Examples of "jacobians"
Exploiting the "division" property of Jacobians yields
The Schottky problem asks which principally polarized abelian varieties are the Jacobians of curves.
where formula_92 are called the flux Jacobians defined as the matrices:
where the state transition and observation matrices are defined to be the following Jacobians
This can be generalized to multiple variables as a matrix product of Jacobians.
One can proceed to Taylor series expansion of the equations, which results in the Jacobians or design matrices: the first one,
That is, the Jacobian of a composite function is the product of the Jacobians of the composed functions (evaluated at the appropriate points).
In algebraic geometry, Fay's trisecant identity is an identity between theta functions of Riemann surfaces introduced by . Fay's identity holds for theta functions of Jacobians of curves, but not for theta functions of general abelian varieties.
The theory of Prym varieties was dormant for a long time, until revived by David Mumford around 1970. It now plays a substantial role in some contemporary theories, for example of the Kadomtsev–Petviashvili equation. One advantage of the method is that it allows one to apply the theory of curves to the study of a wider class of abelian varieties than Jacobians. For example, principally polarized abelian varieties (p.p.a.v.'s) of dimension > 3 are not generally Jacobians, but all p.p.a.v.'s of dimension 5 or less are Prym varieties. It is for this reason that p.p.a.v.'s are fairly well understood up to dimension 5.
In mathematics, Deligne cohomology is the hypercohomology of the Deligne complex of a complex manifold. It was introduced by as a cohomology theory for algebraic varieties that includes both ordinary cohomology and intermediate Jacobians.
Obviously this Jacobian does not exist in discontinuity regions (e.g. contact discontinuities, shock waves in inviscid nonconductive flows). Note that if the flux Jacobians formula_92 are not functions of the state vector formula_43, the equations reveals "linear".
In algebraic geometry, a cubic threefold is a hypersurface of degree 3 in 4-dimensional projective space. Cubic threefolds are all unirational, but used intermediate Jacobians to show that non-singular cubic threefolds are not rational. The space of lines on a non-singular cubic 3-fold is a Fano surface.
The Picard variety, the Albanese variety, and intermediate Jacobians are generalizations of the Jacobian for higher-dimensional varieties. For varieties of higher dimension the construction of the Jacobian variety as a quotient of the space of holomorphic 1-forms generalizes to give the Albanese variety, but in general this need not be isomorphic to the Picard variety.
Perhaps one of the most striking applications of systoles is in the context of the Schottky problem, by P. Buser and P. Sarnak, who distinguished the Jacobians of Riemann surfaces among principally polarized abelian varieties, laying the foundation for systolic arithmetic.
Weil introduced abstract rather than projective varieties partly so that he could construct the Jacobian of a curve. (It was not known at the time that Jacobians are always projective varieties.) It was some time before anyone found any examples of complete abstract varieties that are not projective.
In the topic of iterative methods, he published papers about the Newton method for systems of equations with rectangular or singular Jacobians, directional Newton methods, the quasi-Halley method, Newton and Halley methods for complex roots, and the inverse Newton transform.
Investigations of Niels Henrik Abel and Carl Gustav Jacobi on inversion of elliptic integrals in the first half of 19th century led them to consider special types of complex manifolds, now known as Jacobians. Bernhard Riemann further contributed to their theory, clarifying the geometric meaning of the process of analytic continuation of functions of complex variables.
The 2-torsion points on an Abelian variety admit a symplectic bilinear form called the Weil pairing. In the case of Jacobians of curves of genus two, every nontrivial 2-torsion point is uniquely expressed as a difference between two of the six Weierstrass points of the curve. The Weil pairing is given in this case by
If "C" is a projective, nonsingular curve of genus ≥ 0 over "k", and "J" its Jacobian, then the theta-divisor of "J" induces a principal polarisation of "J", which in this particular case happens to be an isomorphism (see autoduality of Jacobians). Hence, composing the Weil pairing for "J" with the polarisation gives a nondegenerate pairing
More precisely, one should consider algebraic curves "C" of a given genus "g", and their Jacobians "J". There is a moduli space "M" of such curves, and a moduli space "A" of abelian varieties of dimension "g", which are "principally polarized". There is a morphism