Birational Transformations Research Articles

Canonical measures and Kähler-Ricci flow

We show that the Kähler-Ricci flow on a projective manifold of positive Kodaira dimension and semi-ample canonical line bundle converges to a unique canonical metric on its canonical model. It is also shown that there exists a canonical measure of analytic Zariski decomposition on a projective manifold of positive Kodaira dimension. Such a canonical measure is unique and invariant under birational transformations under the assumption of the finite generation of canonical rings.

On the period map for prime Fano threefolds of degree 10

We study, after Logachev, the geometry of smooth complex Fano threefoldsXXwith Picard number11, index11, and degree1010, and their period map to the moduli space of 10-dimensional principally polarized abelian varieties. We prove that a general suchXXhas no nontrival automorphisms. By a simple deformation argument and a parameter count, we show thatXXisnotbirational to a quartic double solid, disproving a conjecture of Tyurin.Through a detailed study of the variety of conics contained inXX, a smooth projective irreducible surface of general type with globally generated cotangent bundle, we construct two smooth projective two-dimensional components of the fiber of the period map through a generalXX: one is isomorphic to the variety of conics inXX, modulo an involution, another is birationally isomorphic to a moduli space of semistable rank-22torsion-free sheaves onXX, modulo an involution. The threefolds corresponding to points of these components are obtained fromXXvia conic and line (birational) transformations. The general fiber of the period map is the disjoint union of an even number of smooth projective surfaces of this type.

Counting walks in a quadrant: a unified approach via boundary value problems

The aim of this article is to introduce a unified method to obtain explicit integral representations of the trivariate generating function counting the walks with small steps which are confined to a quarter plane. For many models, this yields for the first time an explicit expression of the counting generating function. Moreover, the nature of the integrand of the integral formulations is shown to be directly dependent on the finiteness of a naturally attached group of birational transformations as well as on the sign of the covariance of the walk.

Regularization of Local CR-Automorphisms of Real-Analytic CR-Manifolds

Let M be a connected generic real-analytic CR-submanifold of a finite-dimensional complex vector space E. Suppose that for every a∈M the Lie algebra \({\frak{hol}}(M,a)\) of germs of all infinitesimal real-analytic CR-automorphisms of M at a is finite-dimensional and its complexification contains all constant vector fields \(\alpha\frac {\partial}{\partial z} \), α∈E, and the Euler vector field \(z\frac{\partial}{\partial z} \). Under these assumptions we show that: (I) every \({\frak{hol}}(M,a)\) consists of polynomial vector fields, hence coincides with the Lie algebra \({\frak{hol}}(M)\) of all infinitesimal real-analytic CR-automorphisms of M, (II) every local real-analytic CR-automorphism of M extends to a birational transformation of E, and (III) the group Bir (M) generated by such birational transformations is realized as a group of projective transformations upon embedding E as a Zariski open subset into a projective algebraic variety. Under additional assumptions the group Bir (M) is shown to have the structure of a Lie group with at most countably many connected components and Lie algebra \({\frak{hol}}(M)\). All of the above results apply, for instance, to Levi non-degenerate quadrics, as well as a large number of Levi degenerate tube manifolds.

Birational permutations

We prove that every permutation of P n ( K ) , where K is a finite field with odd characteristic, is induced by a birational transformation with no rational indeterminacy point. To cite this article: S. Cantat, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

A birational mapping with a strange attractor: post-critical set and covariant curves

We consider some two-dimensional birational transformations. One of them is a birational deformation of the Hénon map. For some of these birational mappings, the post-critical set (i.e. the iterates of the critical set) is infinite and we show that this gives straightforwardly the algebraic covariant curves of the transformation when they exist. These covariant curves are used to build the preserved meromorphic 2-form. One may also have an infinite post-critical set yielding a covariant curve which is not algebraic (transcendental). For two of the birational mappings considered, the post-critical set is finite and we claim that there is no algebraic covariant curve and no preserved meromorphic 2-form. For these two mappings with finite post-critical sets, attracting sets occur and we show that they pass the usual tests (Lyapunov exponents and the fractal dimension) for being strange attractors. The strange attractor of one of these two mappings is unbounded.

A Conjecture of Mukai Relating Numerical Invariants of Fano Manifolds

A complex manifold X of dimension n such that the anticanonical bundle –K X := det TX is ample is called a Fano manifold. Besides the dimension, other two integers play an essential role in the classification of these manifolds, namely the pseudoindex of X, i X , which is the minimal anticanonical degree of rational curves on X, and the Picard number ρ X , the dimension of N 1(X), the vector space generated by irreducible complex curves modulo numerical equivalence . A (generalization of a) conjecture of Mukai says that ρ X (i X – 1) ≤ n. In this paper we present some partial steps towards the conjecture, we show how one can interpretate and possibly solve it with the use of families of rational curves on a uniruled variety, and more generally with the instruments of Mori theory. We consider also other related problems: the description of some Fano manifolds which are at the border of the Mukai relations and how the pseudoindex changes via (some) birational transformation.

On the Crepant Resolution Conjecture in the Local Case

In this paper we analyze four examples of birational transformations between local Calabi–Yau 3-folds: two crepant resolutions, a crepant partial resolution, and a flop. We study the effect of these transformations on genus-zero Gromov–Witten invariants, proving the Coates–Iritani–Tseng/Ruan form of the Crepant Resolution Conjecture in each case. Our results suggest that this form of the Crepant Resolution Conjecture may also hold for more general crepant birational transformations. They also suggest that Ruan’s original Crepant Resolution Conjecture should be modified, by including appropriate “quantum corrections”, and that there is no straightforward generalization of either Ruan’s original Conjecture or the Cohomological Crepant Resolution Conjecture to the case of crepant partial resolutions. Our methods are based on mirror symmetry for toric orbifolds.

Rational curve with Galois point and extendable Galois automorphism

Let G be the Galois group of a Galois point for a plane curve C. An element of G induces a birational transformation of C. We study if it can be extended to a projective or birational transformation of the plane. In the course of the study we give the defining equation of a rational curve with the Galois point. Furthermore, we introduce a special birational transformation in order to make the defining equation into a simpler form.

Birational mappings and matrix subalgebra from the chiral Potts model

We study birational transformations of the projective space originating from lattice statistical mechanics, specifically from various chiral Potts models. Associating these models with stable patterns and signed patterns, we give general results which allow us to find all chiral q-state spin-edge Potts models when the number of states q is a prime or the square of a prime, as well as several q-dependent family of models. We also prove the absence of monocolor stable signed pattern with more than four states. This demonstrates a conjecture about cyclic Hadamard matrices in a particular case. The birational transformations associated with these lattice spin-edge models show complexity reduction. In particular, we recover a one-parameter family of integrable transformations, for which we give a matrix representation when the parameter has a suitable value.

On birational transformations of pairs in the complex plane

This article deals with the study of the birational transformations of the projective complex plane which leave invariant an irreducible algebraic curve. We try to describe the state of the art and provide some new results on this subject.

Linearisation of finite Abelian subgroups of the Cremona group of the plane

Given a finite Abelian subgroup of the Cremona group of the plane, we provide a way to decide whether it is birationally conjugate to a group of automorphisms of a minimal surface. In particular, we prove that a finite cyclic group of birational transformations of the plane is linearisable if and only if none of its non-trivial elements fix a curve of positive genus. For finite Abelian groups, there exists only one surprising exception, a group isomorphic to ℤ/2ℤ × ℤ/4ℤ , whose non-trivial elements do not fix a curve of positive genus but which is not conjugate to a group of automorphisms of a minimal rational surface. We also give some descriptions of automorphisms (not necessarily of finite order) of del Pezzo surfaces and conic bundles.

Mordell–Weil Groups of a Hyperkähler a Manifold—A Question of F. Campana

Among other things, we show that Mordell–Weil groups of finitely many different abelian fibrations of a hyperkähler manifold have essentially no relation, as its birational transformation. Precise definition of the terms “essentially no relation” will be given in Introduction.

ON j-INVARIANTS OF WEIERSTRASS EQUATIONS

A simple proof of the fact that the j-invariants for Weierstrass equations are invariant under birational transformations which keep the forms of Weierstrass equations is given by finding a non-trivial explicit birational transformation which sends a normalized Weierstrass equation to the same equation.

On a construction of the twistor spaces of Joyce metrics

We explicitly construct the twistor spaces of some self-dual metrics with torus action given by D. Joyce. Starting from a fiber space over a projective line whose fibers are compact singular toric surfaces, we apply a number of birational transformations to obtain the desired twistor spaces. These constructions are based on a detailed analysis of the anticanonical system of the twistor spaces.

On the inertia group of elliptic curves in the Cremona group of the plane

We study the group of birational transformations of the plane that fix (each point of) a curve of geometric genus 1. A precise description of the finite elements is given; it is shown in particular that the order is at most 6, and that if the group contains a non-trivial torsion, the fixed curve is the image of a smooth cubic by a birational transformation of the plane. We show that for a smooth cubic, the group is generated by its elements of degree 3, and prove that it contains a free product of Z/2Z, indexed by the points of the curve.

Elliptic structures on weighted three-dimensional Fano hypersurfaces

We classify birational transformations into elliptic fibrations of a general quasi-smooth hypersurface in of degree that has terminal singularities.

Efficient explicit formulae for genus 3 hyperelliptic curve cryptosystems over binary fields

The ideal class groups of hyperelliptic curves (HECs) can be used in cryptosystems based on the discrete logarithm problem. Recent developments of computational technologies for scalar multiplications of divisor classes have shown that the performance of hyperelliptic curve cryptosystems (HECC) is compatible to that of elliptic curve cryptosystems. Especially, due to short operand sizes, genus 3 HECC are well suited for all kinds of embedded processor architectures, where resources such as storage, time or power are constrained. In the paper, the acceleration of the divisor class doubling for genus 3 HECs over binary fields is investigated and the number of field operations needed is analysed. By constructing birational transformations of variables, four types of curves which can lead to much faster divisor class doubling are found and the corresponding explicit formulae are given. In particular, for special genus 3 HECs over binary fields with h(X)=1, the fastest explicit doubling formula published so far which only requires one field inversion, ten field multiplications and eleven field squarings, is obtained. Furthermore, comparisons with the known results in terms of field operations and implementations of genus 3 HECC over three different binary fields on a Pentium-4 processor are provided.

Twistor lines on Nagata threefold

We give an explicit description of rational curves in the product of three copies of complex projective lines, which are transformed into twistor lines in M. Nagata’s example of non-projective complete algebraic variety, viewed as the twistor space of Eguchi-Hanson metric. In particular, we show that there exist two families of such curves and both of them are parameterized by mutually diffeomorphic, connected real 4-dimensional manifolds. We also give a relationship between these two families through a birational transformation naturally associated to the Nagata’s example.

Effective classes and Lagrangian tori in symplectic four-manifolds

An effective class in a closed symplectic four-manifold is a twodimensional homology class which is realized by a J-holomorphic cycle for every tamed almost complex structure J. We first prove that effective classes are orthogonal to Lagrangian tori with respect to the intersection form. We then deduce an invariant under birational transformations of closed symplectic four-manifolds. We finally prove using the same techniques of symplectic field theory that the unit cotangent bundle of a compact orientable hyperbolic Lagrangian surface does not embed as a hypersurface of contact type in a rational or ruled symplectic four-manifold.