Related papers: Birational rigidity of a three-dimensional double …
We study the group of birational selfmaps of a variety birational to a conic bundle. We prove that it admits a surjective morphism to the direct sum of an uncountable number of copies of $\mathbb Z/2\IZ$.
In this paper we prove criteria for a nonnormal toric variety to be flexible, to be rigid and to be almost rigid. For rigid and almost rigid toric varieties we describe the automorphism group explicitly.
A rigidity theory is developed for bar-joint frameworks in $\mathbb{R}^{d+1}$ whose vertices are constrained to lie on concentric $d$-spheres with independently variable radii. In particular, combinatorial characterisations are established…
It is proved that on a smooth algebraic variety, fibered into cubic surfaces over the projective line and sufficiently ``twisted'' over the base, there is only one pencil of rational surfaces -- that is, this very pencil of cubics. In…
In a recent paper Hodgson and Kerckhoff prove a local rigidity theorem for finite volume, three dimensional hyperbolic cone-manifolds. In this paper we extend this result to geometrically finite cone-manifolds. Our methods also give a new…
We define a complete Riemannian manifold X to be large-scale conformally rigid if all groups that are quasi-isometric to some complete Riemannian manifold of bounded geometry conformal to X are quasi-isometric to X. We prove that many…
One can associate to a bipartite graph a so-called edge ring whose spectrum is an affine normal toric variety. We characterize the faces of the (edge) cone associated to this toric variety in terms of some independent sets of the bipartite…
We construct families of birational involutions on $\mathbb{P}^3$ or a smooth cubic threefold which do not fit into a non-trivial elementary relation of Sarkisov links. As a consequence, we construct new homomorphisms from their group of…
We classify birationally rigid orbifold Fano 3-folds of index one defined by $5 \times 5$ Pfaffians. We give a sharp criterion for birational rigidity of these families based on the type of singularities that the varieties admit. Various…
This is the unabridged web version of the paper that will be published on the American Journal of Mathematics. In this paper, we study the birational geometry of certain examples of mildly singular quartic 3-folds. A quartic 3-fold is an…
This expository paper is concerned with the rationality problems for three-dimensional algebraic varieties with a conic bundle structure. We discuss the main methods of this theory. We sketch the proofs of certain principal results, and…
We study the birational geometry of varieties of maximal Albanese dimension. In particular we discuss criteria for a generically finite morphism of varieties of maximal Albanese dimension to be birational; we give a new characterization of…
A variety is unirational if it is dominated by a rational variety. A variety is rationally connected if two general points can be joined by a rational curve. This paper aims to show that the two notions can cooperate and, building on…
We find all possible isomorphisms and 3-birational maps (i.e., birational maps which induce an isomorphism between open subsets whose respective complements have codimension at least 3) between moduli spaces of parabolic vector bundles with…
Mutations of the cluster variables generating the cluster algebra of type $A^{(2)}_2$ reduce to a two-dimensional discrete integrable system given by a quartic birational map. The invariant curve of the map is a singular quartic curve, and…
Consider a polynomial F such that each variable appears in exactly one monomial. The hypersurface defined by the polynomial F is called a hypersurface with separable variables. A variety is called rigid if there are no nontrivial actions of…
We consider the problem of finding an inductive construction, based on vertex splitting, of triangulated spheres with a fixed number of additional edges (braces). We show that for any positive integer $b$ there is such an inductive…
Suppose $X$ is an irreducible complex variety. We show that when $X$ is ruled, the group of birational transformations $Bir(X)$, as a group, determines $X$ up to birational transformations and automorphisms of the base field. In contrast,…
This paper investigates the rigidity of bordered polyhedral surfaces. Using the variational principle, we show that bordered polyhedral surfaces are determined by boundary value and discrete curvatures on the interior edges. As a corollary,…
An algebraic variety $X$ is called rigid if there is no non-trivial action on $X$ of the additive group of the base field. A trinomial variety is an affine variety that is given by a set of equations consisting of polynomials with three…