Introduction to Algebraic Geometry and Algebraic Groups

AbstractThe well-known fiber dimension theorem in algebraic geometry says that for every morphism f:X→Y of integral schemes of finite type the dimension of every fiber of f is at least dim X−dim Y. This has recently been generalized by Brosnan, Reichstein and Vistoli to certain morphisms of algebraic stacks f:𝒳→𝒴, where the usual dimension is replaced by essential dimension. We will prove a general version for morphisms of categories fibered in groupoids. Moreover, we will prove a variant of this theorem, where essential dimension and canonical dimension are linked. These results let us relate essential dimension to canonical dimension of algebraic groups. In particular, using the recent computation of the essential dimension of algebraic tori by MacDonald, Meyer, Reichstein and the author, we establish a lower bound on the canonical dimension of algebraic tori.

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On the involution fixity of exceptional groups of Lie type

International Journal of Algebra and Computation ◽

10.1142/s0218196718500200 ◽

2018 ◽

Vol 28 (03) ◽

pp. 411-466 ◽

Cited By ~ 3

Author(s):

Timothy C. Burness ◽

Adam R. Thomas

Keyword(s):

Algebraic Geometry ◽

Fixed Points ◽

Algebraic Groups ◽

Natural Analogue ◽

Transitive Action ◽

Exceptional Groups ◽

Closed Fields ◽

Suzuki Group ◽

Algebraically Closed Fields ◽

Groups Of Lie Type

The involution fixity [Formula: see text] of a permutation group [Formula: see text] of degree [Formula: see text] is the maximum number of fixed points of an involution. In this paper we study the involution fixity of primitive almost simple exceptional groups of Lie type. We show that if [Formula: see text] is the socle of such a group, then either [Formula: see text], or [Formula: see text] and [Formula: see text] is a Suzuki group in its natural [Formula: see text]-transitive action of degree [Formula: see text]. This bound is best possible and we present more detailed results for each family of exceptional groups, which allows us to determine the groups with [Formula: see text]. This extends recent work of Liebeck and Shalev, who established the bound [Formula: see text] for every almost simple primitive group of degree [Formula: see text] with socle [Formula: see text] (with a prescribed list of exceptions). Finally, by combining our results with the Lang–Weil estimates from algebraic geometry, we determine bounds on a natural analogue of involution fixity for primitive actions of exceptional algebraic groups over algebraically closed fields.

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AX–SCHANUEL CONDITION IN ARBITRARY CHARACTERISTIC

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748017000378 ◽

2017 ◽

Vol 18 (06) ◽

pp. 1157-1213

Author(s):

Piotr Kowalski

Keyword(s):

Algebraic Geometry ◽

Algebraic Group ◽

Algebraic Variety ◽

Positive Characteristic ◽

Algebraic Groups ◽

General Context ◽

Arbitrary Characteristic ◽

Analytic Subgroup

We prove a positive characteristic version of Ax’s theorem on the intersection of an algebraic subvariety and an analytic subgroup of an algebraic group [Ax, Some topics in differential algebraic geometry. I. Analytic subgroups of algebraic groups, Amer. J. Math. 94 (1972), 1195–1204]. Our result is stated in a more general context of a formal map between an algebraic variety and an algebraic group. We derive transcendence results of Ax–Schanuel type.

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