arXiv subjects
Publications and source records attributed to Yuri Tschinkel.
We determine the Mori cone of holomorphic symplectic varieties deformation equivalent to the punctual Hilbert scheme on a K3 surface. Our description is given in terms of Markman's extended Hodge lattice.
We study irreducibility of families of degree 4 Del Pezzo surface fibrations over curves.
Manin's conjecture predicts an asymptotic formula for the number of rational points of bounded height on a smooth projective variety in terms of its global geometric invariants. The strongest form of the conjecture implies certain inequalities among geometric invariants of the variety and of its subvarieties. We provide a general geometric framework explaining these phenomena, via the notion of balanced line bundles, and prove the required inequalities for a large class of equivariant compactifications of homogeneous spaces.
We analyze morphisms from pointed curves to K3 surfaces with a distinguished rational curve, such that the marked points are taken to the rational curve, perhaps with specified cross ratios. This builds on work of Mukai and others characterizing embeddings of curves into K3 surfaces via non-abelian Brill-Noether theory. Our study leads naturally to enumerative problems, which we solve in several specific cases. These have applications to the existence of sections of del Pezzo fibrations with prescribed invariants.
We classify quartic del Pezzo surface fibrations over the projective line via numerical invariants, giving explicit examples for small values of the invariants. For generic such fibrations, we describe explicitly the geometry of spaces of sections to the fibration, and mappings to the intermediate Jacobian of the total space. We exhibit examples where these are birational, which has applications to arithmetic questions, especially over finite fields.
Using the Kuga-Satake correspondence we provide an effective algorithm for the computation of the Picard and Brauer groups of K3 surfaces of degree 2 over number fields.
We establish asymptotic formulas for the number of integral points of bounded height on toric varieties.
We establish asymptotic formulas for the number of integral points of bounded height on partial equivariant compactifications of vector groups.
We prove an asymptotic formula for the number of rational points of bounded height on projective equivariant compactifications of $H\G$, where $H$ is a connected simple algebraic group embedded diagonally into $G := H^n$.
We explore connections between birational anabelian geometry and abstract projective geometry. One of the applications is a proof of a version of the birational section conjecture.
We consider quadric surface fibrations over curves, defined over algebraically closed and finite fields. Our goal is to understand, in geometric terms, spaces of sections for such fibrations. We analyze varieties of maximal isotropic subspaces in the fibers as P^1-bundles over the discriminant double cover. When the P^1-bundle is suitably stable, we deduce effective estimates for the heights of sections over finite fields satisfying various approximation conditions. We also discuss the behavior of the spaces of sections as the base of the fibration acquires singularities.
We apply the theory of height zeta functions to study the asymptotic distribution of rational points of bounded height on projective equivariant compactifications of semi-direct products.
We study the asymptotic distribution of integral points of bounded height on partial bi-equivariant compactifications of semi-simple groups of adjoint type.
We seek to characterize homology classes of Lagrangian projective spaces embedded in irreducible holomorphic-symplectic manifolds, up to the action of the monodromy group. This paper addresses the case of manifolds deformation-equivalent to the Hilbert scheme of length-three subschemes of a K3 surface. The class of the projective space in the cohomology ring has prescribed intersection properties, which translate into Diophantine equations. Possible homology classes correspond to integral points on an explicit elliptic curve; our proof entails showing that the only such point is two-torsion.
We survey recent developments in the Birational Anabelian Geometry program aimed at the reconstruction of function fields of algebraic varieties over algebraically closed fields from pieces of their absolute Galois groups.
We propose a general framework governing the intersection properties of extremal rays of irreducible holomorphic symplectic manifolds under the Beauville-Bogomolov form. Our main thesis is that extremal rays associated to Lagrangian projective subspaces control the behavior of the cone of curves. We explore implications of this philosophy for examples like Hilbert schemes of points on K3 surfaces and generalized Kummer varieties. We also collect evidence supporting our conjectures in specific cases.
We study Brauer-Manin obstructions to the Hasse principle and to weak approximation on algebraic surfaces over number fields.
We analyze the intersection properties of projective planes embedded in generalized Kummer fourfolds, with a view toward classifying the homology classes represented by these submanifolds.