Equations defining abelian varieties
Encyclopedia
In mathematics
, the concept of abelian variety
is the higher-dimensional generalization of the elliptic curve
. The equations defining abelian varieties are a topic of study because every abelian variety is a projective variety. In dimension d ≥ 2, however, it is no longer as straightforward to discuss such equations.
There is a large classical literature on this question, which in a reformulation is, for complex algebraic geometry, a question of describing relations between theta functions. The modern geometric treatment now refers to some basic papers of David Mumford
, from 1966 to 1967, which reformulated that theory in terms from abstract algebraic geometry valid over general fields.
s.
In general abelian varieties are not complete intersection
s. Computer algebra techniques are now able to have some impact on the direct handling of equations for small values of d > 1.
represented a quotient of an abelian variety with d = 2, by the group of order 2 of automorphisms generated by x → −x on the abelian variety.
L on an abelian variety A. This is a group of self-automorphisms of L, and is a finite analogue of the Heisenberg group. The primary results are on the action of the theta group on the global sections of L. When L is very ample, the linear representation can be described, by means of the structure of the theta group. In fact the theta group is abstractly a simple type of nilpotent group
, a central extension of a group of torsion points on A, and the extension is known (it is in effect given by the Weil pairing
). There is a uniqueness result for irreducible linear representations of the theta group with given central character, or in other words an analogue of the Stone–von Neumann theorem
. (It is assumed for this that the characteristic of the field of coefficients doesn't divide the order of the theta group.)
Mumford showed how this abstract algebraic formulation could account for the classical theory of theta functions with theta characteristic
s, as being the case where the theta group was an extension of the two-torsion of A.
An innovation in this area is to use the Mukai–Fourier transform.
of the embedded abelian variety A, that is, set in a projective space according to a very ample L and its global sections. The graded commutative ring that is formed by the direct sum of the global sections of the
meaning the n-fold tensor product
of itself, is represented as the quotient ring
of a polynomial algebra by a homogeneous ideal I. The graded parts of I have been the subject of intense study.
Quadratic relations were provided by Bernhard Riemann
. Koizumi's theorem states the third power of an ample line bundle is normally generated. The Mumford–Kempf theorem states that the fourth power of an ample line bundle is quadratically presented. For a base field of characteristic zero, Giuseppe Pareschi proved a result including these (as the cases p = 0, 1) which had been conjectured by Lazarsfeld: let L be an ample line bundle on an abelian variety A. If n ≥ p + 3, then the n-th tensor power of L satisfies condition Np. Further results have been proved by Pareschi and Popa, including previous work in the field.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the concept of abelian variety
Abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions...
is the higher-dimensional generalization of the elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
. The equations defining abelian varieties are a topic of study because every abelian variety is a projective variety. In dimension d ≥ 2, however, it is no longer as straightforward to discuss such equations.
There is a large classical literature on this question, which in a reformulation is, for complex algebraic geometry, a question of describing relations between theta functions. The modern geometric treatment now refers to some basic papers of David Mumford
David Mumford
David Bryant Mumford is an American mathematician known for distinguished work in algebraic geometry, and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded the National Medal of Science...
, from 1966 to 1967, which reformulated that theory in terms from abstract algebraic geometry valid over general fields.
Complete intersections
The only 'easy' cases are those for d = 1, for an elliptic curve with linear span the projective plane or projective 3-space. In the plane, every elliptic curve is given by a cubic curve. In P3, an elliptic curve can be obtained as the intersection of two quadricQuadric
In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface in -dimensional space defined as the locus of zeros of a quadratic polynomial...
s.
In general abelian varieties are not complete intersection
Complete intersection
In mathematics, an algebraic variety V in projective space is a complete intersection if it can be defined by the vanishing of the number of homogeneous polynomials indicated by its codimension...
s. Computer algebra techniques are now able to have some impact on the direct handling of equations for small values of d > 1.
Kummer surfaces
The interest in nineteenth century geometry in the Kummer surface came in part from the way a quartic surfaceQuartic surface
In mathematics, especially in algebraic geometry, a quartic surface is a surface defined by an equation of degree 4.More specifically there are two closely related types of quartic surface: affine and projective...
represented a quotient of an abelian variety with d = 2, by the group of order 2 of automorphisms generated by x → −x on the abelian variety.
General case
Mumford defined a theta group associated to an invertible sheafInvertible sheaf
In mathematics, an invertible sheaf is a coherent sheaf S on a ringed space X, for which there is an inverse T with respect to tensor product of OX-modules. It is the equivalent in algebraic geometry of the topological notion of a line bundle...
L on an abelian variety A. This is a group of self-automorphisms of L, and is a finite analogue of the Heisenberg group. The primary results are on the action of the theta group on the global sections of L. When L is very ample, the linear representation can be described, by means of the structure of the theta group. In fact the theta group is abstractly a simple type of nilpotent group
Nilpotent group
In mathematics, more specifically in the field of group theory, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute...
, a central extension of a group of torsion points on A, and the extension is known (it is in effect given by the Weil pairing
Weil pairing
In mathematics, the Weil pairing is a construction of roots of unity by means of functions on an elliptic curve E, in such a way as to constitute a pairing on the torsion subgroup of E...
). There is a uniqueness result for irreducible linear representations of the theta group with given central character, or in other words an analogue of the Stone–von Neumann theorem
Stone–von Neumann theorem
In mathematics and in theoretical physics, the Stone–von Neumann theorem is any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators...
. (It is assumed for this that the characteristic of the field of coefficients doesn't divide the order of the theta group.)
Mumford showed how this abstract algebraic formulation could account for the classical theory of theta functions with theta characteristic
Theta characteristic
In mathematics, a theta characteristic of a non-singular algebraic curve C is a divisor class Θ such that 2Θ is the canonical class, In terms of holomorphic line bundles L on a connected compact Riemann surface, it is therefore L such that L2 is the canonical bundle, here also equivalently the...
s, as being the case where the theta group was an extension of the two-torsion of A.
An innovation in this area is to use the Mukai–Fourier transform.
The coordinate ring
The goal of the theory is to prove results on the homogeneous coordinate ringHomogeneous coordinate ring
In algebraic geometry, the homogeneous coordinate ring R of an algebraic variety V given as a subvariety of projective space of a given dimension N is by definition the quotient ring...
of the embedded abelian variety A, that is, set in a projective space according to a very ample L and its global sections. The graded commutative ring that is formed by the direct sum of the global sections of the
meaning the n-fold tensor product
Tensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
of itself, is represented as the quotient ring
Quotient ring
In ring theory, a branch of modern algebra, a quotient ring, also known as factor ring or residue class ring, is a construction quite similar to the factor groups of group theory and the quotient spaces of linear algebra...
of a polynomial algebra by a homogeneous ideal I. The graded parts of I have been the subject of intense study.
Quadratic relations were provided by Bernhard Riemann
Bernhard Riemann
Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....
. Koizumi's theorem states the third power of an ample line bundle is normally generated. The Mumford–Kempf theorem states that the fourth power of an ample line bundle is quadratically presented. For a base field of characteristic zero, Giuseppe Pareschi proved a result including these (as the cases p = 0, 1) which had been conjectured by Lazarsfeld: let L be an ample line bundle on an abelian variety A. If n ≥ p + 3, then the n-th tensor power of L satisfies condition Np. Further results have been proved by Pareschi and Popa, including previous work in the field.
Further reading
- David MumfordDavid MumfordDavid Bryant Mumford is an American mathematician known for distinguished work in algebraic geometry, and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded the National Medal of Science...
, Selected papers on the classification of varieties and moduli spaces, editorial comment by G. Kempf and H. Lange, pp. 293–5