Subschemes of affine space

AUTHORS:

• David Kohel, William Stein (2005): initial version

• Ben Hutz (2013): affine subschemes

class sage.schemes.affine.affine_subscheme.AlgebraicScheme_subscheme_affine(A, polynomials, embedding_center=None, embedding_codomain=None, embedding_images=None)

Bases: sage.schemes.generic.algebraic_scheme.AlgebraicScheme_subscheme

An algebraic subscheme of affine space.

INPUT:

• A – ambient affine space

• polynomials – single polynomial, ideal or iterable of defining polynomials

EXAMPLES:

sage: A3.<x, y, z> = AffineSpace(QQ, 3)
sage: A3.subscheme([x^2-y*z])
Closed subscheme of Affine Space of dimension 3 over Rational Field defined by:
  x^2 - y*z

dimension()

Return the dimension of the affine algebraic subscheme.

EXAMPLES:

sage: A.<x,y> = AffineSpace(2, QQ)
sage: A.subscheme([]).dimension()
2
sage: A.subscheme([x]).dimension()
1
sage: A.subscheme([x^5]).dimension()
1
sage: A.subscheme([x^2 + y^2 - 1]).dimension()
1
sage: A.subscheme([x*(x-1), y*(y-1)]).dimension()
0

Something less obvious:

sage: A.<x,y,z,w> = AffineSpace(4, QQ)
sage: X = A.subscheme([x^2, x^2*y^2 + z^2, z^2 - w^2, 10*x^2 + w^2 - z^2])
sage: X
Closed subscheme of Affine Space of dimension 4 over Rational Field defined by:
  x^2,
  x^2*y^2 + z^2,
  z^2 - w^2,
  10*x^2 - z^2 + w^2
sage: X.dimension()
1

intersection_multiplicity(X, P)

Return the intersection multiplicity of this subscheme and the subscheme X at the point P.

The intersection of this subscheme with X must be proper, that is \(\mathrm{codim}(self\cap X) = \mathrm{codim}(self) + \mathrm{codim}(X)\), and must also be finite. We use Serre’s Tor formula to compute the intersection multiplicity. If \(I\), \(J\) are the defining ideals of self, X, respectively, then this is \(\sum_{i=0}^{\infty}(-1)^i\mathrm{length}(\mathrm{Tor}_{\mathcal{O}_{A,p}}^{i} (\mathcal{O}_{A,p}/I,\mathcal{O}_{A,p}/J))\) where \(A\) is the affine ambient space of these subschemes.

INPUT:

• X – subscheme in the same ambient space as this subscheme.

• P – a point in the intersection of this subscheme with X.

OUTPUT: An integer.

EXAMPLES:

sage: A.<x,y> = AffineSpace(QQ, 2)
sage: C = Curve([y^2 - x^3 - x^2], A)
sage: D = Curve([y^2 + x^3], A)
sage: Q = A([0,0])
sage: C.intersection_multiplicity(D, Q)
4

sage: R.<a> = QQ[]
sage: K.<b> = NumberField(a^6 - 3*a^5 + 5*a^4 - 5*a^3 + 5*a^2 - 3*a + 1)
sage: A.<x,y,z,w> = AffineSpace(K, 4)
sage: X = A.subscheme([x*y, y*z + 7, w^3 - x^3])
sage: Y = A.subscheme([x - z^3 + z + 1])
sage: Q = A([0, -7*b^5 + 21*b^4 - 28*b^3 + 21*b^2 - 21*b + 14, -b^5 + 2*b^4 - 3*b^3 \
+ 2*b^2 - 2*b, 0])
sage: X.intersection_multiplicity(Y, Q)
3

sage: A.<x,y,z> = AffineSpace(QQ, 3)
sage: X = A.subscheme([z^2 - 1])
sage: Y = A.subscheme([z - 1, y - x^2])
sage: Q = A([1,1,1])
sage: X.intersection_multiplicity(Y, Q)
Traceback (most recent call last):
...
TypeError: the intersection of this subscheme and (=Closed subscheme of Affine Space of dimension 3
over Rational Field defined by: z - 1, -x^2 + y) must be proper and finite

sage: A.<x,y,z,w,t> = AffineSpace(QQ, 5)
sage: X = A.subscheme([x*y, t^2*w, w^3*z])
sage: Y = A.subscheme([y*w + z])
sage: Q = A([0,0,0,0,0])
sage: X.intersection_multiplicity(Y, Q)
Traceback (most recent call last):
...
TypeError: the intersection of this subscheme and (=Closed subscheme of Affine Space of dimension 5
over Rational Field defined by: y*w + z) must be proper and finite

is_smooth(point=None)

Test whether the algebraic subscheme is smooth.

INPUT:

• point – A point or None (default). The point to test smoothness at.

OUTPUT:

Boolean. If no point was specified, returns whether the algebraic subscheme is smooth everywhere. Otherwise, smoothness at the specified point is tested.

EXAMPLES:

sage: A2.<x,y> = AffineSpace(2,QQ)
sage: cuspidal_curve = A2.subscheme([y^2-x^3])
sage: cuspidal_curve
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
  -x^3 + y^2
sage: smooth_point = cuspidal_curve.point([1,1])
sage: smooth_point in cuspidal_curve
True
sage: singular_point = cuspidal_curve.point([0,0])
sage: singular_point in cuspidal_curve
True
sage: cuspidal_curve.is_smooth(smooth_point)
True
sage: cuspidal_curve.is_smooth(singular_point)
False
sage: cuspidal_curve.is_smooth()
False

multiplicity(P)

Return the multiplicity of P on this subscheme.

This is computed as the multiplicity of the local ring of this subscheme corresponding to P. This subscheme must be defined over a field. An error is raised if P is not a point on this subscheme.

INPUT:

• P – a point on this subscheme.

OUTPUT:

An integer.

EXAMPLES:

sage: A.<x,y,z,w> = AffineSpace(QQ, 4)
sage: X = A.subscheme([z*y - x^7, w - 2*z])
sage: Q1 = A([1,1/3,3,6])
sage: X.multiplicity(Q1)
1
sage: Q2 = A([0,0,0,0])
sage: X.multiplicity(Q2)
2

sage: A.<x,y,z,w,v> = AffineSpace(GF(23), 5)
sage: C = A.curve([x^8 - y, y^7 - z, z^3 - 1, w^5 - v^3])
sage: Q = A([22,1,1,0,0])
sage: C.multiplicity(Q)
3

sage: K.<a> = QuadraticField(-1)
sage: A.<x,y,z,w,t> = AffineSpace(K, 5)
sage: X = A.subscheme([y^7 - x^2*z^5 + z^3*t^8 - x^2*y^4*z - t^8])
sage: Q1 = A([1,1,0,1,-1])
sage: X.multiplicity(Q1)
1
sage: Q2 = A([0,0,0,-a,0])
sage: X.multiplicity(Q2)
7

Check that trac ticket #27479 is fixed:

sage: A1.<x> = AffineSpace(QQ, 1)
sage: X = A1.subscheme([x^1789 + x])
sage: Q = X([0])
sage: X.multiplicity(Q)
1

projective_closure(i=None, PP=None)

Return the projective closure of this affine subscheme.

INPUT:

• i – (default: None) determines the embedding to use to compute the projective closure of this affine subscheme. The embedding used is the one which has a 1 in the i-th coordinate, numbered from 0.

• PP – (default: None) ambient projective space, i.e., ambient space of codomain of morphism; this is constructed if it is not given

OUTPUT: a projective subscheme

EXAMPLES:

sage: A.<x,y,z,w> = AffineSpace(QQ,4)
sage: X = A.subscheme([x^2 - y, x*y - z, y^2 - w, x*z - w, y*z - x*w, z^2 - y*w])
sage: X.projective_closure()
Closed subscheme of Projective Space of dimension 4 over Rational Field
defined by:
  x0^2 - x1*x4,
  x0*x1 - x2*x4,
  x1^2 - x3*x4,
  x0*x2 - x3*x4,
  x1*x2 - x0*x3,
  x2^2 - x1*x3

sage: A.<x,y,z> = AffineSpace(QQ, 3)
sage: P.<a,b,c,d> = ProjectiveSpace(QQ, 3)
sage: X = A.subscheme([z - x^2 - y^2])
sage: X.projective_closure(1, P).ambient_space() == P
True

projective_embedding(i=None, PP=None)

Return a morphism from this affine scheme into an ambient projective space of the same dimension.

The codomain of this morphism is the projective closure of this affine scheme in PP, if given, or otherwise in a new projective space that is constructed.

INPUT:

• i – integer (default: dimension of self = last coordinate) determines which projective embedding to compute. The embedding is that which has a 1 in the i-th coordinate, numbered from 0.

• PP – (default: None) ambient projective space, i.e., ambient space

  of codomain of morphism; this is constructed if it is not given.

EXAMPLES:

sage: A.<x, y, z> = AffineSpace(3, ZZ)
sage: S = A.subscheme([x*y-z])
sage: S.projective_embedding()
Scheme morphism:
  From: Closed subscheme of Affine Space of dimension 3 over Integer Ring defined by:
  x*y - z
  To:   Closed subscheme of Projective Space of dimension 3 over Integer Ring defined by:
  x0*x1 - x2*x3
  Defn: Defined on coordinates by sending (x, y, z) to
        (x : y : z : 1)

sage: A.<x, y, z> = AffineSpace(3, ZZ)
sage: P = ProjectiveSpace(3,ZZ,'u')
sage: S = A.subscheme([x^2-y*z])
sage: S.projective_embedding(1,P)
Scheme morphism:
  From: Closed subscheme of Affine Space of dimension 3 over Integer
Ring defined by:
  x^2 - y*z
  To:   Closed subscheme of Projective Space of dimension 3 over Integer
Ring defined by:
  u0^2 - u2*u3
  Defn: Defined on coordinates by sending (x, y, z) to
        (x : 1 : y : z)

sage: A.<x,y,z> = AffineSpace(QQ, 3)
sage: X = A.subscheme([y - x^2, z - x^3])
sage: X.projective_embedding()
Scheme morphism:
  From: Closed subscheme of Affine Space of dimension 3 over Rational
Field defined by:
  -x^2 + y,
  -x^3 + z
  To:   Closed subscheme of Projective Space of dimension 3 over
Rational Field defined by:
  x0^2 - x1*x3,
  x0*x1 - x2*x3,
  x1^2 - x0*x2
  Defn: Defined on coordinates by sending (x, y, z) to
        (x : y : z : 1)

When taking a closed subscheme of an affine space with a projective embedding, the subscheme inherits the embedding:

sage: A.<u,v> = AffineSpace(2, QQ, default_embedding_index=1)
sage: X = A.subscheme(u - v)
sage: X.projective_embedding()
Scheme morphism:
  From: Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
  u - v
  To:   Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
  x0 - x2
  Defn: Defined on coordinates by sending (u, v) to
        (u : 1 : v)
sage: phi = X.projective_embedding()
sage: psi = A.projective_embedding()
sage: phi(X(2, 2)) == psi(A(X(2, 2)))
True

class sage.schemes.affine.affine_subscheme.AlgebraicScheme_subscheme_affine_field(A, polynomials, embedding_center=None, embedding_codomain=None, embedding_images=None)

Bases: sage.schemes.affine.affine_subscheme.AlgebraicScheme_subscheme_affine

Algebraic subschemes of projective spaces defined over fields.

tangent_space(p)

Return the tangent space at the point p.

The points of the tangent space are the tangent vectors at p.

INPUT:

• p – a rational point

EXAMPLES:

sage: A3.<x,y,z> = AffineSpace(3, QQ)
sage: X = A3.subscheme(z-x*y)
sage: X.tangent_space(A3.origin())
Closed subscheme of Affine Space of dimension 3 over Rational Field
defined by:
  z
sage: X.tangent_space(X(1,1,1))
Closed subscheme of Affine Space of dimension 3 over Rational Field
defined by:
  -x - y + z

Tangent space at a point may have higher dimension than the dimension of the point.

sage: C = Curve([x + y + z, x^2 - y^2*z^2 + z^3])
sage: C.singular_points()
[(0, 0, 0)]
sage: p = C(0,0,0)
sage: C.tangent_space(p)
Closed subscheme of Affine Space of dimension 3 over Rational Field
defined by:
  x + y + z
sage: _.dimension()
2
sage: q = C(1,0,-1)
sage: C.tangent_space(q)
Closed subscheme of Affine Space of dimension 3 over Rational Field
defined by:
  x + y + z,
  2*x + 3*z
sage: _.dimension()
1