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ProjectiveSchemes.md

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Projective schemes

Let A be a commutative noetherian base ring and S = A[x_0,\dots, x_n] the standard graded polynomial ring over A. Then X = \mathrm{Proj}(S) = \mathbb P^n_A is a (relative) projective scheme over \mathrm{Spec}(A). Similarly, for a homogeneous ideal I \subset S we have X = \mathrm{Proj}(S/I) \subset \mathbb P^n_A a (relative) projective scheme over \mathrm{Spec}(A) which is a closed subscheme of \mathbb P^n_A in a natural way. The majority of applications will be in the setting where A = \mathbb k is a field, but be aware that we also support different base rings such as the usual four MPolyRing, MPolyQuoRing, MPolyLocRing, and MPolyQuoLocRing.

Abstract types and basic interface

The abstract type for such projective schemes is

AbsProjectiveScheme{CoeffRingType, RingType} where {CoeffRingType<:Ring}

where, in the above notation, CoeffRingType denotes the type of A and RingType the type of either S or S/I, respectively. The abstract type comes with the following interface:

base_ring(X::AbsProjectiveScheme)
base_scheme(X::AbsProjectiveScheme)
homogeneous_coordinate_ring(P::AbsProjectiveScheme)
relative_ambient_dimension(X::AbsProjectiveScheme)
ambient_coordinate_ring(P::AbsProjectiveScheme)
ambient_space(P::AbsProjectiveScheme)
defining_ideal(X::AbsProjectiveScheme)
affine_cone(X::AbsProjectiveScheme)
homogeneous_coordinates_on_affine_cone(X::AbsProjectiveScheme)
covered_scheme(P::AbsProjectiveScheme)

The minimal concrete type realizing this interface is

ProjectiveScheme{CoeffRingType, RingType} <: AbsProjectiveScheme{CoeffRingType, RingType}

Constructors

Besides proj(S) for some graded polynomial ring or a graded affine algebra S, we provide the following constructors:

proj(S::MPolyDecRing)
proj(S::MPolyDecRing, I::MPolyIdeal{T}) where {T<:MPolyDecRingElem}
proj(I::MPolyIdeal{<:MPolyDecRingElem})
proj(Q::MPolyQuoRing{<:MPolyDecRingElem})

Subschemes defined by homogeneous ideals, ring elements, or lists of elements can be created via the respective methods of the subscheme(P::AbsProjectiveScheme, ...) function. Special constructors are provided for projective space itself via the function projective_space and its various methods.

projective_space(A::Ring, var_symb::Vector{VarName})
projective_space(A::Ring, r::Int; var_name::VarName=:s)

Attributes

Besides those attributes already covered by the above general interface we have the following (self-explanatory) ones for projective schemes over a field.

dim(P::AbsProjectiveScheme{<:Field})
hilbert_polynomial(P::AbsProjectiveScheme{<:Field})
degree(P::AbsProjectiveScheme{<:Field})
arithmetic_genus(P::AbsProjectiveScheme{<:Field})

Methods

To facilitate the interplay between an AbsProjectiveScheme and the affine charts of its covered_scheme we provide the following methods:

dehomogenization_map(X::AbsProjectiveScheme, U::AbsAffineScheme)
homogenization_map(P::AbsProjectiveScheme, U::AbsAffineScheme)

Properties

Further properties of projective schemes:

is_smooth(P::AbsProjectiveScheme{<:Any, <:MPolyQuoRing}; algorithm=:default)

is_empty(P::AbsProjectiveScheme{<:Field})
is_irreducible(P::AbsProjectiveScheme)
is_reduced(P::AbsProjectiveScheme)
is_geometrically_reduced(P::AbsProjectiveScheme{<:Field})
is_geometrically_irreducible(P::AbsProjectiveScheme{<:Field})
is_integral(X::AbsProjectiveScheme{<:Field})
is_geometrically_integral(X::AbsProjectiveScheme{<:Field})