Module: sage.schemes.readme
Various parts of schemes were implemented by David Kohel,
David Joyner, and William Stein.
This document:
Author Log:
- David Kohel (2006-01-03): initial version
- William Stein (2006-01-05)
- William Stein (2006-01-20)
- Scheme:
A scheme whose datatype might be not be defined in terms
of algebraic equations: e.g. the Jacobian of a curve may be
represented by means of a Scheme.
- AlgebraicScheme:
A scheme defined by means of polynomial equations, which may be
reducible or defined over a ring other than a field.
In particular, the defining ideal need not be a radical ideal,
and an algebraic scheme may be defined over Spec(R).
- AmbientSpaces: Most effective models of algebraic scheme will be
defined, not by generic gluings, but by embeddings in some fixed
ambient space.
- AffineSpace:
Affine spaces, and their affine subschemes form the most important
universal objects from which algebraic schemes are built.
The affine spaces form universal objects in the sense that a morphism
is uniquely determined by the images of its coordinate functions and
any such images determine a well-defined morphism.
By default affine spaces will embed in some ordinary projective space,
unless it is created as an affine patch of another object.
- ProjectiveSpace:
The projective spaces are the most natural ambient spaces for most
projective objects. They are locally universal objects.
- ProjectiveSpace_ordinary (not implemented)
The ordinary projective spaces have the standard weights
on their coefficients.
- ProjectiveSpace_weighted (not implemented):
A special subtype for non-standard weights.
- ToricSpace (not implemented):
This defines a projective toric variety, which defines a space
equipped with a toral action and certain defining data. These
generalise projective spaces, but it is not envisioned that the
latter should inherit from the
ToricSpace
type.
- AlgebraicScheme_subscheme_affine:
An algebraic scheme defined by means of an embedding in a
fixed ambient affine space.
- AlgebraicScheme_subscheme_projective:
An algebraic scheme defined by means of an embedding in a fixed ambient
projective space.
- QuasiAffineScheme (not yet implemented):
An open subset
of a closed subset
of affine space; note
that this is mathematically a quasi-projective scheme, but its
ambient space is an affine space and its points are represented by
affine rather than projective points.
NOTE: AlgebraicScheme_quasi is implemented, as a base class
for this.
- QuasiProjectiveScheme (not yet implemented):
An open subset of a closed subset of projective space; this datatype
stores the defining polynomial, polynomials, or ideal defining the
projective closure
plus the closed subscheme
of
whose complement
is the quasi-projective scheme.
Note: the quasi-affine and quasi-projective datatype lets one create
schemes like the multiplicative group scheme
and the non-affine scheme
. The
latter is not affine and is not of the form
.
Release 2006.05.25, documentation updated on May 25, 2006.
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