Points for products of projective spaces¶
This class builds on the projective space class and its point and morphism classes.
EXAMPLES:
We construct products projective spaces of various dimensions over the same ring.:
sage: P1xP1.<x,y, u,v> = ProductProjectiveSpaces(QQ, [1, 1])
sage: P1xP1([2, 1, 3, 1])
(2 : 1 , 3 : 1)
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class
sage.schemes.product_projective.point.ProductProjectiveSpaces_point_field(parent, polys, check=True)¶ Bases:
sage.schemes.product_projective.point.ProductProjectiveSpaces_point_ring-
intersection_multiplicity(X)¶ Return the intersection multiplicity of the codomain of this point and subscheme
Xat this point.This uses the subscheme implementation of intersection_multiplicity. This point must be a point on a subscheme of a product of projective spaces.
INPUT:
X– a subscheme in the same ambient space as the codomain of this point.
OUTPUT: An integer.
EXAMPLES:
sage: PP.<x,y,z,u,v> = ProductProjectiveSpaces(QQ, [2,1]) sage: X = PP.subscheme([y^2*z^3*u - x^5*v]) sage: Y = PP.subscheme([u^3 - v^3, x - y]) sage: Q = X([0,0,1,1,1]) sage: Q.intersection_multiplicity(Y) 2
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multiplicity()¶ Return the multiplicity of this point on its codomain.
This uses the subscheme implementation of multiplicity. This point must be a point on a subscheme of a product of projective spaces.
OUTPUT: an integer.
EXAMPLES:
sage: PP.<x,y,z,w,u,v,t> = ProductProjectiveSpaces(QQ, [3,2]) sage: X = PP.subscheme([x^8*t - y^8*t + z^5*w^3*v]) sage: Q1 = X([1,1,0,0,-1,-1,1]) sage: Q1.multiplicity() 1 sage: Q2 = X([0,0,0,1,0,1,1]) sage: Q2.multiplicity() 5 sage: Q3 = X([0,0,0,1,1,0,0]) sage: Q3.multiplicity() 6
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class
sage.schemes.product_projective.point.ProductProjectiveSpaces_point_finite_field(parent, polys, check=True)¶ Bases:
sage.schemes.product_projective.point.ProductProjectiveSpaces_point_field
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class
sage.schemes.product_projective.point.ProductProjectiveSpaces_point_ring(parent, polys, check=True)¶ Bases:
sage.schemes.generic.morphism.SchemeMorphism_pointThe class of points on products of projective spaces.
The components are projective space points.
EXAMPLES:
sage: T.<x,y,z,w,u> = ProductProjectiveSpaces([2, 1], QQ) sage: T.point([1, 2, 3, 4, 5]); (1/3 : 2/3 : 1 , 4/5 : 1)
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change_ring(R, **kwds)¶ Return a new
ProductProjectiveSpaces_pointwhich is this point coerced toR.If the keyword
checkisTrue, then the initialization checks are performed. The user may specify the embedding intoRwith a keyword.INPUT:
R– ring.
kwds:
check– Boolean.embedding– field embedding from the base ring of this point toR.
OUTPUT:
ProductProjectiveSpaces_point.EXAMPLES:
sage: T.<x,y,z,u,v,w> = ProductProjectiveSpaces([1, 1, 1], ZZ) sage: P = T.point([5, 3, 15, 4, 2, 6]); sage: P.change_ring(GF(3)) (1 : 0 , 0 : 1 , 1 : 0)
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dehomogenize(L)¶ Dehomogenize \(k^{th}\) point at \(L[k]^{th}\) coordinate.
This function computes the appropriate affine patch using
Land then returns the dehomogenized point on of this affine space.INPUT:
L- a list of non-negative integers
OUTPUT:
SchemeMorphism_point_affine.
EXAMPLES:
sage: PP = ProductProjectiveSpaces([2, 2, 2], QQ, 'x') sage: A = PP([2, 4, 6, 23, 46, 23, 9, 3, 1]) sage: A.dehomogenize([0, 1, 2]) (2, 3, 1/2, 1/2, 9, 3)
sage: PP.<a,b,x,y,z> = ProductProjectiveSpaces([1, 2], CC) sage: X = PP.subscheme([a^2 + b^2]) sage: P = X([2, 2*i, -3, 6*i, 3 - 6*i]) sage: P.dehomogenize([1,0]) (-1.00000000000000*I, -2.00000000000000*I, -1.00000000000000 + 2.00000000000000*I)
sage: PP = ProductProjectiveSpaces([1, 1], ZZ) sage: A = PP([0,1,2,4]) sage: A.dehomogenize([0,0]) Traceback (most recent call last): ... ValueError: can...t dehomogenize at 0 coordinate
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global_height(prec=None)¶ Return the absolute logarithmic height of the point.
This function computes the maximum of global height of each component point in the product. Global height of component point is computed using function for projective point.
INPUT:
prec– desired floating point precision (default: default RealField precision).
OUTPUT:
a real number.
EXAMPLES:
sage: PP = ProductProjectiveSpaces(QQ, [2,2], 'x') sage: Q = PP([1, 7, 5, 18, 2, 3]) sage: Q.global_height() 1.94591014905531
sage: PP = ProductProjectiveSpaces(ZZ, [1,1], 'x') sage: A = PP([-30, 2, 1, 6]) sage: A.global_height() 3.40119738166216
sage: R.<x> = PolynomialRing(QQ) sage: k.<w> = NumberField(x^2 + 5) sage: PP = ProductProjectiveSpaces(k, [1, 2], 'y') sage: Q = PP([3, 5*w+1, 1, 7*w, 10]) sage: Q.global_height() 2.30258509299405
sage: PP = ProductProjectiveSpaces(QQbar, [1, 1], 'x') sage: Q = PP([1, QQbar(sqrt(2)), QQbar(5^(1/3)), QQbar(3^(1/3))]) sage: Q.global_height() 0.536479304144700
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local_height(v, prec=None)¶ Return the maximum of the local height of the coordinates of this point.
This function computes the maximum of local height of each component point in the product. Local height of component point is computed using function for projective point.
INPUT:
v– a prime or prime ideal of the base ring.prec– desired floating point precision (default: default RealField precision).
OUTPUT:
a real number.
EXAMPLES:
sage: PP = ProductProjectiveSpaces(QQ, [1, 1], 'x') sage: A = PP([11, 5, 10, 2]) sage: A.local_height(5) 1.60943791243410
sage: P = ProductProjectiveSpaces(QQ, [1,2], 'x') sage: Q = P([1, 4, 1/2, 2, 32]) sage: Q.local_height(2) 4.15888308335967
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normalize_coordinates()¶ Remove common factors (componentwise) from the coordinates of this point (including \(-1\)).
OUTPUT: None.
EXAMPLES:
sage: T.<x,y,z,u,v,w> = ProductProjectiveSpaces([2, 2], ZZ) sage: P = T.point([5, 10, 15, 4, 2, 6]); sage: P.normalize_coordinates() sage: P (1 : 2 : 3 , 2 : 1 : 3)
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scale_by(t)¶ Scale the coordinates of the point by
t, done componentwise.A
TypeErroroccurs if the point is not in the base ring of the codomain after scaling.INPUT:
t– a ring element
EXAMPLES:
sage: T.<x, y, z, u, v, w> = ProductProjectiveSpaces([1, 1, 1], ZZ) sage: P = T.point([5, 10, 15, 4, 2, 6]); sage: P.scale_by([2, 1, 1]) sage: P (10 : 20 , 15 : 4 , 2 : 6)
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