def usecase__neron_severi_lattice():
'''
Compute NS-lattice of a linear series and
the dimension of complete linear with base points.
'''
# Blowup of projective plane in 3 colinear points
# and 2 infinitly near points. The image of the
# map associated to the linear series is a quartic
# del pezzo surface with 5 families of conics. Moreover
# the surface contains 8 straight lines.
#
ring = PolyRing('x,y,z', True)
p1 = (-1, 0)
p2 = (0, 0)
p3 = (1, 0)
p4 = (0, 1)
p5 = (2, 0)
bp_tree = BasePointTree()
bp_tree.add('z', p1, 1)
bp_tree.add('z', p2, 1)
bp_tree.add('z', p3, 1)
bp = bp_tree.add('z', p4, 1)
bp.add('t', p5, 1)
ls = LinearSeries.get([3], bp_tree)
LSTools.p('ls = ', ls)
LSTools.p(ls.get_bp_tree(), '\n\n ', ls.get_implicit_image())
# Detects that 3 base points lie on a line.
#
bp_tree = BasePointTree()
bp_tree.add('z', p1, 1)
bp_tree.add('z', p2, 1)
bp_tree.add('z', p3, 1)
ls123 = LinearSeries.get([1], bp_tree)
LSTools.p('ls123 =', ls123)
assert ls123.pol_lst == ring.coerce('[y]')
# example of infinitly near base points
# that is colinear with another simple base point.
#
bp_tree = BasePointTree()
bp_tree.add('z', p1, 1)
bp = bp_tree.add('z', p4, 1)
bp.add('t', (1, 0), 1)
ls1 = LinearSeries.get([1], bp_tree)
LSTools.p('ls1 =', ls1)
assert ls1.pol_lst == ring.coerce('[x-y+z]')
# Detects that an infinitly near base points
# is not colinear with other point.
#
for p in [p1, p2, p3]:
bp_tree = BasePointTree()
bp = bp_tree.add('z', p4, 1)
bp.add('t', p5, 1)
bp_tree.add('z', p, 1)
ls45i = LinearSeries.get([1], bp_tree)
assert ls45i.pol_lst == []
# line with predefined tangent
#
bp_tree = BasePointTree()
bp = bp_tree.add('z', p4, 1)
bp.add('t', p5, 1)
ls45 = LinearSeries.get([1], bp_tree)
LSTools.p('ls45 =', ls45)
assert ls45.pol_lst == ring.coerce('[x-2*y+2*z]')
# class of conics through 5 basepoints
#
bp_tree = BasePointTree()
bp_tree.add('z', p1, 1)
bp_tree.add('z', p2, 1)
bp_tree.add('z', p3, 1)
bp = bp_tree.add('z', p4, 1)
bp.add('t', p5, 1)
ls1234 = LinearSeries.get([2], bp_tree)
LSTools.p('ls1234 =', ls1234)
LSTools.p('\t\t', sage_factor(ls1234.pol_lst[0]))
assert ring.coerce('(x - 2*y + 2*z, 1)') in ring.aux_gcd(ls1234.pol_lst)[0]
assert ring.coerce('(y, 1)') in ring.aux_gcd(ls1234.pol_lst)[0]
# class of conics through 4 basepoints
# has a fixed component
#
bp_tree = BasePointTree()
bp_tree.add('z', p4, 1)
ls4 = LinearSeries.get([1], bp_tree)
LSTools.p('ls4 =', ls4)
bp_tree = BasePointTree()
bp_tree.add('z', p1, 1)
bp_tree.add('z', p2, 1)
bp_tree.add('z', p3, 1)
bp_tree.add('z', p4, 1)
ls1234 = LinearSeries.get([2], bp_tree)
LSTools.p('ls1234 =', ls1234)
# return
# Compose map associated to linear series "ls"
# with the inverse stereographic projection "Pinv".
#
R = sage_PolynomialRing(sage_QQ, 'y0,y1,y2,y3,y4,x,y,z')
y0, y1, y2, y3, y4, x, y, z = R.gens()
delta = y1**2 + y2**2 + y3**2 + y4**2
Pinv = [
y0**2 + delta, 2 * y0 * y1, 2 * y0 * y2, 2 * y0 * y3, 2 * y0 * y4,
-y0**2 + delta
]
H = sage__eval(str(ls.pol_lst), R.gens_dict())
PinvH = [
elt.subs({
y0: H[0],
y1: H[1],
y2: H[2],
y3: H[3],
y4: H[4]
}) for elt in Pinv
]
LSTools.p('Pinv =', Pinv)
LSTools.p('H =', H)
LSTools.p('Pinv o H =', PinvH)
# In order to compute the NS-lattice of the image
# of "PinvH" we do a base point analysis.
#
ls_PinvH = LinearSeries([str(elt) for elt in PinvH], ring)
LSTools.p('ls_PinvH =', ls_PinvH)
LSTools.p('\t\t', ls_PinvH.get_implicit_image())
if False: # takes a long time
LSTools.p(ls_PinvH.get_bp_tree())
def get_linear_series(deg_lst, bp_tree):
'''
INPUT:
- "deg_lst" -- A list of either one or two integers representing the degree of polynomials.
- "bp_tree" -- BasePointTree where base points might be in
overlapping charts.
We require that "bp_tree.chart_lst" equals either
['z', 'x', 'y'] or ['xv', 'xw', 'yv', 'yw'].
OUTPUT:
- Return A LinearSeries "ls" with base points defined by "bp_tree".
The polynomials in "ls.pol_lst" are either
* homogeneous in (x:y:z) and of degree "deg_tup[0]" or
* bi-homogenous in (x:y)(v:w) and of bi-degree ("deg_tup[0]","deg_tup[1]")
Note that there might unassigned base points.
'''
#
# Obtain generators of polynomial ring from "bp_tree.chart_lst"
# Note that for initializing PolyRing, it is important that the
# base field is not reset to the rationals QQ. The input "bp_tree"
# could already be defined over some extension field.
#
if bp_tree.chart_lst == ['z', 'x', 'y']:
ring = PolyRing('x,y,z', False)
elif bp_tree.chart_lst == ['xv', 'xw', 'yv', 'yw']:
ring = PolyRing('x,y,v,w', False)
else:
raise ValueError('Expect "bp_tree.chart_lst" in ',
(['z', 'x', 'y'], ['xv', 'xw', 'yv', 'yw']))
#
# Obtain linear series defined by
# monomials of degree "deg_lst[0]" or bidegree ("deg_lst[0]","deg_lst[1]").
#
mon_lst = get_mon_lst(deg_lst, ring.gens())
LSTools.p('mon_lst =', mon_lst)
ls = class_linear_series.LinearSeries(mon_lst, ring)
#
# For each chart "c" we obtain a list of linear series.
#
ls_lst = []
for c in bp_tree.chart_lst:
ls_lst += get_ls_lst(ls.copy().chart(c), bp_tree[c])
#
# Create matrix whose coefficients are obtained
# by evaluating each polynomial in linear series
# in "ls_lst" at (0,0).
#
row_lst = []
for ls in ls_lst:
LSTools.p(ls)
for g in ls.ring.gens():
ls.subs({g: 0})
row = sage__eval(str(ls.pol_lst), PolyRing.num_field.gens_dict())
row_lst += [row]
LSTools.p('matrix =', row_lst)
#
# Compute kernel.
#
kern = list(sage_matrix(row_lst).right_kernel().matrix())
LSTools.p('kernel =', kern)
LSTools.p(mon_lst)
#
# Obtain linear series from linear conditions on "mon_lst".
#
if kern != []:
mon_lst = ring.coerce(mon_lst) # update w.r.t. PolyRing.num_field
kern = ring.coerce(kern)
pol_lst = sage_matrix(kern) * sage_vector(mon_lst)
else:
pol_lst = []
LSTools.p(pol_lst)
return class_linear_series.LinearSeries(pol_lst, ring)
def copy(self):
cp_ring = PolyRing(self.gens())
cp_pol_lst = cp_ring.coerce(self.pol_lst)
return LinearSeries(cp_pol_lst, cp_ring)