the third part, we study the problem of determining the minimum number of generators of an etale algebra over a ring. The minimum of number of generators of an etale algebra is a natural measure of its complexity.

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This thesis consists of three parts. The common theme is finite group actions on algebraic curves defined over an arbitrary field k.In Part I we classify finite group actions on irreducible conic curves defined over k. Equivalently, we classify finite (constant) subgroups of SO(q) up to conjugacy, where q is a nondegenerate quadratic form of rank 3 defined over k. In the case where k is the field of complex numbers, these groups were classified by F. Klein at the end of the 19th century. In recent papers of A. Beauville and X. Faber, this classification is extended to the case where k is arbitrary, but q is split. We further extend their results by classifying finite subgroups of SO(q) for any base field k of characteristic ≠ 2 and any nondegenerate ternary quadratic form q.In Part II we address the Hyperelliptic Lifting Problem (or HLP): Given a faithful G-action on ℙ¹ defined over k and an exact sequence 1 → μ₂ → Gʹ→ G → 1, determine the conditions for the existence of a hyperelliptic curve C/k endowed with a faithful Gʹ-action that lifts the prescribed G-action on the projective line. Alternatively, this problem may be regarded as the Galois embedding problem given by the surjection Gʹ ↠ G and the G-Galois extension k(ℙ¹)/k(ℙ¹)G. In this thesis, we find a complete solution to the HLP in characteristic 0 for every faithful group action on ℙ¹ and every exact sequence as above.In Part III we determine whether, given a finite group G and a base field k of characteristic 0, there exists a strongly incompressible G-curve defined over k. Recall that a G-curve is an algebraic curve endowed with the action of a finite group G. A faithful G-curve C is called strongly incompressible if every dominant G-equivariant rational map of C onto a faithful G-variety is birational. We prove that strongly incompressible G-curves exist if G cannot act faithfully on the projective line over k. On the other hand, if G does embed into PGL₂ over k, we show that the existence of strongly incompressible G-curves depends on finer arithmetic properties of k.

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The essential dimension of an algebraic group G is a measure of the complexity of G-torsors. One of the central open problems in the theory of essential dimension is to compute the essential dimension of PGL_n, whose torsors correspond to central simple algebras up to isomorphism. In this thesis, we study the essential dimension of groups of the form G/μ, where G is a reductive algebraic group satisfying certain properties, and μ is a central subgroup of G. In particular, we consider the case G=GL_(n₁) × ⋯ × GL_(n_r ) where each n_i is a power of a single prime p, which is a generalization of the group PGL_(p^a )=GL_(p^a )/G_m. We will see that torsors for G/μ correspond to tuples of central simple algebras satisfying certain properties. Surprisingly, computing the essential dimension of G/μ becomes easier when r≥3.Using techniques from Galois cohomology, representation theory and the essential dimension of stacks, we give upper and lower bounds for the essential dimension of G/μ. To do this, we first attach a linear ‘code’ to the central subgroup μ, and define a weight function on this code. Our upper and lower bounds are given in terms of a minimal weight generator matrix for the code. In some cases we can determine the exact value of the essential dimension of G/μ.

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Informally, essential dimension is the minimal number of parameters required to define an algebraic object. This invariant has numerous connections to Galois cohomology, linear algebraic groups and birational geometry. In particular, the essential dimension of finite groups has connections to the Noether problem, inverse Galois theory and the simplification of polynomials via Tschirnhaus transformations.This thesis studies finite groups of low essential dimension using methods from birational geometry. Specifically, the main results are a classification of finite groups of essential dimension 2, and a proof that the alternating and symmetric groups on 7 letters have essential dimension 4.

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We study the essential dimension of linear algebraic groups. For a group G, essential dimension is a measure for the complexity of G-torsors or, more generally, the complexity of any algebraic or geometric structure with automorphism group G. This makes essential dimension a powerful invariant with many interesting and surprising connections to problems in algebra and geometry.We show that for various classes of groups, including finite (algebraic) groups and algebraic tori, the essential dimension is related to minimal faithful representations. In many cases this renders the exact value of the essential dimension computable and we explore several of its consequences. An important open problem is the essential dimension of the projective linear group PGLn. This topic is closely related to the structure theory of central simple algebras, which may be viewed as twisted forms of the algebra of n x n matrices.We study central simple algebras with additional structure such as a distinguished Galois subfield. We prove new bounds on the essential dimension of these algebras and, as a corollary, of the group PGLn.

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##### Master's Student Supervision (2010 - 2018)

In this thesis, we study the problem of finding birational models of projective G-varieties with tame stabilizers. This is done with linearizations, so that each birational model may be considered as a (modular) compactification of an orbit space (of properly stable points).The main portion of the thesis is a re-working of a result in Kirwan's paper "Partial Desingularisations of Quotients of Nonsingular Varieties and their Betti Numbers", written in a purely algebro-geometric language. As such, the proofs are novel and require the Luna Slice Theorem as their primary tool.Chapter 1 is devoted to preliminary material on Geometric Invariant Theory and the Luna Slice Theorem.In Chapter 2, we present and prove a version of "Kirwan's procedure". This chapter concludes with an outline of some differences between the current thesis and Kirwan's original paper.In Chapter 3, we combine the results from Chapter 2 and a result from a paper by Reichstein and Youssin to provide another type of birational model with tame stabilizers (again, with respect to an original linearization).

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Let E/F be a field extension of degree n. A classical problem is to find a generating element in E whose characteristic polynomial over F is as simple is possible. An 1861 theorem of Ch. Hermite [5] asserts that for every separable field E/F of degree n there exists an element a ∈ E whose characteristic polynomial is of the formf(x) = x⁵ +b₂x³ +b₄x+b₅or equivalently, tr{E/F}(a) = tr{E/F}(a³) = 0. A similar result for extensions of degree 6 was proven by P. Joubert in 1867; see [6].In this thesis we ask if these results can be extended to field extensions of larger degree. Specifically, we give a necessary and sufficient condition for a field F, a prime p and an integer n ≥ 3 to have the following property: Every separable field extension E/F of degree n contains an element a ∈ E such that a generates E over F, and tr{E/F}(a) = tr{E/F}(a^p) = 0.As a corollary we show for infinitely many new values of n that the theorems of Hermite and Joubert do not extend to field extensions of degree n. We conjecture the same for more values of n and provide computational evidence for a large number of these.

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In this thesis we first prove that the algebra of invariants for reductive groups over the base field complex numbers are finitely generated. Then we focus on invariant algebras of finite groups. After showing the natural relation between invariant algebras and reflections in the group we prove the Chevalley-Shephard-Todd theorem. We conclude with classification of complex finite reflection groups and some examples. Throughout the thesis we follow similar arguments as in Springer[8] and Kane[6] where we give full details on the arguments.

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Galois cohomology is an important tool in algebra that can be used to classify isomorphism classes of algebraic objects over a field. In this thesis, we show that many objects of interest in algebra can be described in cohomological terms. Some objects that we discuss include quadratic forms, Pfister forms, G-crossed product algebras, and tuples of central simple algebras. We also provide cohomological interpretation to some induced maps that naturally occur in short exact sequences.

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