def usecase__get_linear_series__P1P1_DP6():
'''
Construct linear series of curves in P^1xP^1,
with a given tree of (infinitely near) base points.
'''
# construct ring over Gaussian rationals
#
ring = PolyRing('x,y,v,w', True)
ring.ext_num_field('t^2 + 1')
a0 = ring.root_gens()[0]
# setup base point tree for 2 simple complex conjugate base points.
#
bp_tree = BasePointTree(['xv', 'xw', 'yv', 'yw'])
bp = bp_tree.add('xv', (-a0, a0), 1)
bp = bp_tree.add('xv', (a0, -a0), 1)
LSTools.p(
'We consider linear series of curves in P^1xP^1 with the following base point tree:'
)
LSTools.p(bp_tree)
#
# Construct linear series is defined by a list of polynomials
# in (x:y)(v:w) of bi-degree (2,2) with base points as in "bp_tree".
# The defining polynomials of this linear series, correspond to a parametric map
# of an anticanonical model of a Del Pezzo surface of degree 6 in P^6.
#
# We expect that the linear series is defined by the following set of polynomials:
#
# [ 'x^2*v^2 - y^2*w^2', 'x^2*v*w + y^2*v*w', 'x^2*w^2 + y^2*w^2', 'x*y*v^2 - y^2*v*w',
# 'x*y*v*w - y^2*w^2', 'y^2*v*w + x*y*w^2', 'y^2*v^2 + y^2*w^2' ]
#
ls = LinearSeries.get([2, 2], bp_tree)
LSTools.p(
'The linear series of bi-degree (2,2) corresponding to this base point tree is as follows:'
)
LSTools.p(ls.get_bp_tree())
#
# construct corresponding linear series of bi-degree (1,1)
# and 2 simple complex conjugate base points
#
# We expect that the linear series is defined by the following set of polynomials:
#
# ['x * v - y * w', 'y * v + x * w' ]
#
#
ls = LinearSeries.get([1, 1], bp_tree)
LSTools.p(
'The linear series of bi-degree (1,1) corresponding to this base point tree is as follows:'
)
LSTools.p(ls.get_bp_tree())
def usecase__get_base_points__P2():
'''
We obtain (infinitely near) base points of a
linear series defined over a number field.
'''
ring = PolyRing('x,y,z', True)
ring.ext_num_field('t^2 + 1')
ring.ext_num_field('t^3 + a0')
a0, a1 = ring.root_gens()
x, y, z = ring.gens()
pol_lst = [x**2 + a0 * y * z, y + a1 * z + x]
ls = LinearSeries(pol_lst, ring)
LSTools.p(ls)
bp_tree = ls.get_bp_tree()
LSTools.p(bp_tree)