Stable rationality in smooth families of threefolds
We exhibit families of smooth projective threefolds with both stably rational and non stably rational fibers.
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arXiv subjects
Yuri Tschinkel
Publications and source records attributed to Yuri Tschinkel.
Noether's problem and descent
We study Noether's problem from the perspective of torsors under linear algebraic groups and descent.
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Stable rationality of Brauer-Severi surface bundles
For sufficiently ample linear systems on rational surfaces we show that a very general associated Brauer-Severi surface bundle is not stably rational.
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Homomorphisms of multiplicative groups of fields preserving algebraic dependence
We study homomorphisms of multiplicative groups of fields preserving algebraic dependence and show that such homomorphisms give rise to valuations.
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Models of Brauer-Severi surface bundles
We study Brauer-Severi surface bundles over smooth projective varieties via root stacks, with a view towards applications to failure of stable rationality.
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Specialization of birational types
We introduce a simplified version of the Grothendieck group of algebraic varieties and use it to show that birational types specialize in families with mild singularities of the central fiber.
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Torsion of elliptic curves and unlikely intersections
We study effective versions of unlikely intersections of images of torsion points of elliptic curves on the projective line.
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Intersections of three quadrics in $\mathbb{P}^7$
We study rationality properties of smooth complete intersections of three quadrics in $\mathbb{P}^7$. We exhibit a smooth family of such intersections with both rational and non-rational fibers.
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A very general quartic double fourfold is not stably rational
We prove that a very general double cover of the projective four-space, ramified in a quartic threefold, is not stably rational.
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Stable rationality of quadric surface bundles over surfaces
We study rationality properties of quadric surface bundles over the projective plane. We exhibit families of smooth projective complex fourfolds of this type over connected bases, containing both rational and non-rational fibers.
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On stable rationality of Fano threefolds and del Pezzo fibrations
We prove that very general non-rational Fano threefolds which are not birational to cubic threefolds are not stably rational.
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Rational points on K3 surfaces and derived equivalence
We study K3 surfaces over non-closed fields and relate the notion of derived equivalence to arithmetic problems.
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Extremal rays and automorphisms of holomorphic symplectic varieties
We survey recent results on ample cones and birational contractions of holomorphic symplectic varieties of K3 type, focusing on explicit constructions and concrete examples.
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Stable rationality and conic bundles
We study stable rationality properties of conic bundles over rational surfaces.
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Height zeta functions of equivariant compactifications of unipotent groups
We prove Manin's conjecture for bi-equivariant compactifications of unipotent groups.
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Balanced line bundles on Fano varieties
A conjecture of Batyrev and Manin relates arithmetic properties of varieties with ample anticanonical class to geometric invariants; in particular, counting functions defined by metrized ample line bundles and the corresponding asymptotics of rational points of bounded height are interpreted in terms of cones of effective divisors and certain thresholds with respect to these cones. This framework leads to the notion of balanced line bundles, whose counting functions, conjecturally, capture generic distributions of rational points. We investigate balanced line bundles in the context of the Minimal Model Program, with special regard to the classification of Fano threefolds.
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Universal spaces for unramified Galois cohomology
We construct and study universal spaces for birational invariants of algebraic varieties over algebraic closures of finite fields.
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Variation of Neron-Severi ranks of reductions of K3 surfaces
We study the behavior of geometric Picard ranks of K3 surfaces over the rationals under reduction modulo primes. We compute these ranks for reductions of smooth quartic surfaces modulo all primes $p<2^{16}$ in several representative examples and investigate the resulting statistics.
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