arXiv subjects
Publications and source records attributed to Yuri Tschinkel.
We study linearizability of actions of finite groups on singular cubic threefolds, using cohomological tools, intermediate Jacobians, Burnside invariants, and the equivariant Minimal Model Program.
We study rationality properties of real singular cubic threefolds.
We provide new stable linearizability constructions for regular actions of finite groups on homogeneous spaces and low-dimensional quadrics.
We develop an equivariant version of the Pfaffian-Grassmannian correspondence and apply it to produce examples of nontrivial twisted equivariant stable birationalities between cubic threefolds and degree 14 Fano threefolds.
We study involutions on K3 surfaces under conjugation by derived equivalence and more general relations, together with applications to equivariant birational geometry.
We study equivariant geometry and rationality of moduli spaces of points on the projective line, for twists associated with permutations of the points.
We provide a general algorithm for the computation of the unramified Brauer group of quotients of rational varieties by finite groups.
We investigate the birational geometry of Deligne-Mumford stacks and define new birational invariants in this context.
We investigate equivariant birational geometry of rational surfaces and threefolds from the perspective of derived categories.
We study linearizability and stable linearizability of actions of finite groups on the Segre cubic and Burkhardt quartic, using techniques from group cohomology, birational rigidity, and the Burnside formalism.
We study equivariant birationality from the perspective of derived categories. We produce examples of nonlinearizable but stably linearizable actions of finite groups on smooth cubic fourfolds.
We study arithmetic properties of derived equivalent K3 surfaces over the field of Laurent power series, using the equivariant geometry of K3 surfaces with cyclic groups actions.
We study linear actions of finite groups in small dimensions, up to equivariant birationality.
We construct new invariants of equivariant birational isomorphisms taking values in equivariant Burnside groups.
We develop the formalism of universal torsors in equivariant birational geometry and apply it to produce new examples of nonbirational but stably birational actions of finite groups.
An equivariant stable birational invariant of an action of a finite group on a smooth projective variety is the first cohomology group of the Picard module. Bogomolov-Prokhorov and Shinder computed this for actions of cyclic groups on rational surfaces, with maximal stabilizers, in terms of the geometry of the fixed point locus. Using the Brauer group of the quotient stack, we extend the computation to more general actions and relate it to the equivariant Burnside group formalism.
Fix a finite group $G$. We seek to classify varieties with $G$-action equivariantly birational to a representation of $G$ on affine or projective space. Our focus is odd-dimensional smooth complete intersections of two quadrics, relating the equivariant rationality problem with analogous Diophantine questions over nonclosed fields. We explore how invariants -- both classical cohomological invariants and recent symbol constructions -- control rationality in some cases.
We study the structure of combinatorial Burnside groups, which receive equivariant birational invariants of actions of finite groups on algebraic varieties.