def dual(self):
r"""
Return the projective dual of the given subscheme of projective space.
INPUT:
- ``X`` -- A subscheme of projective space. At present, ``X`` is
required to be an irreducible and reduced hypersurface defined
over `\QQ` or a finite field.
OUTPUT:
- The dual of ``X`` as a subscheme of the dual projective space.
EXAMPLES:
The dual of a smooth conic in the plane is also a smooth conic::
sage: R.<x, y, z> = QQ[]
sage: P.<x, y, z> = ProjectiveSpace(2, QQ)
sage: I = R.ideal(x^2 + y^2 + z^2)
sage: X = P.subscheme(I)
sage: X.dual()
Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
y0^2 + y1^2 + y2^2
The dual of the twisted cubic curve in projective 3-space is a singular
quartic surface. In the following example, we compute the dual of this
surface, which by double duality is equal to the twisted cubic itself.
The output is the twisted cubic as an intersection of three quadrics::
sage: R.<x, y, z, w> = QQ[]
sage: P.<x, y, z, w> = ProjectiveSpace(3, QQ)
sage: I = R.ideal(y^2*z^2 - 4*x*z^3 - 4*y^3*w + 18*x*y*z*w - 27*x^2*w^2)
sage: X = P.subscheme(I)
sage: X.dual()
Closed subscheme of Projective Space of dimension 3 over
Rational Field defined by:
y2^2 - y1*y3,
y1*y2 - y0*y3,
y1^2 - y0*y2
The singular locus of the quartic surface in the last example
is itself supported on a twisted cubic::
sage: X.Jacobian().radical()
Ideal (z^2 - 3*y*w, y*z - 9*x*w, y^2 - 3*x*z) of Multivariate
Polynomial Ring in x, y, z, w over Rational Field
An example over a finite field::
sage: R = PolynomialRing(GF(61), 'a,b,c')
sage: P.<a, b, c> = ProjectiveSpace(2, R.base_ring())
sage: X = P.subscheme(R.ideal(a*a+2*b*b+3*c*c))
sage: X.dual()
Closed subscheme of Projective Space of dimension 2 over
Finite Field of size 61 defined by:
y0^2 - 30*y1^2 - 20*y2^2
TESTS::
sage: R = PolynomialRing(Qp(3), 'a,b,c')
sage: P.<a, b, c> = ProjectiveSpace(2, R.base_ring())
sage: X = P.subscheme(R.ideal(a*a+2*b*b+3*c*c))
sage: X.dual()
Traceback (most recent call last):
...
NotImplementedError: base ring must be QQ or a finite field
"""
from sage.libs.singular.function_factory import ff
K = self.base_ring()
if not (is_RationalField(K) or is_FiniteField(K)):
raise NotImplementedError("base ring must be QQ or a finite field")
I = self.defining_ideal()
m = I.ngens()
n = I.ring().ngens() - 1
if (m != 1 or (n < 1) or I.is_zero() or I.is_trivial()
or not I.is_prime()):
raise NotImplementedError("At the present, the method is only"
" implemented for irreducible and"
" reduced hypersurfaces and the given"
" list of generators for the ideal must"
" have exactly one element.")
R = PolynomialRing(K, 'x', n + 1)
from sage.schemes.projective.projective_space import ProjectiveSpace
Pd = ProjectiveSpace(n, K, 'y')
Rd = Pd.coordinate_ring()
x = R.variable_names()
y = Rd.variable_names()
S = PolynomialRing(K, x + y + ('t', ))
if S.has_coerce_map_from(I.ring()):
T = PolynomialRing(K, 'w', n + 1)
I_S = (I.change_ring(T)).change_ring(S)
else:
I_S = I.change_ring(S)
f_S = I_S.gens()[0]
z = S.gens()
J = I_S
for i in range(n + 1):
J = J + S.ideal(z[-1] * f_S.derivative(z[i]) - z[i + n + 1])
sat = ff.elim__lib.sat
max_ideal = S.ideal(z[n + 1:2 * n + 2])
J_sat_gens = sat(J, max_ideal)[0]
J_sat = S.ideal(J_sat_gens)
L = J_sat.elimination_ideal(z[0:n + 1] + (z[-1], ))
return Pd.subscheme(L.change_ring(Rd))
def dual(self):
r"""
Return the projective dual of the given subscheme of projective space.
INPUT:
- ``X`` -- A subscheme of projective space. At present, ``X`` is
required to be an irreducible and reduced hypersurface defined
over `\QQ` or a finite field.
OUTPUT:
- The dual of ``X`` as a subscheme of the dual projective space.
EXAMPLES:
The dual of a smooth conic in the plane is also a smooth conic::
sage: R.<x, y, z> = QQ[]
sage: P.<x, y, z> = ProjectiveSpace(2, QQ)
sage: I = R.ideal(x^2 + y^2 + z^2)
sage: X = P.subscheme(I)
sage: X.dual()
Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
y0^2 + y1^2 + y2^2
The dual of the twisted cubic curve in projective 3-space is a singular
quartic surface. In the following example, we compute the dual of this
surface, which by double duality is equal to the twisted cubic itself.
The output is the twisted cubic as an intersection of three quadrics::
sage: R.<x, y, z, w> = QQ[]
sage: P.<x, y, z, w> = ProjectiveSpace(3, QQ)
sage: I = R.ideal(y^2*z^2 - 4*x*z^3 - 4*y^3*w + 18*x*y*z*w - 27*x^2*w^2)
sage: X = P.subscheme(I)
sage: X.dual()
Closed subscheme of Projective Space of dimension 3 over
Rational Field defined by:
y2^2 - y1*y3,
y1*y2 - y0*y3,
y1^2 - y0*y2
The singular locus of the quartic surface in the last example
is itself supported on a twisted cubic::
sage: X.Jacobian().radical()
Ideal (z^2 - 3*y*w, y*z - 9*x*w, y^2 - 3*x*z) of Multivariate
Polynomial Ring in x, y, z, w over Rational Field
An example over a finite field::
sage: R = PolynomialRing(GF(61), 'a,b,c')
sage: P.<a, b, c> = ProjectiveSpace(2, R.base_ring())
sage: X = P.subscheme(R.ideal(a*a+2*b*b+3*c*c))
sage: X.dual()
Closed subscheme of Projective Space of dimension 2 over
Finite Field of size 61 defined by:
y0^2 - 30*y1^2 - 20*y2^2
TESTS::
sage: R = PolynomialRing(Qp(3), 'a,b,c')
sage: P.<a, b, c> = ProjectiveSpace(2, R.base_ring())
sage: X = P.subscheme(R.ideal(a*a+2*b*b+3*c*c))
sage: X.dual()
Traceback (most recent call last):
...
NotImplementedError: base ring must be QQ or a finite field
"""
from sage.libs.singular.function_factory import ff
K = self.base_ring()
if not(is_RationalField(K) or is_FiniteField(K)):
raise NotImplementedError("base ring must be QQ or a finite field")
I = self.defining_ideal()
m = I.ngens()
n = I.ring().ngens() - 1
if (m != 1 or (n < 1) or I.is_zero()
or I.is_trivial() or not I.is_prime()):
raise NotImplementedError("At the present, the method is only"
" implemented for irreducible and"
" reduced hypersurfaces and the given"
" list of generators for the ideal must"
" have exactly one element.")
R = PolynomialRing(K, 'x', n + 1)
from sage.schemes.projective.projective_space import ProjectiveSpace
Pd = ProjectiveSpace(n, K, 'y')
Rd = Pd.coordinate_ring()
x = R.variable_names()
y = Rd.variable_names()
S = PolynomialRing(K, x + y + ('t',))
if S.has_coerce_map_from(I.ring()):
T = PolynomialRing(K, 'w', n + 1)
I_S = (I.change_ring(T)).change_ring(S)
else:
I_S = I.change_ring(S)
f_S = I_S.gens()[0]
z = S.gens()
J = I_S
for i in range(n + 1):
J = J + S.ideal(z[-1] * f_S.derivative(z[i]) - z[i + n + 1])
sat = ff.elim__lib.sat
max_ideal = S.ideal(z[n + 1: 2 * n + 2])
J_sat_gens = sat(J, max_ideal)[0]
J_sat = S.ideal(J_sat_gens)
L = J_sat.elimination_ideal(z[0: n + 1] + (z[-1],))
return Pd.subscheme(L.change_ring(Rd))
def Chow_form(self):
r"""
Returns the Chow form associated to this subscheme.
For a `k`-dimensional subvariety of `\mathbb{P}^N` of degree `D`.
The `(N-k-1)`-dimensional projective linear subspaces of `\mathbb{P}^N`
meeting `X` form a hypersurface in the Grassmannian `G(N-k-1,N)`.
The homogeneous form of degree `D` defining this hypersurface in Plucker
coordinates is called the Chow form of `X`.
The base ring needs to be a number field, finite field, or `\QQbar`.
ALGORITHM:
For a `k`-dimension subscheme `X` consider the `k+1` linear forms
`l_i = u_{i0}x_0 + \cdots + u_{in}x_n`. Let `J` be the ideal in the
polynomial ring `K[x_i,u_{ij}]` defined by the equations of `X` and the `l_i`.
Let `J'` be the saturation of `J` with respect to the irrelevant ideal of
the ambient projective space of `X`. The elimination ideal `I = J' \cap K[u_{ij}]`
is a principal ideal, let `R` be its generator. The Chow form is obtained by
writing `R` as a polynomial in Plucker coordinates (i.e. bracket polynomials).
[DalbecSturmfels]_.
OUTPUT: a homogeneous polynomial.
REFERENCES:
.. [DalbecSturmfels] J. Dalbec and B. Sturmfels. Invariant methods in discrete and computational geometry,
chapter Introduction to Chow forms, pages 37-58. Springer Netherlands, 1994.
EXAMPLES::
sage: P.<x0,x1,x2,x3> = ProjectiveSpace(GF(17), 3)
sage: X = P.subscheme([x3+x1,x2-x0,x2-x3])
sage: X.Chow_form()
t0 - t1 + t2 + t3
::
sage: P.<x0,x1,x2,x3> = ProjectiveSpace(QQ,3)
sage: X = P.subscheme([x3^2 -101*x1^2 - 3*x2*x0])
sage: X.Chow_form()
t0^2 - 101*t2^2 - 3*t1*t3
::
sage: P.<x0,x1,x2,x3>=ProjectiveSpace(QQ,3)
sage: X = P.subscheme([x0*x2-x1^2, x0*x3-x1*x2, x1*x3-x2^2])
sage: Ch = X.Chow_form(); Ch
t2^3 + 2*t2^2*t3 + t2*t3^2 - 3*t1*t2*t4 - t1*t3*t4 + t0*t4^2 + t1^2*t5
sage: Y = P.subscheme_from_Chow_form(Ch, 1); Y
Closed subscheme of Projective Space of dimension 3 over Rational Field
defined by:
x2^2*x3 - x1*x3^2,
-x2^3 + x0*x3^2,
-x2^2*x3 + x1*x3^2,
x1*x2*x3 - x0*x3^2,
3*x1*x2^2 - 3*x0*x2*x3,
-2*x1^2*x3 + 2*x0*x2*x3,
-3*x1^2*x2 + 3*x0*x1*x3,
x1^3 - x0^2*x3,
x2^3 - x1*x2*x3,
-3*x1*x2^2 + 2*x1^2*x3 + x0*x2*x3,
2*x0*x2^2 - 2*x0*x1*x3,
3*x1^2*x2 - 2*x0*x2^2 - x0*x1*x3,
-x0*x1*x2 + x0^2*x3,
-x0*x1^2 + x0^2*x2,
-x1^3 + x0*x1*x2,
x0*x1^2 - x0^2*x2
sage: I = Y.defining_ideal()
sage: I.saturation(I.ring().ideal(list(I.ring().gens())))[0]
Ideal (x2^2 - x1*x3, x1*x2 - x0*x3, x1^2 - x0*x2) of Multivariate
Polynomial Ring in x0, x1, x2, x3 over Rational Field
"""
I = self.defining_ideal()
P = self.ambient_space()
R = P.coordinate_ring()
N = P.dimension() + 1
d = self.dimension()
# create the ring for the generic linear hyperplanes
# u0x0 + u1x1 + ...
SS = PolynomialRing(R.base_ring(), 'u', N * (d + 1), order='lex')
vars = SS.variable_names() + R.variable_names()
S = PolynomialRing(R.base_ring(), vars, order='lex')
n = S.ngens()
newcoords = [S.gen(n - N + t) for t in range(N)]
# map the generators of the subscheme into the ring with the hyperplane variables
phi = R.hom(newcoords, S)
phi(self.defining_polynomials()[0])
# create the dim(X)+1 linear hyperplanes
l = []
for i in range(d + 1):
t = 0
for j in range(N):
t += S.gen(N * i + j) * newcoords[j]
l.append(t)
# intersect the hyperplanes with X
J = phi(I) + S.ideal(l)
# saturate the ideal with respect to the irrelevant ideal
J2 = J.saturation(S.ideal([phi(u) for u in R.gens()]))[0]
# eliminate the original variables to be left with the hyperplane coefficients 'u'
E = J2.elimination_ideal(newcoords)
# create the plucker coordinates
D = binomial(N, N - d - 1) #number of plucker coordinates
tvars = [str('t') + str(i) for i in range(D)] #plucker coordinates
T = PolynomialRing(R.base_ring(),
tvars + list(S.variable_names()),
order='lex')
L = []
coeffs = [
T.gen(i) for i in range(0 + len(tvars),
N * (d + 1) + len(tvars))
]
M = matrix(T, d + 1, N, coeffs)
i = 0
for c in M.minors(d + 1):
L.append(T.gen(i) - c)
i += 1
# create the ideal that we can use for eliminating to get a polynomial
# in the plucker coordinates (brackets)
br = T.ideal(L)
# create a mapping into a polynomial ring over the plucker coordinates
# and the hyperplane coefficients
psi = S.hom(coeffs + [0 for _ in range(N)], T)
E2 = T.ideal([psi(u) for u in E.gens()] + br)
# eliminate the hyperplane coefficients
CH = E2.elimination_ideal(coeffs)
# CH should be a principal ideal, but because of the relations among
# the plucker coordinates, the elimination will probably have several generators
# get the relations among the plucker coordinates
rel = br.elimination_ideal(coeffs)
# reduce CH with respect to the relations
reduced = []
for f in CH.gens():
reduced.append(f.reduce(rel))
# find the principal generator
# polynomial ring in just the plucker coordinates
T2 = PolynomialRing(R.base_ring(), tvars)
alp = T.hom(tvars + (N * (d + 1) + N) * [0], T2)
# get the degrees of the reduced generators of CH
degs = [u.degree() for u in reduced]
mind = max(degs)
# need the smallest degree form that did not reduce to 0
for d in degs:
if d < mind and d > 0:
mind = d
ind = degs.index(mind)
CF = reduced[ind] #this should be the Chow form of X
# check that it is correct (i.e., it is a principal generator for CH + the relations)
rel2 = rel + [CF]
assert all(f in rel2
for f in CH.gens()), "did not find a principal generator"
return alp(CF)
def Chow_form(self):
r"""
Returns the Chow form associated to this subscheme.
For a `k`-dimensional subvariety of `\mathbb{P}^N` of degree `D`.
The `(N-k-1)`-dimensional projective linear subspaces of `\mathbb{P}^N`
meeting `X` form a hypersurface in the Grassmannian `G(N-k-1,N)`.
The homogeneous form of degree `D` defining this hypersurface in Plucker
coordinates is called the Chow form of `X`.
The base ring needs to be a number field, finite field, or `\QQbar`.
ALGORITHM:
For a `k`-dimension subscheme `X` consider the `k+1` linear forms
`l_i = u_{i0}x_0 + \cdots + u_{in}x_n`. Let `J` be the ideal in the
polynomial ring `K[x_i,u_{ij}]` defined by the equations of `X` and the `l_i`.
Let `J'` be the saturation of `J` with respect to the irrelevant ideal of
the ambient projective space of `X`. The elimination ideal `I = J' \cap K[u_{ij}]`
is a principal ideal, let `R` be its generator. The Chow form is obtained by
writing `R` as a polynomial in Plucker coordinates (i.e. bracket polynomials).
[DalbecSturmfels]_.
OUTPUT: a homogeneous polynomial.
REFERENCES:
.. [DalbecSturmfels] J. Dalbec and B. Sturmfels. Invariant methods in discrete and computational geometry,
chapter Introduction to Chow forms, pages 37-58. Springer Netherlands, 1994.
EXAMPLES::
sage: P.<x0,x1,x2,x3> = ProjectiveSpace(GF(17), 3)
sage: X = P.subscheme([x3+x1,x2-x0,x2-x3])
sage: X.Chow_form()
t0 - t1 + t2 + t3
::
sage: P.<x0,x1,x2,x3> = ProjectiveSpace(QQ,3)
sage: X = P.subscheme([x3^2 -101*x1^2 - 3*x2*x0])
sage: X.Chow_form()
t0^2 - 101*t2^2 - 3*t1*t3
::
sage: P.<x0,x1,x2,x3>=ProjectiveSpace(QQ,3)
sage: X = P.subscheme([x0*x2-x1^2, x0*x3-x1*x2, x1*x3-x2^2])
sage: Ch = X.Chow_form(); Ch
t2^3 + 2*t2^2*t3 + t2*t3^2 - 3*t1*t2*t4 - t1*t3*t4 + t0*t4^2 + t1^2*t5
sage: Y = P.subscheme_from_Chow_form(Ch, 1); Y
Closed subscheme of Projective Space of dimension 3 over Rational Field
defined by:
x2^2*x3 - x1*x3^2,
-x2^3 + x0*x3^2,
-x2^2*x3 + x1*x3^2,
x1*x2*x3 - x0*x3^2,
3*x1*x2^2 - 3*x0*x2*x3,
-2*x1^2*x3 + 2*x0*x2*x3,
-3*x1^2*x2 + 3*x0*x1*x3,
x1^3 - x0^2*x3,
x2^3 - x1*x2*x3,
-3*x1*x2^2 + 2*x1^2*x3 + x0*x2*x3,
2*x0*x2^2 - 2*x0*x1*x3,
3*x1^2*x2 - 2*x0*x2^2 - x0*x1*x3,
-x0*x1*x2 + x0^2*x3,
-x0*x1^2 + x0^2*x2,
-x1^3 + x0*x1*x2,
x0*x1^2 - x0^2*x2
sage: I = Y.defining_ideal()
sage: I.saturation(I.ring().ideal(list(I.ring().gens())))[0]
Ideal (x2^2 - x1*x3, x1*x2 - x0*x3, x1^2 - x0*x2) of Multivariate
Polynomial Ring in x0, x1, x2, x3 over Rational Field
"""
I = self.defining_ideal()
P = self.ambient_space()
R = P.coordinate_ring()
N = P.dimension()+1
d = self.dimension()
#create the ring for the generic linear hyperplanes
# u0x0 + u1x1 + ...
SS = PolynomialRing(R.base_ring(), 'u', N*(d+1), order='lex')
vars = SS.variable_names() + R.variable_names()
S = PolynomialRing(R.base_ring(), vars, order='lex')
n = S.ngens()
newcoords = [S.gen(n-N+t) for t in range(N)]
#map the generators of the subscheme into the ring with the hyperplane variables
phi = R.hom(newcoords,S)
phi(self.defining_polynomials()[0])
#create the dim(X)+1 linear hyperplanes
l = []
for i in range(d+1):
t = 0
for j in range(N):
t += S.gen(N*i + j)*newcoords[j]
l.append(t)
#intersect the hyperplanes with X
J = phi(I) + S.ideal(l)
#saturate the ideal with respect to the irrelevant ideal
J2 = J.saturation(S.ideal([phi(t) for t in R.gens()]))[0]
#eliminate the original variables to be left with the hyperplane coefficients 'u'
E = J2.elimination_ideal(newcoords)
#create the plucker coordinates
D = binomial(N,N-d-1) #number of plucker coordinates
tvars = [str('t') + str(i) for i in range(D)] #plucker coordinates
T = PolynomialRing(R.base_ring(), tvars+list(S.variable_names()), order='lex')
L = []
coeffs = [T.gen(i) for i in range(0+len(tvars), N*(d+1)+len(tvars))]
M = matrix(T,d+1,N,coeffs)
i = 0
for c in M.minors(d+1):
L.append(T.gen(i)-c)
i += 1
#create the ideal that we can use for eliminating to get a polynomial
#in the plucker coordinates (brackets)
br = T.ideal(L)
#create a mapping into a polynomial ring over the plucker coordinates
#and the hyperplane coefficients
psi = S.hom(coeffs + [0 for i in range(N)],T)
E2 = T.ideal([psi(u) for u in E.gens()] +br)
#eliminate the hyperplane coefficients
CH = E2.elimination_ideal(coeffs)
#CH should be a principal ideal, but because of the relations among
#the plucker coordinates, the elimination will probably have several generators
#get the relations among the plucker coordinates
rel = br.elimination_ideal(coeffs)
#reduce CH with respect to the relations
reduced = []
for f in CH.gens():
reduced.append(f.reduce(rel))
#find the principal generator
#polynomial ring in just the plucker coordinates
T2 = PolynomialRing(R.base_ring(), tvars)
alp = T.hom(tvars + (N*(d+1) +N)*[0], T2)
#get the degrees of the reduced generators of CH
degs = [u.degree() for u in reduced]
mind = max(degs)
#need the smallest degree form that did not reduce to 0
for d in degs:
if d < mind and d >0:
mind = d
ind = degs.index(mind)
CF = reduced[ind] #this should be the Chow form of X
#check that it is correct (i.e., it is a principal generator for CH + the relations)
rel2 = rel + [CF]
assert all([f in rel2 for f in CH.gens()]), "did not find a principal generator"
return(alp(CF))
