14 Algebraic Geometry

14.1.1 Ambient Spaces

A scheme is contained in some ambient space, either an affine space or one of a small number of standard projective spaces. A characteristic of these spaces is that they have some kind of polynomial ring as their coordinate ring.

• Affine and projective spaces including weighted projective space

• Ruled surfaces

• Rational scrolls

• Direct products of ambient spaces

• Points of schemes with coefficients in k-algebras

14.1.2 Creation and Properties

• Creation of schemes by equations or implicit methods

• Saturation of the defining ideal of a scheme in the projective cases

• Changing the base ring

• Point set operations

• Projective closure and affine patches

• Global properties: Dimension, reducibility, singularity

• Prime components, primary components

• Local geometric analysis: singularities, tangent spaces

• Zero-dimensional schemes

• Determining whether a scheme over a number field is locally solvable

• Searching for points (over the rationals).

• Listing of all points oevr a finite field.

14.1.3 Mappings

• Construction of maps between spaces

• Coordinate manipulation functions, including birational transformations of the projective plane

• Calculation of pullbacks of schemes by maps

• Tools to enable the calculation of images of schemes by maps

• A large number of specialised types of schemes, especially curves, have additional map functionality

• Availability of maps with multiple definitions

• Alternative map type for ordinary projective schemes defined by the "graph" of the map in the product scheme.

14.1.4 Automorphisms

Automorphisms of schemes defined over a field may be constructed. The main cases where there is significant functionality is for automorphisms of affine and projective spaces and algebraic curves.

• Construction of general automorphisms of affine space in terms of functions or matrices

• Construction of special automorphisms of affine space: translation, permutation automorphisms, Nagata automorphism, projectivities

• Construction of general automorphisms of projective space in terms of polynomials or matrices

• Construction of special automorphisms of projective space: translation, quadratic transformation

• Automorphism group of a projective space defined over a finite field

• Automorphism groups of algebraic curves of genus at least 2 (see below).

14.1.5 Isomorphic Projections

• Computation of the tangent and secant varieties of schemes

• Faster tests for whether a specified point lies in the tangent or the secant varieties of a scheme.

• Isomorphic projection of projective schemes to smaller dimensional ambient spaces

• Birational embedding of plane curves as non-singular projective space curves in 3-dimensional space

14.1.6 Linear Systems

The complete linear system on P of degree d is the collection of all homogeneous polynomials of degree d on P, or equivalently, the degree d hypersurfaces thereby defined. A general linear system corresponds to some vector subspace of the coefficient space of a complete linear system.

• Creation of a linear system explicitly

• Creation of a linear system satisfying geometric conditions

• Properties: Sections, degree, dimension, base scheme

• Properties: Base component, base points, generic multiplicity at a point

• Complement with respect to a subsystem or scheme

• Intersection

• Maps of a linear system

14.3.1 Construction and General Properties

• Creation of affine and projective curves together with the ambient spaces

• Basic manoeuvres between affine and projective curves: projective closure, affine patches and so on

• Basic scheme-type functions: e.g., irreducibility

• Specialised data types for distinct classes of curves such as elliptic curves

• Linear systems of curves in the projective plane with assigned basepoints

• Implicitization of parametric curves

• Parametrization of rational curves

• Point searches on plane curves using a p-adic method of Elkies

• Global invariants of curves, such as genus and dimension

• Function field and divisor computations (see below)

• Random curves by genus over the rationals or finite fields for all genera up to 13.

• Index calculus for discrete logs on the Jacobians of plane curves.

14.3.2 Mappings

• Creation of the basic automorphisms of the affine plane (translation, flip, automorphism) and general maps between curves.

• Evaluation of the image of a point under a rational map as a power series expansion

• Ramification divisor of a non-constant dominant map

• Pullbacks and pushforwards of functions, differentials or divisors along maps between curves.

• Explicit computation of the map into ordinary projective space associated to a divisor (class) - in particular the canonical map. Optimised computation of canonical images.

14.3.3 Automorphism groups

• Computation of the full automorphism group (over the base field) of an (integral) algebraic curve of genus at least 2.

• Structures for defining and working with groups of automorphisms of such curves. In particular, efficient arithmetic of automorphisms (products, powers, etc.) and subgroup definition.

• Abstract realisation of an automorphism group as a permutation group with 2-way correspondence. This allows the full range of abstract group constructions (eg p-Sylow subgroups) to be applied.

• Computation of the quotient of a curve by a group of automorphisms as a new curve with an explicit quotient scheme map.

• Tests for isomorphism and construction of explicit isomorphisms as scheme maps.

14.3.4 Local Analysis

• Calculation of tangent spaces and cones for plane curves

• Identification of all singularities of a curve, together with the basic analysis

• Blowups, including weighted blowups, of points on plane curves

• Local intersections on plane curves

14.3.5 Ordinary Plane Curves

• Tests for only ordinary or only nodal singularities of plane curves.

• Special computation of the canonical or more general adjoint maps for an ordinary plane curve.

• Random nodal curves of specified degree and number of singularities.

• Random ordinary curves of specified degree and with specified number of ordinary singularities of given multiplicities.

• Special parametrization routines for ordinary rational plane curves and rational normal curves.

14.3.6 Function Field

The following functions apply to integral curves in general.

• Construction of the function field of a curve

• Function field arithmetic

• Exact constant field

• Enumeration of places of degree m (over a finite field)

• Residue class field

• Class number

• Divisor class group of a curve defined over a finite field

• Group of global units of the function field of a curve

14.3.7 Divisors and the Riemann-Roch Theorem

The following functions apply to integral curves in general.

• Creation of places and divisors and their arithmetic

• Compute the divisor of a function on a curve and other constructors

• Determine whether a divisor is principal and if so find a function with the given divisor

• Ramification divisor

• Riemann–Roch space of a divisor

• Computation of Weierstrass places on a curve

14.3.8 Differentials

The following functions apply to integral curves in general.

• Arithmetic and various, related operations

• Valuation of a differential at a place

• Divisor of a differential

• Differential spaces for given divisors (e.g. holomorphic differentials)

• Higher differentiations

• Residue of a differential at a place of degree one

• Cartier operator and representation matrix of the Cartier operator (curves over finite fields)

14.3.9 Resolution Graphs and Splice Diagrams

• Creation of resolution graphs and their vertices, either implicitly using resolution routines or Newton Polygons, or explicitly by listing the required data

• Calculation of numerical data associated to blowups, e.g., the canonical class of certain rational surfaces

• The Cartan matrix of a graph and associated calculations such as the contribution to the genus of a plane curve of a singularity having a given graph as its resolution

• Surgery on resolution graphs such as cutting an edge of a graph

• Creation of splice diagrams implicitly and explicitly

• Edge determinants and linking numbers of splice diagrams

• Test for regularity of splice diagrams

• Translation between resolution graphs and splice diagrams

These decorated graphs encode data generated by the resolution machinery. At present they are only attached to resolutions of plane curve singularities, but in due course they may be extended to resolutions of linear systems, surface singularities, special fibres in curve fibrations or other geometrical contexts. The package includes a resolution function for curves adapted to present its output in resolution graph format. This does not supersede other resolution machinery such as Puiseux expansions or genus calculations, but is intended as a complementary tool for those users with this kind of geometric background.

14.4.1 Formal Desingularisation and Classification

• Formal resolution of singularities of hypersurfaces in 3-D projective space over a field of characteristic zero, following the Hirzebruch-Jung method.

• Result given using algebraic power series for the formal completions of components over the singular locus.

• Geometric genus, arithmetic genus and higher plurigenera of (any) desingularisation.

• General K_Xⁿ(m) adjoint linear systems.

• Tests for a hypersurface being rational or (birationally) ruled.

• Classification of rational surfaces by terminal type. Construction of birational map to terminal model following Schicho's algorithm.

14.4.2 Parametrization Of Rational Surfaces

• Test for the parametrizability over the rational numbers of a rational hypersurface in 3-D projective space and construction of an explicit parametrization.

• Routines for the direct parametrization of singular or non-singular anticanonically embedded Del Pezzo surfaces of degree at least 5 over the rationals. Uses the Lie algebra method for degree greater than 5.

• General routine for direct parametrization of any degree Del Pezzo (singular or non-singular) in its standard weighted anticanonical ample embedding by blowing down lines to reduce to degree at least 5.

• Routines for the direct parametrization of quadric hypersurfaces, rational pencils and conic bundles.

14.5.1 Toric lattices

One often begins to study toric geometry by considering a lattice L = ℤⁿ and discussing polygons or cones in its overlying rational vector space L_ℚ = L⊗ℚ. In Magma we bind these two spaces together in a single object, a toric lattice. Although these spaces lie underneath all the combinatorics, it is seldom necessary to be explicit about them; they often remain invisible. Magma supports many operations for constructing and manipulating toric lattices, including:

• Creation of a toric lattice and its dual;

• Construction of direct sums, sublattice, and (torsion free) quotient lattices;

• The sublattice fixed under a group action;

• Maps between toric lattices;

• Standard operations on points in a toric lattice.

14.5.2 Cones and Polyhedra

A cone (in a toric lattice L) is the convex hull of finitely many rays. A (rational) polytope is the convex hull of finitely many points of L. More generally a (rational) polyhedron is given by the Minkowski sum of a polytope and a cone. There exist dual definitions in terms of the intersection of finitely many half-spaces. There is no requirement that polyhedra are of maximum dimension in the ambient toric lattice. Important operations include:

• Standard constructions of cones, half-spaces, cross polytopes, cyclic polytopes, etc.;

• Creation of cones and polyhedrons via a system of inequalities;

• The defining inequalities of a cone or polyhedron;

• The dual cone or polyhedron;

• Taking the (affine) linear space spanned by, or contained in, a polyhedron;

• Standard properties such as dimension, volume, and boundary volume;

• Triangulations of a polytope and of its boundary;

• Information on whether a polyhedron is simplicial, reflexive, terminal, etc.;

• Construction of the cone in L×ℤ spanned by a polyhedron P in L;

• The polyhedron defined by taking the convex hull of P∩ℤⁿ for any polyhedron P;

• The infinite and compact parts of a polyhedron;

• Calculation of intersections, Minkowski sums, translations, etc.;

• Taking hyperplane slices;

• The minimum ℚ-generators and ℤ-generators of a cone;

• Point membership;

• Point counting and enumeration of lattice points;

• The Ehrhart series, Ehrhart polynomial, and δ-vector of a polytope;

• Faces of a cone or polyhedron, the f-vector and h-vector;

• Construction of the face-poset;

• The group of lattice automorphisms fixing a polytope.

14.5.3 Fans

A fan F (in a toric lattice L) is a collection of cones in L satisfying the usual conditions of a cell decomposition. Magma allows for the construction of any fan; it will check that the cones intersect correctly, and will add the lower-dimensional cones as necessary. There are also several constructors for well-known fans, and standard methods for modifying existing fans.

• Creation of a fan from a collection of cones;

• Constructions for the fan of affine space, projective space, and weighted projective space;

• Construction of the product of two (or more) fans;

• The dual fan and spanning fan generated by a polyhedron;

• The normal fan to a cone C, given by the quotient of L by the span of C;

• Support and point membership;

• Extraction of the rays, one-skeleton, singular cones, maximum dimensional cones, and so forth;

• The blowup of F at a point in L;

• Answers to questions such as whether F is projective, or nonsingular, or ℚ-factorial;

• The simplicial subdivision of a fan;

• A refinement of F such that the resulting fan is nonsingular;

• Maps between fans.

14.5.4 Basic operations on toric varieties

There exist many standard constructors for toric varieties. They can also be defined directly from a fan, or in terms of a Cox ring. However one constructs a toric variety, the corresponding fan can always be recovered, although this is seldom necessary. Once you have constructed your toric variety you can treat it as a purely geometric object, as you would any scheme in Magma. Operations include:

• Standard constructions for affine, projective, and weighted projective space;

• Construction of scrolls, or of any toric variety specified by a collection of gradings;

• The toric variety given by the product X×Y;

• Recovery of a fan from any toric variety, and of the toric variety associated to any fan;

• The variety corresponding to a Cox ring can be constructed, and the Cox ring of any toric variety recovered;

• Extract the i-th affine patch of a toric variety X, along with the inclusion map;

• A resolution, ℚ-factorialisation, terminalisation, etc., along with the corresponding maps;

• Extract the one parameter subgroup lattice, monomial lattice, Cox monomial lattice, or divisor class lattice;

• Is the variety complete? Is it projective? Is it Fano?

• Determine whether the toric variety is nonsingular, Gorenstein, ℚ-factorial, terminal, or canonical.

14.5.5 Maps of toric varieties

Magma supports the creation of very general maps between toric varieties. These can be derived from maps between the associated toric lattices, or presented in terms of the variables of the Cox ring. More generally one can define all maps between toric varieties using an appropriate notion of 'rational radical function' defined in terms of the polynomial Cox coordinates. Common questions Magma can answer include:

• Are two toric varieties naturally isomorphic?

• Is a given map regular?

• The indeterminancy locus of a map.

14.5.6 Invariant divisors and Riemann-Roch spaces

Divisors on toric varieties work in the same way as on other varieties, except that within each linear equivalence class it is possible to choose torus invariant representatives. Divisors on a toric variety have a single divisor group as their parent; divisors can be constructed by coercing the appropriate data into this group, or via several standard constructions. Once you have constructed your divisor, Magma supports all the operations you would expect. Examples are:

• Construction of the zero divisor and canonical divisor;

• Construction of a "nice" representative of an element in the Picard latice or divisor class lattice, chosen to be as effective as possible;

• The combinatorial data defining a Weil or Cartier divisor;

• Is a given divisor principal or ℚ-principal? Is it linearly equivalent to a Cartier divisor?

• Are two divisors linearly equivalent?

• Answers to questions such as whether a divisor D is Cartier, ℚ-Cartier, Weil, ample, very ample, nef, or big;

• The Picard class or movable part of a divisor D;

• Proj of the ring of sections of D;

• The relative (sheaf) Proj of sections of the divisor D;

• The Riemann-Roch polytope associated to a divisor;

• The basis of the associated Riemann-Roch space;

• The graded cone of sections of multiples of D;

• The Hilbert series and Hilbert polynomial of (X,D).

14.5.7 Mori theory

The Mori cone and nef cone of a toric variety can be computed, extremal rays calculated, and so on. Magma can answer questions such as whether a given extremal ray corresponds to a Mori fibre space, is a divisorial contraction, or gives a small contraction. Standard constructions and operations are also supported:

• The images of extremal contractions of rays in the nef cone;

• A description of the extremal contractions;

• The (generalised) flip of the morphism given by the ℚ-Cartier divisor;

• The weights of the G_m action whose variation would give the flip;

• An implementation of the Minimal Model Programme.