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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.
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 approximation of the change of f in a neighborhood of x, if f(x) is differentiable at x.
Given a projective scheme /, its cotangent sheaf can be computed from the sheafification of the cotangent module on the underlying graded algebra. For example, consider the complex curve For example, consider the complex curve
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 ...
In mathematics, the Jacobian conjecture is a famous unsolved problem concerning polynomials in several variables. It states that if a polynomial function from an n -dimensional space to itself has Jacobian determinant which is a non-zero constant, then the function has a polynomial inverse.
In algebraic geometry, the Quot scheme is a scheme parametrizing sheaves on a projective scheme.More specifically, if X is a projective scheme over a Noetherian scheme S and if F is a coherent sheaf on X, then there is a scheme whose set of T-points () = (, ()) is the set of isomorphism classes of the quotients of that are flat over T.
If S is projective (or equivalently, compact), then the degree of L is determined by the holomorphic Euler characteristics of X and S: deg(L) = χ(X,O X) − 2χ(S,O S). The canonical bundle formula implies that K X is Q-linearly equivalent to the pullback of some Q-divisor on S; it is essential here that the elliptic surface X → S is minimal.
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.