With the projected specialised functionality for surfaces in mind, in contrast to curves, the dimension-2 schemes covered by the Magma surface subtype are assumed to be a variety (reduced and irreducible) defined over a field.
In the classification of algebraic curves, the geometric genus of the non-singular projective model is the unique discrete invariant. For surfaces, there are a number of independent discrete invariants besides the geometric genus and Magma has functionality to compute these which will be described below. If X is a non-singular projective surface then KX denotes the canonical sheaf or a canonical divisor. The notation nKX, n ≥ 1, denotes the n-canonical sheaf or divisor and the associated linear system is also termed n-canonical. The adjective pluricanonical is used to refer generally to one of these sheaves, divisors, or linear systems. The nth plurigenus, Pn, is the dimension of the space of global sections of the n-canonical sheaf. The first plurigenus is just the geometric genus. The higher plurigenera are also important surface invariants that cannot be automatically deduced from the genus as in the curve case.
An important classifier of a projective non-singular surface is its Kodaira–Enriques type. This refines the most basic classification by Kodaira dimension d, where -1 ≤ d ≤ 2 can be thought of as the maximum dimension of the image of the surface under birational divisor maps coming from positive multiples of the canonical divisor/sheaf (multi-canonical maps). Kodaira dimension d is -1 if and only there are no effective multi-canonical divisors of any degree. The notation -1 will be used in this section rather than the conventional -∞.
Another big difference between curves and surfaces is non-existence of a unique (up to isomorphism) non-singular projective surface in a birational equivalence class of surfaces. However, apart from surfaces of Kodaira dimension -1, there is a unique minimal model up to isomorphism in the equivalence class. The minimal models are characterised by having no exceptional curves (or (-1)-curves), which are non-singular rational curves on the surface with self-intersection number -1.
If Y is a minimal model of Kodaira dimension ≥ 0 then any birational map from a non-singular projective X to Y is a morphism (everywhere defined) and is purely a sequence of blow-downs. Thus, Y can be thought of as the unique surface at the base of an infinite blow-up graph representing the surfaces in a birational equivalence class. The minimal model is therefore an important surface that provides a canonical birational representative of a given class. Minimal models for surfaces can be computed with some restrictions on the surface presentation as described below.
As with curves, the ideal situation is to be able to get information about and perform operations on the non-singular projective surfaces that are birational desingularisations of an input surface X. This is a more difficult problem for surfaces. Function field methods based on normalising Dedekind domains are not available for varieties of dimension greater than 1. However, Magma does have code for computing the formal desingularisation of a surface in Proj3 using the Hirzebruch–Jung resolution procedure. The information returned by the algorithm does not give the full resolution graph of curves over a singular point or singular curve. It gives the reverse inclusion map between the formal power series rings which are the completions of the local rings at the generic points of a curve lying over the singular locus and its image in X, a point or curve.
In a similar way, a function is provided to compute the more general adjoint linear systems, which correspond to the sheaves mKX + nH where H is a hyperplane divisor on X or its pullback to the desingularisation. This allows the computation of the plurigenera as well as important canonical, anticanonical, pluricanonical, or adjoint maps on the desingularisation as explicit birational maps on X and their images such as pluricanonical models, etc.
This provides enough information to explicitly compute the m-canonical linear systems of the desingularisation as a sublinear system of the full linear system of homogeneous forms of degree m(d – 4) on X where d is the degree of X. In a similar way, there is a function to compute the more general adjoint linear systems, which correspond to the sheaves mKX + nH where H is a hyperplane divisor on X (or its pullback to the desingularisation), as sublinear systems of the degree n + m(d – 4) forms. This allows the computation of the plurigenera as well as important canonical, anticanonical, pluricanonical, or adjoint maps on the desingularisation as explicit birational maps on X and their images (pluricanonical models, etc.). The computed (formal) desingularisation is a non-uniquely defined member of a birational equivalence class which is often not minimal. However, all of the things listed above are birational invariants that are independent of whichever desingularisation is constructed.
The formal desingularisation package can be used for projective surfaces in higher-dimensional ambients by first projecting birationally down to Proj3. There are some problems with this approach, however. An alternative approach taken in Magma is to provide functions that can efficiently compute properties of interest for surfaces lying in higher-dimensional ambients with only singularities of restricted type. A suite of general functions based on the sheaf machinery is provided; these functions work on (mainly ordinary projective) surfaces that only have simple singularities.
An algorithm for desingularising surfaces by local blow-up was developed and implemented by M. Harrison (CAG). This complements the existing machinery for formally desingularising projective hypersurfaces (surfaces in Proj3) using the Hirzebruch–Jung method. The issue with the older method is that it is much too slow when applied to many practical examples, often hanging.
The local blow-up procedure, applied to a scheme-theoretic singular point p/k (p corresponding to a Galois-conjugate set of singular points on S) returns:
The blow-up desingularisation data includes the same functionality as formal desingularisation and also extra functionality which is impossible with the formal version. This comes in two flavours: functions which relate to the canonical divisor class on the desingularisation and others which are unrelated. The former have only been implemented for surfaces in Proj3 because it is relatively easy to handle canonical sections in the hypersurface case. A natural basic pseudo-differential is available on which the computation of sections of pluri-canonical, pluri-anticanonical and, more generally, multi-adjoint sheaves on the desingularisation can be based.
The non-canonical functionality includes the computation of the intersection matrix of the IEDs above each singular point, the multiplicities w.r.t. IEDs of the pullbacks of divisors on S to the desingularisation X and the intersections with IEDs of the strict transforms of S-divisors to X (the pullbacks with all IED components removed). A function is also available to compute the restriction of a linear system on S by imposing vanishing (with multiplicity) conditions at the IEDs. This is extremely important for computing Riemann–Roch spaces and divisor maps of the strict transforms of divisors on S. For example, it is used to compute multi-adjoint maps and can also be used to compute the blow-down of exceptional lines on degenerate Del Pezzo surfaces.
One of the examples that was analysed rigorously using the new machinery was a degree 9 hypersurface S over ℚ that was sent in by a Magma user. S has only point singularities but these are quite nasty. The formal desingularisation method hung at an early stage when applied to S. In contrast, the blow-up method is able to resolve all of the singularities in about 2 seconds. The singularities of S consist of 2 conjugate A3 singularities, 2 pairs of conjugate A1 singularities, and two non-simple singularities s1 and s2 defined over ℚ. The minimal resolution of s1 has 11 rational IEDs above it, 10 (-2)-curves and a (-3)-curve. The minimal resolution of s2 has 38 rational IEDs above it, 33 (-2)-curves, 4 (-3)-curves and a (-4)-curve.
We were able to work out for the user that the surface was (birationally) an elliptic fibration of general type (Kodaira dimension 1) with geometric genus 3 and that the canonical map gave an explicit elliptic fibration map. These further computations took about 1 second in total. The intersection matrices — giving the graphs of IEDs over the singular points — also took less than a second except for the singularity s2, where the computation of the self-intersections of the 38 IEDs took about 5 minutes. However, the non-self-intersections for s2 were all computed in a fraction of a second.
We used the full suite of functions to go further and locate 4 (-1)-curves and another 23 curves on the desingularisation that needed to be contracted to give a minimal model. We were also able to work out the entire structure of the set of bad fibres of the minimal elliptic fibration. Apart from the fibres containing IEDs over the simple singularities, the IEDs over s1 and s2 along with a number of linking curves in S gave 4 distinct singular fibres in the minimal model each of Kodaira/Neron type I8.
As described elsewhere, structures have been created for testing whether an isolated singular point on a scheme (over a field) is analytically isomorphic to a (isolated) analytic hypersurface singularity and, if so, to compute an equivalent analytic equation and transformation to this up to an arbitrary precision. This precision can be increased at a later stage if required.
Isolated hypersurface singularities up to analytic equivalence can be transformed into a sum of a diagonal quadric part x12+… +xr2 in the first r variables and a residual part F(xr+1,…,xn) in the remaining variables. The residual part is a power series with all terms of total degree ≥ 3. Two singularities are equivalent if and only if their residual parts are equivalent as (isolated) hypersurface singularities in the ambient space given by the final variables. This is all true formally over any (algebraically closed) field of characteristic not equal to 2. The corank of the singularity is m = n – r, the number of variables of the residual part. Classification up to analytic equivalence comes down to the classification of the residual parts in all possible coranks and transformation to normal form consists of first transformation to diagonal plus residual form and then transformation of the residual form to an appropriate normal form.
Functionality has been implemented which determines whether an isolated analytical hypersurface singularity lies in one of Arnold's families and, if so, computes the exact family, normal form and the analytic transformation to that normal form to any required degree of precision. The explicit transformation allows the user to transfer any geometric problem that can be defined in terms of local functions at the singularity over to the corresponding one on the normal hypersurface form, which is usually of a much simpler shape. As opposed to other computer algebra packages that deal with Arnold normal forms, the original singularity is not required to be on an actual hypersurface. It only has to be analytically equivalent to a hypersurface singularity. Another useful feature is that this machinery for singularities applies over fields of characteristic p > 0. Arnold's classification and analysis applies to characteristic 0. However, all of his families make sense in characteristic p > p0, where p0 is a small prime depending on the family (and its indices).
The base field over which the input singularity is defined will usually be a finite field or number field k. To effect the full transformation to normal form, which is only possible in general over an algebraically-closed field, requires an extension of k. To avoid unnecessarily large field extensions, certain trivial final steps are ignored in the normalisation and also slightly different but easily equivalent normal forms are used in places. The major changes have been to avoid normalising coefficients of terms of the form axn to 1 by taking an nth root of a, and to remove binomial terms of the form ax2yn rather than axy2n via x→x + αyn transformations. The latter avoids making unnecessary cubic field extensions. The differences between the normal forms used in this approach and Arnold's are all documented, and the user can easily make their own final transformation to Arnold normal form.
Other features of the implementation include a one or two-stage reduction to normal form and a uniformised computation of the transformation that works through one of only 3 auxiliary functions which deal with the semiquasihomogeneous cases, the non-sqh cases where Arnold's condition A is satisfied, and the non-sqh cases where the full ‘spectral sequence’ processing is required.
The functions that are described below apply to either the case of surfaces in Proj3, where formal desingularisation is used, or to surfaces in ambients of arbitrary dimension when the simple singularity assumption is required.
Again, once the Kodaira–Enriques class has been determined (see below), these can all be computed from the basic invariants and class or with the computation of only a few of the smaller plurigenera. For example, for surfaces of general type, only the basic invariants and second plurigenus P2 are needed to compute all Pn by formula, and only the basic invariants are required if the surface is known to be minimal.
For surfaces of Kodaira dimension ≥ 0 there are functions to compute a minimal model and the birational morphism from the original surface to it. For Kodaira dimensions 0 and 1 these functions can currently only be applied to non-singular (ordinary projective) surfaces. They use repeated adjunction maps, iterating until the termination condition c12 = 0 applies to the image. If H is a hyperplane divisor of X in its projective embedding, the adjunction map is the divisor map of the H + KX divisor. Adjunction theory says that this map is birational, everywhere defined (a morphism), and that its image is again a non-singular surface if X is not rational or ruled. Furthermore, it blows down all exceptional curves of degree 1 in the projective embedding and it reduces the degree of higher degree exceptional curves or maps them to non-exceptional curves. It is therefore clear that iteration will eventually blow down all exceptional curves and give a minimal model. For Kodaira dimension 1 the function also returns the fibration map to a curve that presents the minimal model as an elliptic fibration.
There are parametrisation functions for rational surfaces. These construct a birational map from X to Proj2 by first mapping to a terminal surface X1 and then calling special parametrisation routines for Del Pezzo surfaces and line and conic bundles which will be described below. For rational surfaces with only simple singularities, the map to the terminal surface is constructed as described above. For rational surfaces in Proj3 with general singularities, a special routine is used that finds an appropriate multi-adjunction map, associated to a nH + mKX divisor, which maps X to a terminal surface in one step. This first map can be constructed over an arbitrary base field k. Parametrisation of the terminal surfaces is an arithmetic question, though, and the existence of such a parametrisation depends on k. The routines for parametrising the terminal surfaces assume that k is ℚ or a number field.
There are a number of functions to construct random surfaces from interesting families of non-singular projective surfaces in Proj4. These families cover a number of different Kodaira–Enriques types. The constructions use ‘codimension 2’ commutative algebra techniques and are based on functions in the Macaulay Computer Algebra System for the same families.
An example of a rational surface in Proj3 parametrisable over ℚ is the non-normal surface with defining polynomial w3y2z + (xz + w2)3. The classification routine maps it to a degenerate degree-6 Del Pezzo surface in Proj6 with an A2 singularity. Then the parametrisation routines for Del Pezzo surfaces construct a parametrisation map Proj2→X given by degree-6 polynomials with rational coefficients.
A (strong) Del Pezzo surface X is a rational surface for which the anticanonical divisor -KX is ample. This means that the divisor map of a multiple of the divisor gives an isomorphic embedding into ordinary projective space. The degree d of X is equal to its first Chern number KX.KX and takes values between 1 and 9 inclusive. For d ≥ 3, -KX gives an embedding of X as a non-singular surface of degree d in Projd. For degrees 1 and 2 it is necessary to take the multiple (4 – d)(-KX) to get an ordinary projective embedding. In these last two cases there is an anticanonical embedding as a hypersurface in an appropriately-weighted projective space of dimension 3. A weak Del Pezzo surface is one where -KX is not necessarily ample but is close to ampleness; technically, it is big and nef. These differ only slightly from the (strong) Del Pezzo surfaces.
Del Pezzo surfaces are represented in Magma by their anticanonical models. Both strong and weak Del Pezzo surfaces are allowed but some of the current functionality only applies to strong Del Pezzo surfaces. A Del Pezzo surface of degree d can also be defined as a surface which is isomorphic over the projective closure of its base field k to the projective plane Proj2 blown up at 9 – d points. The points have to be distinct and satisfy certain conditions for a strong Del Pezzo surface. For weak Del Pezzo surfaces, the blow-up points can be infinitely-near (i.e., lying on the exceptional curve of a point that has already been blown up) and the set of points has to satisfy weaker conditions. There are functions to check the set of conditions on a given set of plane points and construct the corresponding Del Pezzo surface. There is also a function to test whether a given non-singular projective surface X is abstractly a strong Del Pezzo surface and return the anticanonical model and isomorphism from X to it.
There is a suite of functions that constructs a parametrisation of a Del Pezzo surface over ℚ and number fields. This covers all degrees and also the weak case. For high degree Del Pezzo surfaces the functions use a method based on explicitly computing the Lie algebra of the automorphism group of the surface. Lower degree cases are reduced to the higher degree case by blowing down Galois-stable sets of disjoint exceptional curves. For the lower degree degenerate cases, special algorithms based on projecting from singular points are used. The functions return whether the input Del Pezzo surface X is parametrisable over the base field k and also returns an explicit parametrisation (birational map Proj2→X) over k in the affirmative case.
Other functionality includes functions for computing and working with the lines on a cubic (degree 3 Del Pezzo) surface in Proj3. There are also functions to compute the classical invariants and covariants of a cubic surface and to transform its equation to pentahedral form (that is, the sum of 5 cubes of linear forms) if possible.
On the arithmetic side, minimisation and reduction routines are provided for degree 3 and degree 4 Del Pezzo surfaces over ℚ which find isomorphisms over ℚ to models with simpler defining equations containing smaller integer coefficients.