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...

divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors...

Alexander von Brill and Max Noether (1874), is the study of special divisors, certain divisors on a curve C that determine more compatible functions than...

special divisors is a result of William K. Clifford (1878) on algebraic curves, showing the constraints on special linear systems on a curve C. A divisor on...

broadest of the "Bertini theorems" applying to a linear system of divisors; simplest because there is no restriction on the characteristic of the underlying...

generalization of a sequence Net, a linear system of divisors of dimension 2 Net (polyhedron), an arrangement of polygons that can be folded up to form...

separation of variables in the Hamilton–Jacobi equation Web, a linear system of divisors of dimension 3 Web Entertainment, a record label Web (album), a...

divisors and line bundles, the first twisting sheaf O ( 1 ) {\displaystyle {\mathcal {O}}(1)} is equivalent to hyperplane divisors. Since the ring of...

of the interpretation of J as the linear equivalence classes of divisors on C of degree 0. That is, Q on C maps to the class of Q − P. Then since J is...

defined n-th plurigenus of V. The pluricanonical divisor K n {\displaystyle K^{n}} , via the corresponding linear system of divisors, gives a map to projective...

Function field of an algebraic variety Ample line bundle Ample vector bundle Linear system of divisors Birational geometry Blowing up Resolution of singularities...

of the points are collinear and no seven points lie on a conic. The Cayley–Bacharach theorem can be deduced from this fact. Linear system of divisors...

condition, so determined a quadric in P5. However the linear system of divisors consisting of all such quadrics is not without a base locus. In fact...

1.  A homaloidal linear system of divisors is a linear system of grade 1, such as the image of the linear system of hyperplanes of projective space under...

adjoint linear system of a linear system of divisors in classical algebraic geometry. This was re-expressed, with the advent of sheaf theory, in a way that...

{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 linear system of divisors on V cut out by the dual of the tautological line bundle on projective space...

the linear system of divisors defining the embedding of V can be related to the line bundle or invertible sheaf defining the embedding by its space of sections...

} Computing all divisors of the two numbers in this way is usually not efficient, especially for large numbers that have many divisors. Much more efficient...

reduced row echelon form to solve a system of linear equations over a field. Using matrix notation every system of linear Diophantine equations may be written...

pencil is the special case of a linear system of divisors in which the parameter space is a projective line. Typical pencils of curves in the projective...

algebraic geometry, an algebraic generalization is given by the notion of a linear system of divisors. Weisstein, Eric W. "Family of Curves". MathWorld....

sophisticated theory of handling a linear system of divisors was developed (in effect, a line bundle theory for hyperplane sections of putative embeddings...

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...

analyse the algebraic topology of an algebraic variety V. A pencil is a particular kind of linear system of divisors on V, namely a one-parameter family...

qualitative study of dynamical systems, that is, properties that do not change under coordinate changes. Linear dynamical systems and systems that have two...

a permutation of the Jordan blocks Index of nilpotence Elementary divisors, which form a complete set of invariants for similarity of matrices over a...

1 subvarieties), rather than to sets of points, and regular divisors are contrasted with superabundant divisors, as discussed in the Riemann–Roch theorem...

that the divisors are pairwise coprime (no two divisors share a common factor other than 1). For example, if we know that the remainder of n divided...

with a single divisor (itself), where on the other hand, 0 {\displaystyle 0} is the only number to have an infinite number of divisors, since dividing...