Magma has a wide range of functionality for performing hard arithmetic and geometric computations with algebraic varieties or more general schemes of finite-type over a base ring. A scheme is defined by a finite set of multivariate polynomials on an ambient space or the corresponding polynomial ideal generated by these. It can be thought of as the space of simultaneous solutions of this system of polynomials; however, schemes may be non-reduced. The reader should recall that the scheme defined by an ideal is not considered the same as the scheme defined by the radical of that ideal. Amongst other cases, this is important for questions involving multiplicity of intersections or map fibres.
A subscheme Y of a scheme X is one in the same ambient space whose defining ideal I(Y) contains the defining ideal I(X) of X, up to a redundant ideal factor. In the literature this is often referred to as a closed subscheme, but we use the more concise terminology here. A variety is a scheme that is reduced and irreducible whose base ring is a field. Thus, a variety cannot be decomposed into a finite union of subschemes none of which contain the other. The dimension of a scheme X can be defined as being one more than the length of a maximal totally ordered (by inclusion) chain of subvarieties of X. It is equal to the maximum of the dimensions of the irreducible components of X. The dimension is the most important property of a scheme. Curves are one-dimensional schemes and surfaces are two-dimensional.
The ambient spaces A are generally defined by a multivariate polynomial ring R over the base ring k with a single or multiple ℤ-grading of the n variables. If there is no grading, this is taken to mean n-dimensional affine space kn. The ring R, referred to as the coordinate ring of A, gives the allowable global algebraic functions on A in the scheme category; that is, polynomials in the variables. Quotients of polynomials give rational functions on A. With a single grading where all variables have weight 1, A is an ordinary projective space over k, ℙn-1(k).
The defining data for a scheme X lying in ambient A are polynomials or ideals of R that are jointly homogeneous with respect to each grading (there is no extra condition in affine spaces). The points of X over a field (or ring) L are the points of A on which the defining polynomials all vanish.
As well as the importance of schemes/varieties for a large number of algebraic and arithmetic problems in pure mathematics and its applications, many of the real or complex manifolds arising in analysis are spaces of real or complex points of varieties defined over a number-field. In the case of compact complex manifolds coming from a projective scheme, rational functions and maps on the manifolds correspond precisely to the corresponding algebraic objects on the schemes by the GAGA principle. Other objects like vector bundles, coherent sheaves and their cohomology groups, and global divisors also correspond in a one-to-one fashion to those on the algebraic side. Thus, problems like the computation of cohomological invariants, intersection numbers, global embeddings into projective space, or isomorphisms can be tackled with algebraic geometry packages such as Magma's.
Magma schemes are all globally represented and the ones that can be defined are quasi-projective as abstract schemes — mainly affine or projective. However, as described in the introduction, there is a wide range of projective space ambients in which they may lie and any closed subscheme of one of these ambients is representable. Most of the deeper functionality is only available for schemes whose base ring k is an (exact) field. The ring k may be an arbitrary Magma ring, however, and it can be useful to define schemes over the integer ring ℤ, for example, and maps over ℤ in order to conveniently specialise to multiple prime fields by reduction mod p. Gröbner basis computations over ℤ can be convenient to determine a finite set of primes outside of which there is good reduction. The geometry machinery is designed for exact algebraic computation and very little beyond the basic constructors will work for inexact fields like ℝ and ℂ.
As an example, consider the homogeneous polynomialIn projective ambients, a one-to-one correspondence between ideals and schemes does not exist. Two ideals define the same scheme if and only if their primary components differ by a redundant ideal that depends on the gradings of the ambient. There is a maximal (saturated) ideal that defines a given scheme and it can be necessary to compute this to make correct computations of dimension, images of maps, tests for being reduced or irreducible, and a number of other properties.
Projective schemes are covered by standard affine patches which are open subsets for the Zariski toplogy. These are needed internally for certain computations where it is necessary to reduce to commutative algebra operations in an affine setting and they can be useful to the user for the same reasons. It should be noted, however, that most of the algorithms work with homogeneous coordinate rings and ideals, which is generally more efficient.
Currently in Magma not all of the standard affine patches are constructible in some weighted projective spaces. However, all of the patches are available in the commonest cases of ordinary and product projective spaces. One place where affine patches are essential is in working with individual points in pointsets on schemes. A point is definable only if it lies in one of the constructible affine patches although it is represented by its homogeneous coordinates in the projective space. In bad cases, it is always possible to consider the point as a zero-dimensional subscheme of the base scheme X and work with that.
There are functions to return the affine patches of a projective scheme and to restrict various structures like maps to patches. There is also a projective closure map to compute the projective closure of an affine scheme. If the affine scheme is the patch of an existing projective one, that is returned (the information linking patches and projective schemes is stored internally). Otherwise, the usual projective closure in ordinary projective space is returned.
There is a lot of functionality for working with rational maps between schemes. Two very useful functions for rational maps are the following: An extension function is provided which attempts to find alternative equations for the map so as to define it on a larger open subset of the domain. If the codomain is affine, ordinary projective, or product projective, then this will extend the map to its maximal domain of definition. There is also a function to test whether a map is invertible (as a rational map) and compute and add inverse defining equations to explicitly present it as a birational map.
Magma can compute the image scheme f(Z) for a subscheme Z of X and a rational map f : X→Y. It can also compute inverse images (pullbacks) of subschemes of Y. The same is true for points on X or Y. The image computations will return an error if Z or the point lies in the complement of the open set on which f is defined by its current set of defining equations or alternative defining equations. This is a situation in which the extension function is often very useful because more complicated rational maps are usually not maximally defined by a single set of mapping equations.
A variant on standard maps are graph maps, which can only be defined for maps between ordinary projective schemes. Here the map is defined by the scheme-theoretic closure of its graph in X×Y. The advantage of graph maps is that they are automatically defined on their maximal domain of definition and testing for invertibility as well as computing the inverse (which will also be maximally defined) is much quicker and more efficient.
Of the particular rational maps for which special construction functions are available in Magma, blow-ups should be mentioned. A function is provided to compute the (global) blow-up scheme Z of scheme X by an arbitrary subscheme Y of X together with the blow-up map φ of Z to X. The map φ gives an isomorphism of the Zariski-open subsets Z/φ-1(Y) and X/Y and φ-1(Y) is of codimension 1 in Z whatever the codimension of Y in X. If Y is a non-singular subvariety of a non-singular variety X, the points of Z above a point P in Y correspond to the tangent directions in X that are normal to Y, which explains the name ‘blow-up’. Blow-ups of points (as reduced subschemes) and their inverses, blow-downs, are extremely important for surfaces in particular. Any birational map between two non-singular projective surfaces can be factored into a sequence of blow-ups and blow-downs.
Properties and objects that can be computed for a scheme X are listed below:
The following constructions are available:
More sophisticated functionality for schemes of arbitrary dimension is available in the sheaves and divisors package.
Zero-dimensional schemes are discrete schemes that only contain a finite number of points over any extension of the base field. These are the solutions of algebraic systems of equations with only finitely many solutions. Some special functionality is provided for these schemes. The total number of solutions with multiplicity may be computed with the degree function (which does not require the scheme to be ordinary projective) and the total number of distinct solutions is given by the degree of the reduced subscheme. There are functions that list all points (solutions) over a given extension of the base field or list all of them over an algebraic closure of the base field.
Code is being developed in Magma to classify isolated hypersurface singular points according to the classification scheme of V. I. Arnold and to compute their normal form. The analytic transformation to the normal form can also be computed to arbitrary precision.
To extend the range of applicability of Magma's singularity analysis, a function is provided that transforms a singular point P on a scheme X to an analytically equivalent hypersurface singularity if this is possible. When it is, the singularity is referred to as an analytic hypersurface singularity or AHS for short. When X is reduced and all irreducible components of X through P have the same dimension d, the necessary and sufficient condition for P to be singular and an AHS is that the tangent space of X at P has dimension d + 1.
Let P on X be an isolated singular point (that is, P does not lie on a positive dimensional component of the singular subscheme of X) and assume that X is a hypersurface or that P is an AHS on X. The classification function determines whether P is a singularity that lies in one of the families of Arnold's classification. If so, the function returns the name of the family and also computes Arnold's polynomial normal form of the singularity. Optionally, the user may also ask for an analytic automorphism R→R, computed to desired precision, that maps the analytic hypersurface equation F to the normal form.
An example of a singularity that is not on a hypersurface but is analytically isomorphic to a hyperplane singularity is the origin O of the surface S in 4-space defined by the two polynomials x2+yt+t2 and z3+xt+y2 – t. The above functions first determine that O is an AHS and then that it is an E7 singularity with normal form (in three variables) x2+y3+yz3. In this case, projection onto the 3-dimensional tangent space at O does not give an isomorphism to a hypersurface singularity (this global projection technique works in some examples). Instead, it produces a non-isomorphic singularity that is no longer isolated.
Full testing for singularity/non-singularity of schemes lying in ambients other than affine, ordinary projective, or product projective space has been a major omission of the geometry package since the introduction of schemes into Magma. The basic singularity test formally applies the Jacobian criterion to the affine or projective coordinate ring of scheme X. In the projective cases given above, the coordinate rings are those of faithfully-flat affine cones over X for which the fibres are non-singular. By standard results on faithfully-flat scheme coverings, applying the Jacobian criterion to the projective coordinate ring gives correct results. However, for projective ambients with different weightings, the result will generally be incorrect. Essentially, the criterion does not apply where X meets the singular locus of the ambient, where it will usually give the conclusion of singularity, even when X is non-singular there.
The most useful ambients apart from those mentioned above are the weighted projective spaces (WPSs) with positive integral weights. The singularities of the ambient occur in the affine patches that are undefined internally in Magma; that is, the patches corresponding to the localisation of a variable whose weight does not divide all other weights (i.e., of weight > 1 in the usual case where the set of weights has been normalised to have no common divisor). These affine patches exist but are not isomorphic to affine space with one fewer variable than the WPS.
A facility for constructing “user-level” affine patches for WPS at indices of variable weight greater than 1. While not yet integrated into the low-level scheme structures, they can be obtained by a function call which also returns a projective closure map to give the birational restriction from the original ambient to the patch and the inclusion map of the patch into the original ambient. Another difference is that the inclusion map is also a birational map that is not everywhere-defined. Mathematically, it is everywhere defined of course, but it cannot be everywhere defined using ordinary Magma scheme maps.
The affine patches are computed in a natural way using Magma's invariant theory functionality applied to a finite cyclic group action on a full polynomial ring. This gives the patch as an affine variety with relations. For weights of even a medium size, the number of variables of the patch is often much larger than the original number of projective variables, but it is usually still much smaller than the number of variables need to re-embed the WPS into ordinary projective space by d-uple embedding, say.
Routines have also been implemented to embed product projective schemes into ordinary projective space using Segre embeddings. This increases ambient dimension but can still be a useful tool in order to apply sheaf or divisor functionality that is only available at present for ordinary projective schemes. The functions work for an arbitrary number of components of the product projective space and also return isomorphisms from the original to the re-embedded scheme that have alternative inverse defining equations giving the inverse everywhere.