Rational points on certain homogeneous varieties

Rational points on varieties

On Tangential Varieties of Rational Homogeneous Varieties

We determine which tangential varieties of homogeneously embedded rational homogeneous varieties are spherical. We determine the homogeneous coordinate rings and rings of covariants of the tangential varieties of homogenously embedded compact Hermitian symmetric spaces (CHSS). We give bounds on the degrees of generators of the ideals of tangential varieties of CHSS and obtain more explicit info...

Homogeneous varieties - zero cycles of degree one versus rational points

Examples of projective homogeneous varieties over the field of Laurent series over p-adic fields which admit zero cycles of degree one and which do not have rational points are constructed. Let k be a field and X a quasi-projective variety over k. Let Z0(X) denote the group of zero cycles on X and deg : Z0(X) → Z the degree homomorphism which associates to a closed point x of X, the degree [k(x...

Counting 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...

Counting Rational Points on Ruled Varieties

In this paper, we prove a general result computing the number of rational points of bounded height on a projective variety V which is covered by lines. The main technical result used to achieve this is an upper bound on the number of rational points of bounded height on a line. This upper bound is such that it can be easily controlled as the line varies, and hence is used to sum the counting fu...