Versality of algebraic group actions and rational points on twisted varieties
نویسندگان
چکیده
منابع مشابه
Versality of Algebraic Group Actions and Rational Points on Twisted Varieties
We formalize and study several competing notions of versality for an action of a linear algebraic group on an algebraic variety X. Our main result is that these notions of versality are equivalent to various statements concerning rational points on twisted forms of X (existence of rational points, existence of a dense set of rational points, etc.) We give applications of this equivalence in bot...
متن کاملCounting Rational Points on Algebraic Varieties
In these lectures we will be interested in solutions to Diophantine equations F (x1, . . . , xn) = 0, where F is an absolutely irreducible polynomial with integer coefficients, and the solutions are to satisfy (x1, . . . , xn) ∈ Z. Such an equation represents a hypersurface in A, and we may prefer to talk of integer points on this hypersurface, rather than solutions to the corresponding Diophan...
متن کاملCounting Rational Points on Algebraic Varieties
For any N ≥ 2, let Z ⊂ P be a geometrically integral algebraic variety of degree d. This paper is concerned with the number NZ(B) of Q-rational points on Z which have height at most B. For any ε > 0 we establish the estimate NZ(B) = Od,ε,N (B ), provided that d ≥ 6. As indicated, the implied constant depends at most upon d, ε and N . Mathematics Subject Classification (2000): 11G35 (14G05)
متن کاملAlgebraic varieties with many rational points
sometimes we want to understand a typical equation, i.e., a general equation in some family. To draw inspiration (and techniques) from different branches of algebra it is necessary to consider solutions with values in other rings and fields, most importantly, finite fields Fq, finite extensions of Q, or the function fields Fp(t) and C(t). While there is a wealth of ad hoc elementary tricks to d...
متن کاملRational Points and Coxeter Group Actions on the Cohomology of Toric Varieties
We derive a simple formula for the action of a finite crystallographic Coxeter group on the cohomology of its associated complex toric variety, using the method of counting rational points over finite fields, and the Hodge structure of the cohomology. Various applications are given, including the determination of the graded multiplicity of the reflection representation.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Journal of Algebraic Geometry
سال: 2015
ISSN: 1056-3911,1534-7486
DOI: 10.1090/s1056-3911-2015-00644-0