Affine Curve Research Articles

Serre’s generalization of Nagao’s theorem: An elementary approach

Let C C be a smooth projective curve over a field k k . For each closed point Q Q of C C let C = C ( C , Q , k ) \mathcal {C} = \mathcal {C}(C, Q, k) be the coordinate ring of the affine curve obtained by removing Q Q from C C . Serre has proved that G L 2 ( C ) GL_2(\mathcal {C}) is isomorphic to the fundamental group, π 1 ( G , T ) \pi _1(G, T) , of a graph of groups ( G , T ) (G, T) , where T T is a tree with at most one non-terminal vertex. Moreover the subgroups of G L 2 ( C ) GL_2(\mathcal {C}) attached to the terminal vertices of T T are in one-one correspondence with the elements of Cl ⁡ ( C ) \operatorname {Cl}(\mathcal {C}) , the ideal class group of C \mathcal {C} . This extends an earlier result of Nagao for the simplest case C = k [ t ] \mathcal {C} = k[t] . Serre’s proof is based on applying the theory of groups acting on trees to the quotient graph X ¯ = G L 2 ( C ) ∖ X \overline {X} = GL_2(\mathcal {C}) \backslash X , where X X is the associated Bruhat-Tits building. To determine X ¯ \overline {X} he makes extensive use of the theory of vector bundles (of rank 2) over C C . In this paper we determine X ¯ \overline {X} using a more elementary approach which involves substantially less algebraic geometry. The subgroups attached to the edges of T T are determined (in part) by a set of positive integers S \mathcal {S} , say. In this paper we prove that S \mathcal {S} is bounded, even when Cl ( C ) (\mathcal {C}) is infinite. This leads, for example, to new free product decomposition results for certain principal congruence subgroups of G L 2 ( C ) GL_2(\mathcal {C}) , involving unipotent and elementary matrices.

An affine monomial curve with irrational Poincaré–Betti series

We show that there exists an affine monomial curve with non-rational Poincaré–Betti series. The proofs depend on an earlier paper by Roos and Sturmfels which treated the projective case. Some open problems are formulated.

SK1 of Affine Curves over Finite Fields

We show that SK1(X)=0 for every affine curve X over a finite field.

On the distribution of integer points on curves of genus zero

Let C⊂ A n be a geometrically irreducible affine curve of (geometric) genus 0 defined over Z such that C( Z) is an infinite set. We use elementary methods to describe the distribution of C( Z) in C( R) relative to the real topology.

Local Triviality of Proper Ga Actions

Regular actions of the additive group of complex numbers on complex surfaces and on complex affine space are considered. A proper action on an affine surface admits a geometric quotient which is an affine curve. A proper action on a normal quasiaffine surface is equivariantly trivial. New criteria for local and “global” triviality of proper actions on a complex affine space of arbitrary dimension are presented.

Classification of rings of differential operators on affine curves

Classification of rings of differential operators on affine curves

Topology of families of affine plane curves

We determine bifurcation sets of families of affine curves and study the topology of such families.

Finiteness theorems for the cohomology of an overconvergent isocrystal on a curve

Let M be an overconvergent isocrystal on a smooth affine curve X/ k over a perfect field of characteristic p > 0, realized as a module on a suitable lifting of X with connection. We give a topological condition on the connection which guarantees that the rigid cohomology of M is finite-dimensional. As a result, one sees that M has finite-dimensional cohomology if it satisfies an analogue of Grothendieck's local monodromy theorem. Some arithmetic applications are given.

Borel-Weil-Bott theory on the moduli stack of G -bundles over a curve

Let G be a semi-simple group and M the moduli stack of G-bundles over a smooth, complex, projective curve. Using representation-theoretic methods, I prove the pure-dimensionality of sheaf cohomology for certain “evaluation vector bundles” over M, twisted by powers of the fundamental line bundle. This result is used to prove a Borel-Weil-Bott theorem, conjectured by G. Segal, for certain generalized flag varieties of loop groups. Along the way, the homotopy type of the group of algebraic maps from an affine curve to G, and the homotopy type, the Hodge theory and the Picard group of M are described. One auxiliary result, in Appendix A, is the Alexander cohomology theorem conjectured in [Gro2]. A self-contained account of the “uniformization theorem” of [LS] for the stack M is given, including a proof of a key result of Drinfeld and Simpson (in characteristic 0). A basic survey of the simplicial theory of stacks is outlined in Appendix B.

On the Orbit Method for the Lie Algebra of Vector Fields on a Curve

Let g be the Lie algebra of vector fields on an affine smooth curve Σ. Our goal is to establish an orbit method for g. Since g is infinite-dimensional, we face some technical problems. Without having groups acting on g, we try nevertheless to define the notion of “orbits.” So, we focus our attention to a subspace g*fof g*. This subspace consists of the “finite-dimensional orbits.” To almost all λ in g*fit corresponds a simple induced representation of g whose annihilator is a primitive ideal. We conjecture that this ideal has a finite Gelfand–Kirillov codimension. What we are actually looking for is a bijection similar to Dixmier's bijection (in the finite-dimensional case) between the “orbits” of g*fand certain primitive ideals of the enveloping algebra of g.

Algebraic integrability of the two-body Ruijsenaars operator

We study the algebra of difference operators that commute with the two-body Ruijsenaars operator, a $q$-deformation of the Lam\'e differential operator, for generic values of the deformation parameter. The algebra is commutative. It is the algebra of polynomial functions on an affine hyperelliptic curve $Y^2=P(X^2)$. We also compute the difference Galois group of the eigenvalue problem.

The determination of implicit polynomial canonical curves

A new method is presented for identifying and comparing closed, bounded, free-form curves that are defined by even implicit polynomial (IP) equations in the X-Y Cartesian coordinates. The method provides a new expression for an IP involving a product of conic factors with unique conic factor centers. The critical points for an IP curve are also defined. The conic factor centers and the critical points are shown to be useful related points that directly map to one another under affine transformations. In particular, the explicit determination of such points implies both a canonical form for the curves and the transformation matrix which relates affine equivalent curves.

EQUILIBRIUM ANALYSIS OF SYNTHESIS GAS WITH COMPOSITION-DEPENDENT DEGREES OF FREEDOM

An analysis is presented of the equilibrium composition of syngas mixtures parametrized in terms of the syngas ratio and the hydrogen mole fraction, for fixed values of pressure and temperature. Conditions for the existence of single or multiple carbon-free solutions are determined by analysis of the carbon affinity equation, formulated by the method of atomic potentials. The system is characterized by two curves describing the maximum possible mole fraction of hydrogen, and the mole fraction of hydrogen at unit carbon affinity. These curves are tangent to each other at a single contact point. The unit carbon affinity curve consists of two branches, corresponding either to the higher or to the lower carbon affinity root becoming equal to one. These branches join together at the contact point of the maximum bounding and unit carbon affinity curves. The syngas ratio at the contact point is pressure-independent and is expressed as a function of temperature. A single carbon-free solution is obtained in the region below both branches of the unit carbon affinity curve. It is shown, however, that two distinct carbon-free solutions can be constructed in the region enclosed between the maximum bounding curve and the right branch of the unit carbon affinity curve.

Generalized conic curves and their applications in curve approximation

The appriximation properties of generalized conic curves are studied in this paper. A generalized conic curve is defined as one of the following curves or their affine and translation equivalent curves: (i) conic curves, including parabolas, hyperbolas and ellipses; (ii) generalized monomial curves, including curves of the form x=yγ, γ∈R, γ≠0,1, in the x−y Cartesian coordinate system; (iii) exponential spiral curves of the form ρ(ϑ)=Aeγϑ, A>0, γ≠0, in the ρ-ϑ polar coordinate system.

Integer points on algebraic curves with exceptional units

AbstractLet F(X, Y) be an absolutely irreducible polynomial with coefficients in an algebraic number field K. Denote by C the algebraic curve defined by the equation F(X, Y) = 0 and by K[C] the ring of regular functions on Cover K. Assume that there is a unit ϕ in K[C] − K such that 1 − ϕ is also a unit. Then we establish an explicit upper bound for the size of integral solutions of the equation F(X, Y) = 0, defined over K. Using this result we establish improved explicit upper bounds on the size of integral solutions to the equations defining non-singular affine curves of genus zero, with at least three points at ‘infinity’, the elliptic equations and a class of equations containing the Thue curves.

Affine curve moment invariants for shape recognition

This paper presents a method of affine curve moment invariants for shape recognition. The proposed method extends affine moment invariants from an area domain to a curve domain. First, a new type of curve moments is defined on a parameterized boundary description of an object. A set of affine moment invariants are then derived based on the theory of algebraic invariants. The proposed moment invariants only deal with pixels on a shape boundary and are mathematically proven to be invariant under arbitrary affine transformations. Furthermore, after a parameterized object boundary description is acquired through curve fitting, the newly defined curve moments can be represented by an integration of the fitting function, so that the computation of moment invariants can be obtained from either one-by-one computation of boundary pixels or a direct integration of the fitting function, which makes the computation efficient while at the same time reducing noise effect. Affine curve moment invariants work well in recognition of affine-deformed objects, which is demonstrated by processing various 2-D shapes under affine transformations.

The Grothendieck conjecture for affine curves

We prove that the isomorphism class of an affine hyperbolic curve defined over a field finitely generated over is completely determined by its arithmetic fundamental group. We also prove a similar result for an affine curve defined over a finite field.

Affine Geometry, Curve Flows, and Invariant Numerical Approximations

A new geometric approach to the affine geometry of curves in the plane and to affine-invariant curve shortening is presented. We describe methods of approximating the affine curvature with discrete finite difference approximations, based on a general theory of approximating differential invariants of Lie group actions by joint invariants. Applications to computer vision are indicated.

Algebraic integrability of the two-body Ruijsenaars operator

We study the algebra of difference operators that commute with the two-body Ruijsenaars operator, a $q$-deformation of the Lam\'e differential operator, for generic values of the deformation parameter. The algebra is commutative. It is the algebra of polynomial functions on an affine hyperelliptic curve $Y^2=P(X^2)$. We also compute the difference Galois group of the eigenvalue problem.

Uniqueness of plane embeddings of special curves

For a family of special affine plane curves, it is shown that their embeddings in the affine plane are unique up to automorphisms of the affine plane. Examples are also given for which the embedding is not unique. We also discuss the Lin-Zaidenberg estimate of the number of singular points of an irreducible curve in terms of its rank. Formulas concerning the rank of the curve lead to an alternate simpler version of the proof of the Epimorphism Theorem.