Scheme implementation overview - Schemes

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# Scheme implementation overview

Various parts of schemes were implemented by different authors. This document aims to give an overview of the different classes of schemes working together coherently.

## Generic

- 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 \(\mathrm{Spec}(R)\).

- AmbientSpaces: Most effective models of algebraic scheme will be defined not by generic gluings, but by embeddings in some fixed ambient space.

## Ambients

- 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 \((\CC^*)^n\) 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`.

## Subschemes

- 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 \(\mathrm{Spec}(R)\).

## Point sets

- PointSets and points over a ring (to do): 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 \(S\).

  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

      sage: PP.<X,Y,Z> = ProjectiveSpace(2, QQ)
      sage: PP
      Projective Space of dimension 2 over Rational Field
      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)

  Python

      >>> from sage.all import *
      >>> PP = ProjectiveSpace(Integer(2), QQ, names=('X', 'Y', 'Z',)); (X, Y, Z,) = PP._first_ngens(3)
      >>> PP
      Projective Space of dimension 2 over Rational Field
      >>> PP(QQ)
      Set of rational points of Projective Space
      of dimension 2 over Rational Field
      >>> PP(QQ)([-Integer(2), Integer(3), Integer(5)])
      (-2/5 : 3/5 : 1)

  Over a ring, this is not true anymore. For example, even over an integral domain which is not a PID, there may be no single affine patch which covers a point.

  Sage

      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

  [Python]

      >>> from sage.all import *
      >>> R = ZZ['x']; (x,) = R._first_ngens(1)
      >>> S = R.quo(x**Integer(2) + Integer(5), names=('t',)); (t,) = S._first_ngens(1)
      >>> P = ProjectiveSpace(Integer(2), S, names=('X', 'Y', 'Z',)); (X, Y, Z,) = P._first_ngens(3)
      >>> 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 \(p = (2,1+t)\) and the second element is not well-defined at the prime \(q = (1-t,3)\), but that \(p + q = (1)\), so globally the pair of coordinate representatives is well-defined.

  Sage

      sage: P([2, 1 + t])
      (2 : t + 1 : 1)

  Python

      >>> from sage.all import *
      >>> P([Integer(2), Integer(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.

## Berkovich Analytic Spaces

- Berkovich Analytic Space (not yet implemented) The construction of analytic spaces from schemes due to Berkovich. Any Berkovich space should inherit from `Berkovich`

- Berkovich Analytic Space over Cp A special case of the general Berkovich analytic space construction. Affine Berkovich space over \(\CC_p\) is the set of seminorms on the polynomial ring \(\CC_p[x]\), while projective Berkovich space over \(\CC_p\) is the one-point compactification of affine Berkovich space \(\CC_p\). Points are represented using the classification (due to Berkovich) of a corresponding decreasing sequence of disks in \(\CC_p\).

AUTHORS:

- David Kohel, William Stein (2006-01-03): initial version

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