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We also construct a sheaf on , called the “structure sheaf” as in the affine case, which makes it into a scheme.As in the case of the Spec construction there are many ways to proceed: the most direct one, which is also highly suggestive of the construction of regular functions on a projective variety in classical algebraic geometry, is the following.
At the remaining critical point (0, 0) the second derivative test is insufficient, and one must use higher order tests or other tools to determine the behavior of the function at this point. (In fact, one can show that f takes both positive and negative values in small neighborhoods around (0, 0) and so this point is a saddle point of f .)
[8] [9] Some authors define the Jacobian as the transpose of the form given above. The Jacobian matrix represents the differential of f at every point where f is differentiable. In detail, if h is a displacement vector represented by a column matrix , the matrix product J ( x ) ⋅ h is another displacement vector, that is the best linear ...
For various applications, it is necessary to consider more general algebro-geometric objects than projective varieties, namely projective schemes. The first step towards projective schemes is to endow projective space with a scheme structure, in a way refining the above description of projective space as an algebraic variety, i.e., () is a ...
For m = 0 the generalized Jacobian J m is just the usual Jacobian J, an abelian variety of dimension g, the genus of C. For m a nonzero effective divisor the generalized Jacobian is an extension of J by a connected commutative affine algebraic group L m of dimension deg(m)−1. So we have an exact sequence 0 → L m → J m → J → 0
The Abel–Jacobi theorem implies that the Albanese variety of a compact complex curve (dual of holomorphic 1-forms modulo periods) is isomorphic to its Jacobian variety (divisors of degree 0 modulo equivalence). For higher-dimensional compact projective varieties the Albanese variety and the Picard variety are dual but need not be isomorphic.
Another example of a singular variety is the projective cone of a smooth variety: ... The non-singularity of this scheme can also be checked using the Jacobian ...
The ones constructed by Weil have natural polarizations if M is projective, and so are abelian varieties, while the ones constructed by Griffiths behave well under holomorphic deformations. A complex structure on a real vector space is given by an automorphism I with square − 1 {\displaystyle -1} .