"Kodaira map" redirects here. Not to be confused with Kodaira–Spencer map from cohomology theory.
A linear system of divisors algebraicizes the classic geometric notion of a family of curves, as in the Apollonian circles.
In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family.
These arose first in the form of a linear system of algebraic curves in the projective plane. It assumed a more general form, through gradual generalisation, so that one could speak of linear equivalence of divisors D on a general scheme or even a ringed space .[1]
Linear system of dimension 1, 2, or 3 are called a pencil, a net, or a web, respectively.
A map determined by a linear system is sometimes called the Kodaira map.
^Grothendieck, Alexandre; Dieudonné, Jean. EGA IV, 21.3.
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divisors and line bundles, the first twisting sheaf O ( 1 ) {\displaystyle {\mathcal {O}}(1)} is equivalent to hyperplane divisors. Since the ring of...
defined n-th plurigenus of V. The pluricanonical divisor K n {\displaystyle K^{n}} , via the corresponding linearsystemofdivisors, gives a map to projective...
of the interpretation of J as the linear equivalence classes ofdivisors on C of degree 0. That is, Q on C maps to the class of Q − P. Then since J is...
Function field of an algebraic variety Ample line bundle Ample vector bundle Linearsystemofdivisors Birational geometry Blowing up Resolution of singularities...
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{O}}(D)\end{cases}}} from the group of (Weil) divisors modulo principal divisors to the group of isomorphism classes of line bundles. A divisor corresponding to ωX is...
dimensions. Another equivalent condition is in terms of the linearsystemofdivisors on V cut out by the dual of the tautological line bundle on projective space...
adjoint linear system of a linearsystemofdivisors in classical algebraic geometry. This was re-expressed, with the advent of sheaf theory, in a way that...
algebraic geometry, an algebraic generalization is given by the notion of a linearsystemofdivisors. Weisstein, Eric W. "Family of Curves". MathWorld....
analyse the algebraic topology of an algebraic variety V. A pencil is a particular kind oflinearsystemofdivisors on V, namely a one-parameter family...
non-hyperelliptic case of g at least 3), Riemann-Roch, and the theory of special divisors is rather close. Effective divisors D on C consisting of distinct points...
} Computing all divisorsof the two numbers in this way is usually not efficient, especially for large numbers that have many divisors. Much more efficient...
sophisticated theory of handling a linearsystemofdivisors was developed (in effect, a line bundle theory for hyperplane sections of putative embeddings...
sheaf of differential one-forms on X. Linearsystemofdivisors (bundles of principal parts can be used to study the oscillating behaviors of a linear system...
1 subvarieties), rather than to sets of points, and regular divisors are contrasted with superabundant divisors, as discussed in the Riemann–Roch theorem...
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with a single divisor (itself), where on the other hand, 0 {\displaystyle 0} is the only number to have an infinite number ofdivisors, since dividing...
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