# 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.
• 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.

Affine $$n$$ space over a ring