Birational Automorphisms Research Articles

On conjugacy of additive actions in the affine Cremona group

An additive action on an irreducible algebraic variety X is an effective action with an open orbit of the vector group . Any two additive actions on X are conjugate by a birational automorphism of X. We prove that, if X is the projective space, the conjugating element can be chosen in the affine Cremona group and it is given by so-called basic polynomials of the corresponding local algebra.

Remark on a theorem of Oguiso

For a Calabi–Yau variety X, Oguiso (Math Res Lett 25(1):181–198, 2018) gave a useful criterion for primitivity of a self-map of X in terms of the associated linear map on the Néron–Severi space of X. In this short note, we prove a variant of Oguiso’s criterion and use it to verify primitivity of a certain birational automorphism of a Calabi–Yau threefold, to which Oguiso’s original criterion does not apply.

Finite abelian subgroups in the groups of birational and bimeromorphic selfmaps

Let $X$ be a complex projective variety. Suppose that the group of birational automorphisms of $X$ contains finite subgroups isomorphic to $(\mathbb{Z}/N\mathbb{Z})^r$ for $r$ fixed and $N$ arbitrarily large. We show that $r$ does not exceed $2\dim(X)$. Moreover, the equality holds if and only if $X$ is birational to an abelian variety. We also show that an analogous result holds for groups of bimeromorphic automorphisms of compact Kähler spaces under some additional assumptions.

Tits-type alternative for certain groups acting on algebraic surfaces

According to a theorem of Cantat [Ann. of Math. (2) 174 (2011), pp. 299–340] and Urech [J. Reine Angew. Math. 770 (2021), pp. 27–57], an analog of the classical Tits alternative holds for the group of birational automorphisms of a compact complex Kähler surface. We established in Arzhantsev and Zaidenberg [Int. Math. Res. Not. IMRN 2022 (2022), pp. 8162–8195] the following Tits-type alternative: if X X is a toric affine variety and G ⊂ Aut ⁡ ( X ) G\subset \operatorname {Aut}(X) is a subgroup generated by a finite set of unipotent subgroups normalized by the acting torus then either G G contains a nonabelian free subgroup or G G is a unipotent affine algebraic group. In the present paper we extend the latter result to any group G G of automorphisms of a complex affine surface generated by a finite collection of unipotent algebraic subgroups. It occurs that either G G contains a nonabelian free subgroup or G G is a metabelian unipotent algebraic group.

On Numerical Dimensions of Calabi–Yau Varieties

Abstract Let $X$ be a Calabi–Yau variety of Picard number two with infinite birational automorphism group. We show that the numerical dimension $\kappa ^{\mathbb{R}}_{\sigma }$ of the extremal rays of the closed movable cone of $X$ is $\dim X/2$. More generally, we investigate the relation between the two numerical dimensions $\kappa ^{\mathbb{R}}_{\sigma }$ and $\kappa ^{\mathbb{R}}_{\textrm{vol}}$ for Calabi–Yau varieties. We also compute $\kappa ^{\mathbb{R}}_{\sigma }$ for non-big divisors in the closed movable cone of a projective hyperkähler manifold.

Regularizations of pseudo-automorphisms with positive algebraic entropy

We study birational automorphisms of threefolds which have positive algebraic entropy. We identify some conditions which imply that such an automorphism is non-regularizable. We show that this criterion applies in the example of a birational automorphism of $  \mathbb{P}^3 $ of positive algebraic entropy constructed by Blanc, thus showing that for a general choice of parameters it is non-regularizable. Additionally, we establish a criterion which proves that the automorphism in this example does not preserve a structure of a fibration over a surface.

Higher index Fano varieties with finitely many birational automorphisms

A famous problem in birational geometry is to determine when the birational automorphism group of a Fano variety is finite. The Noether–Fano method has been the main approach to this problem. The purpose of this paper is to give a new approach to the problem by showing that in every positive characteristic, there are Fano varieties of arbitrarily large index with finite (or even trivial) birational automorphism group. To do this, we prove that these varieties admit ample and birationally equivariant line bundles. Our result applies the differential forms that Kollár produces on $p$-cyclic covers in characteristic $p > 0$.

Remarks on nef and movable cones of hypersurfaces in Mori dream spaces

We investigate nef and movable cones of hypersurfaces in Mori dream spaces. The first result is: Let Z be a smooth Mori dream space of dimension at least four whose extremal contractions are of fiber type of relative dimension at least two and let X be a smooth ample divisor in Z, then X is a Mori dream space as well. The second result is: Let Z be a Fano manifold of dimension at least four whose extremal contractions are of fiber type and let X be a smooth anti-canonical hypersurface in Z, which is a smooth Calabi–Yau variety, then the unique minimal model of X up to isomorphism is X itself, and moreover, the movable cone conjecture holds for X, namely, there exists a rational polyhedral cone which is a fundamental domain for the action of birational automorphisms on the effective movable cone of X. The third result is: Let P:=Pn×⋯×Pn be the N-fold self-product of the n-dimensional projective space. Let X be a general complete intersection of n+1 hypersurfaces of multidegree (1,…,1) in P with dim⁡X≥3. Then X has only finitely many minimal models up to isomorphism, and moreover, the movable cone conjecture holds for X.

Birational automorphism groups and the movable cone theorem for Calabi–Yau complete intersections of products of projective spaces

For a Calabi-Yau manifold X, the Kawamata – Morrison movable cone conjecture connects the convex geometry of the movable cone Mov‾(X) to the birational automorphism group. Using the theory of Coxeter groups, Cantat and Oguiso proved that the conjecture is true for general varieties of Wehler type, and they described explicitly Bir(X). We generalize their argument to prove the conjecture and describe Bir(X) for general complete intersections of ample divisors in arbitrary products of projective spaces. Then, under a certain condition, we give a description of the boundary of Mov‾(X) and an application connected to the numerical dimension of divisors.

Diagonals of Rational Functions: From Differential Algebra to Effective Algebraic Geometry

We show that the results we had previously obtained on diagonals of 9- and 10-parameter families of rational functions in three variables x, y, and z, using creative telescoping, yielding modular forms expressed as pullbacked 2F1 hypergeometric functions, can be obtained much more efficiently by calculating the j-invariant of an elliptic curve canonically associated with the denominator of the rational functions. These results can be drastically generalized by changing the parameters into arbitrary rational functions of the product p=xyz. In other cases where creative telescoping yields pullbacked 2F1 hypergeometric functions, we extend this algebraic geometry approach to other families of rational functions in three or more variables. In particular, we generalize this approach to rational functions in more than three variables when the denominator can be associated to an algebraic variety corresponding to products of elliptic curves, or foliations in elliptic curves. We also extend this approach to rational functions in three variables when the denominator is associated with a genus-two curve such that its Jacobian is a split Jacobian, corresponding to the product of two elliptic curves. We sketch the situation where the denominator of the rational function is associated with algebraic varieties that are not of the general type, having an infinite set of birational automorphisms. We finally provide some examples of rational functions in more than three variables, where the telescopers have pullbacked 2F1 hypergeometric solutions, because the denominator corresponds to an algebraic variety that has a selected elliptic curve.

On the Numerical Dimension of Calabi–Yau 3-Folds of Picard Number 2

Abstract We show that for any smooth Calabi–Yau threefold $X$ of Picard number $2$ with infinite birational automorphism group, the numerical dimension $\kappa _\sigma $ of the extremal rays of the movable cone of $X$ is $\frac {3}{2}$. Furthermore, we provide new examples of Calabi–Yau threefolds of Picard number $2$ with infinite birational automorphism group.

On the Wall-Crossing Formula for Quadratic Differentials

Abstract We prove an analytic version of the Kontsevich–Soibelman wall-crossing formula describing how the number of finite-length trajectories of a quadratic differential jumps as the differential is varied. We characterize certain birational automorphisms of an algebraic torus appearing in this wall-crossing formula using Fock–Goncharov coordinates. As an application, we compute the Stokes automorphisms for the Voros symbols appearing in the exact WKB analysis of Schrödinger’s equation.

On birational automorphisms of Severi-Brauer surfaces

The generators of the group of birational automorphisms of any Severi-Brauer surface non-isomorphic over an algebraically non-closed field to the projective plane are explicitly described.

On birational transformations of Hilbert schemes of points on K3 surfaces

We classify the group of birational automorphisms of Hilbert schemes of points on algebraic K3 surfaces of Picard rank one. We study whether these automorphisms are symplectic or non-symplectic and if there exists a hyperk\"ahler birational model on which they become biregular. We also present new geometrical constructions of these automorphisms.

Algebraic intermediate hyperbolicities

We extend Lang's conjectures to the setting of intermediate hyperbolicity and prove two new results motivated by these conjectures. More precisely, we first extend the notion of algebraic hyperbolicity (originally introduced by Demailly) to the setting of intermediate hyperbolicity and show that this property holds if the appropriate exterior power of the cotangent bundle is ample. Then, we prove that this intermediate algebraic hyperbolicity implies the finiteness of the group of birational automorphisms and of the set of surjective maps from a given projective variety. Our work answers the algebraic analogue of a question of Kobayashi on analytic hyperbolicity.

Finite groups acting on Severi–Brauer surfaces

We classify finite groups that can act by automorphisms and birational automorphisms on non-trivial Severi–Brauer surfaces over fields of characteristic zero.

Singularities of linear systems and boundedness of Fano varieties

We study log canonical thresholds (also called global log canonicalthreshold or $\alpha$-invariant) of $\mathbb{R}$-linear systems. We prove existence of positive lower bounds in different settings, in particular, proving a conjecture of Ambro. We then show that the Borisov-Alexeev-Borisov conjecture holds; that is, given a natural number $d$ and a positive real number $\epsilon$, the set of Fano varieties of dimension $d$ with $\epsilon$-log canonical singularities forms a bounded family. This implies that birational automorphism groups of rationally connected varieties are Jordan which, in particular, answers a question of Serre. Next we show that if the log canonical threshold of the anti-canonical system of a Fano variety is at most one, then it is computed by some divisor, answering a question of Tian in this case.

A remark on the rank of finitep-groups of birational automorphisms

We improve a result of Prokhorov and Shramov on the rank of finite $p$-subgroups of the birational automorphism group of a rationally connected variety. Known examples show that they are sharp in many cases.

Finite 3-Subgroups in the Cremona Group of Rank 3

We consider 3-subgroups in groups of birational automorphisms of rationally connected threefolds and show that any 3-subgroup can be generated by at most five elements. Moreover, we study groups of regular automorphisms of terminal Fano threefolds and prove that, in all cases which are not among several explicitly described exceptions any 3-subgroup of such group can be generated by at most four elements.

On Calculation of the Group of Automorphisms of Hyperelliptic Curves

We suggest an algorithm for finding the group of birational automorphisms of a hyperelliptic curve y2 = p(x), p ∈ ℚ[x], over the field of complex numbers based on power series expansion. We present an implementation of this algorithm in the computer algebra system Sage and examples of its application. Computer experiments show that the algorithm does not lead to unexpectedly cumbersome calculations. The format of the group representation allows one to use the embedded Sage tools for dealing with finite groups of low order. Bibliography: 20 titles.