Coverage for local/lib/python2.7/site-packages/sage/schemes/readme.py : 100%

r""" 

Scheme implementation overview 

Various parts of schemes were implemented by 

Volker Braun, 

David Joyner, 

David Kohel, 

Andrey Novoseltsev, 

and 

William Stein. 

AUTHORS: 

- David Kohel (2006-01-03): initial version 

- William Stein (2006-01-05) 

- William Stein (2006-01-20) 

- Andrey Novoseltsev (2010-09-24): update due to addition of toric varieties. 

.. comment separator 

- **Scheme:** 

A scheme whose datatype might 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:** 

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 `[1,..,1]` 

on their coefficients. 

- **ProjectiveSpace_weighted (not implemented):** 

A special subtype for non-standard weights. 

- **ToricVariety:** 

Toric varieties are (partial) compactifications of algebraic tori 

`\left(\CC^*\right)^n` compatible with torus action. Affine and projective 

spaces are examples of toric varieties, but it is not envisioned that these 

special cases should inherit from ``ToricVariety``. 

- **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 `U = X \setminus Z` of a closed subset `X` 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 `X` plus the closed subscheme `Z` of `X` whose complement 

`U = X \setminus Z` is the quasi-projective scheme. 

.. NOTE:: 

The quasi-affine and quasi-projective datatype lets one create schemes 

like the multiplicative group scheme 

`\mathbb{G}_m = \mathbb{A}^1\setminus \{(0)\}` 

and the non-affine scheme `\mathbb{A}^2\setminus \{(0,0)\}`. The latter 

is not affine and is not of the form `\mathrm{Spec}(R)`. 

TODO List 

--------- 

- **PointSets and points over a ring:** 

For algebraic schemes `X/S` and `T/S` over `S`, one can form 

the point set `X(T)` of morphisms from `T\to X` over `S`. 

:: 

sage: PP.<X,Y,Z> = ProjectiveSpace(2, QQ) 

sage: PP 

Projective Space of dimension 2 over Rational Field 

The first line is an abuse of language -- returning the generators 

of the coordinate ring by ``gens()``. 

A projective space object in the category of schemes is a locally 

free object -- the images of the generator functions *locally* 

determine a point. Over a field, one can choose one of the standard 

affine patches by the condition that a coordinate function `X_i \ne 0` 

:: 

sage: PP(QQ) 

Set of rational points of Projective Space 

of dimension 2 over Rational Field 

sage: PP(QQ)([-2,3,5]) 

(-2/5 : 3/5 : 1) 

Over a ring, this is not true, e.g. even over an integral domain which is not 

a PID, there may be no *single* affine patch which covers a point. 

:: 

sage: R.<x> = ZZ[] 

sage: S.<t> = R.quo(x^2+5) 

sage: P.<X,Y,Z> = ProjectiveSpace(2, S) 

sage: P(S) 

Set of rational points of Projective Space of dimension 2 over 

Univariate Quotient Polynomial Ring in t over Integer Ring with 

modulus x^2 + 5 

In order to represent the projective point `(2:1+t) = (1-t:3)` we 

note that the first representative is not well-defined at the 

prime `pp = (2,1+t)` and the second element is not well-defined at 

the prime `qq = (1-t,3)`, but that `pp + qq = (1)`, so globally the 

pair of coordinate representatives is well-defined. 

:: 

sage: P( [2, 1+t] ) 

(2 : t + 1 : 1) 

In fact, we need a test ``R.ideal([2,1+t]) == R.ideal([1])`` in order 

to make this meaningful. 

"""