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Ulrich Derenthal

Publications and source records attributed to Ulrich Derenthal.

At least 37 records · Page 2Linked to original sources
arXiv2014

Universal torsors and values of quadratic polynomials represented by norms

Let $K/k$ be an extension of number fields, and let $P(t)$ be a quadratic polynomial over $k$. Let $X$ be the affine variety defined by $P(t) = N_{K/k}(\mathbf{z})$. We study the Hasse principle and weak approximation for $X$ in three cases. For $[K:k]=4$ and $P(t)$ irreducible over $k$ and split in $K$, we prove the Hasse principle and weak approximation. For $k=\mathbb{Q}$ with arbitrary $K$, we show that the Brauer-Manin obstruction to the Hasse principle and weak approximation is the only one. For $[K:k]=4$ and $P(t)$ irreducible over $k$, we determine the Brauer group of smooth proper models of $X$. In a case where it is non-trivial, we exhibit a counterexample to weak approximation.

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arXiv2013

Singular del Pezzo surfaces whose universal torsors are hypersurfaces

We classify all generalized del Pezzo surfaces (i.e., minimal desingularizations of singular del Pezzo surfaces containing only rational double points) whose universal torsors are open subsets of hypersurfaces in affine space. Equivalently, their Cox rings are polynomial rings with exactly one relation. For all 30 types with this property, we describe the Cox rings in detail. These explicit descriptions can be applied to study Manin's conjecture on the asymptotic behavior of the number of rational points of bounded height for singular del Pezzo surfaces, using the universal torsor method.

math.AG↗
arXiv2013

Equivariant compactifications of two-dimensional algebraic groups

We classify generically transitive actions of semidirect products of an additive and a multiplicative group on the projective plane. Motivated by the program to study the distribution of rational points on del Pezzo surfaces (Manin's conjecture), we determine all (possibly singular) del Pezzo surfaces that are equivariant compactifications of homogeneous spaces for such semidirect products.

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arXiv2013

Counting imaginary quadratic points via universal torsors

A conjecture of Manin predicts the distribution of rational points on Fano varieties. We provide a framework for proofs of Manin's conjecture for del Pezzo surfaces over imaginary quadratic fields, using universal torsors. Some of our tools are formulated over arbitrary number fields. As an application, we prove Manin's conjecture over imaginary quadratic fields K for the quartic del Pezzo surface S of singularity type A_3 with five lines given in P^4 by the equations vw - xy = vy + wy + xz = 0.

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arXiv2013

On the factor alpha in Peyre's constant

For an arbitrary del Pezzo surface S, we compute alpha(S), which is the volume of a certain polytope in the dual of the effective cone of S, using Magma and Polymake. The constant alpha(S) appears in Peyre's conjecture for the leading term in the asymptotic formula for the number of rational points of bounded height on S over number fields.

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arXiv2012

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.

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arXiv2010

Cox Rings

This is the first chapter of an introductory text under construction; further chapters are available via the authors' web pages. Our aim is to provide an elementary access to Cox rings and their applications in algebraic and arithmetic geometry. Any comments and suggestions on this draft will be highly appreciated.

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arXiv2010

Singular del Pezzo surfaces that are equivariant compactifications

We determine which singular del Pezzo surfaces are equivariant compactifications of G_a^2, to assist with proofs of Manin's conjecture for such surfaces. Additionally, we give an example of a singular quartic del Pezzo surface that is an equivariant compactification of a semidirect product of G_a and G_m.

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arXiv2009

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.

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arXiv2007

Looking for rational curves on cubic hypersurfaces

These are the substantially expanded notes of the lectures of JK at the summer school "Higher-Dimensional Geometry over Finite Fields" in Göttingen, June 2007. The first part gives an overview of the methods. The main new result is the construction of rational curves passing through a given collection of points on smooth cubic hypersurfaces over finite fields.

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arXiv2007

The nef cone volume of generalized Del Pezzo surfaces

We compute a naturally defined measure of the size of the nef cone of a Del Pezzo surface. The resulting number appears in a conjecture of Manin on the asymptotic behavior of the number of rational points of bounded height on the surface. The nef cone volume of a Del Pezzo surface Y with (-2)-curves defined over an algebraically closed field is equal to the nef cone volume of a smooth Del Pezzo surface of the same degree divided by the order of the Weyl group of a simply-laced root system associated to the configuration of (-2)-curves on Y. When Y is defined over a non-closed field of characteristic 0, a similar result holds, except that the associated root system is no longer necessarily simply-laced.

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arXiv2007

On a constant arising in Manin's conjecture for Del Pezzo surfaces

For split smooth Del Pezzo surfaces, we analyse the structure of the effective cone and prove a recursive formula for the value of alpha, appearing in the leading constant as predicted by Peyre of Manin's conjecture on the number of rational points of bounded height on the surface. Furthermore, we calculate alpha for all singular Del Pezzo surfaces of degree at least 3.

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arXiv2007

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.

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arXiv2006

Universal torsors over Del Pezzo surfaces and rational points

We discuss Manin's conjecture concerning the distribution of rational points of bounded height on Del Pezzo surfaces, and its refinement by Peyre, and explain applications of universal torsors to counting problems. To illustrate the method, we provide a proof of Manin's conjecture for the unique split singular quartic Del Pezzo surface with a singularity of type D4. (Lectures at the summer school "Equidistribution in number theory", Montreal, August 2005)

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arXiv2006

On the Cox ring of Del Pezzo surfaces

Let S_r be the blow-up of P^2 in r general points, i.e., a smooth Del Pezzo surface of degree 9-r. For r <= 7, we determine the quadratic equations defining its Cox ring explicitly. The ideal of the relations in Cox(S_8) is calculated up to radical. As conjectured by Batyrev and Popov, all the generating relations are quadratic.

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