43 research outputs found
Counting integral points on universal torsors
Manin's conjecture for the asymptotic behavior of the number of rational
points of bounded height on del Pezzo surfaces can be approached through
universal torsors. We prove several auxiliary results for the estimation of the
number of integral points in certain regions on universal torsors. As an
application, we prove Manin's conjecture for a singular quartic del Pezzo
surface.Comment: 48 page
Universal torsors of Del Pezzo surfaces and homogeneous spaces
Let Cox(S) be the homogeneous coordinate ring of the blow-up S of P^2 in r
general points, i.e., a smooth Del Pezzo surface of degree 9-r. We prove that
for r=6 and 7, Proj(Cox(S)) can be embedded into G/P, where G is an algebraic
group with root system given by the primitive Picard lattice of S and P is a
certain maximal parabolic subgroup in G.Comment: 15 page
Gaussian rational points on a singular cubic surface
Manin's conjecture predicts the asymptotic behavior of the number of rational
points of bounded height on algebraic varieties. For toric varieties, it was
proved by Batyrev and Tschinkel via height zeta functions and an application of
the Poisson formula. An alternative approach to Manin's conjecture via
universal torsors was used so far mainly over the field Q of rational numbers.
In this note, we give a proof of Manin's conjecture over the Gaussian rational
numbers Q(i) and over other imaginary quadratic number fields with class number
1 for the singular toric cubic surface defined by t^3=xyz.Comment: 16 page
Counting Integral Points on Universal Torsors
Manin's conjecture for the asymptotic behavior of the number of rational points of bounded height on del Pezzo surfaces can be approached through universal torsors. We prove several auxiliary results for the estimation of the number of integral points in certain regions on universal torsors. As an application, we prove Manin's conjecture for a singular quartic del Pezzo surfac