CoCalc -- 1529-last-week.ipynb

Kernel: SageMath (stable)

In [1]:

# The fastest way to get the ambient space in some "normalized" representation.
Delta = ReflexivePolytope(3, 1529)
P = CPRFanoToricVariety(Delta)
P

3-d CPR-Fano toric variety covered by 12 affine patches

In [2]:

Delta.nef_partitions()

[
Nef-partition {0, 1, 2, 3, 4, 7} U {5, 6, 8, 9, 10, 11},
Nef-partition {0, 1, 2, 3, 4, 5, 6, 8, 9} U {7, 10, 11} (projection)
]

In [3]:

Delta.polar()

3-d reflexive polytope in 3-d lattice N

In [4]:

Delta.polar().nef_partitions()

[
Nef-partition {0, 1, 4, 7, 11, 12, 13} U {2, 3, 5, 6, 8, 9, 10}
]

In [5]:

Delta.plot3d()

In [6]:

#Delta.faces()

In [7]:

P.fan().rays()

N(-1, -1, -1),
N(-1,  0,  0),
N( 1,  0,  1),
N( 1,  0,  0),
N( 0, -1,  0),
N( 1,  1,  1),
N( 0,  0, -1),
N( 0, -1,  1),
N( 0,  1, -1),
N( 0,  1,  0),
N( 1,  1,  0),
N( 0,  0,  1),
N(-1,  0, -1),
N(-1, -1,  0)
in 3-d lattice N

In [8]:

# Successive blowups are more convenient to see "P3 coordinates" inside
P3 = toric_varieties.P(3)
Sigma = P3.fan().subdivide(sum(sigma.rays()) for sigma in P3.fan())    # Blow up points
Sigma = Sigma.subdivide(sum(sigma.rays()) for sigma in P3.fan(2))      # Blow up lines
P = CPRFanoToricVariety(
        Delta_polar=LatticePolytope(Sigma.rays()),
        charts=[sigma.ambient_ray_indices() for sigma in Sigma])
P.plot(mode="generators", wall_label=None)

In [9]:

P.Delta().index()

1529

In [10]:

X = P.anticanonical_hypersurface()
show(X)

In [11]:

# "Standalone" homogeneous polynomial of the family.
# fraction_field is needed to use these parameters in toric setup later
Rm = PolynomialRing(QQ, "m0, m1, m2, m3, p").fraction_field()
Rm.inject_variables()
m = Rm.gens()[:-1]
Rx = PolynomialRing(Rm, 4, "x")
x = Rx.gens()
I = range(4)
A3 = (sum(m[i]^2 * x[i] for i in I) * sum(1 / x[i] for i in I) - p^2) * prod(x)
show(A3)

Defining m0, m1, m2, m3, p

In [12]:

# Antocanonical hypersurface of the blowup with only "original" P3 coordinates visible.
show(X.defining_polynomials()[0].subs(dict(zip(X.coordinate_ring().gens()[4:], [1]*10))))

In [13]:

# The matching is "mi^2 coefficients for monomials with zi^2"
a = [1]*13
a[3] = a[7] = a[8] = m0^2
a[5] = a[6] = a[9] = m1^2
a[4] = a[10] = a[11] = m2^2
a[0] = a[1] = a[2] = m3^2
a[12] = m0^2 + m1^2 + m2^2 + m3^2 - p^2
# This is the family of interest realized in a toric variety.
X = P.anticanonical_hypersurface(coefficients=a)
# X lives in a slightly different space - its coordinate ring includes variables for coefficients
P = X.ambient_space()
show(X.defining_polynomials()[0].subs(dict(zip(X.coordinate_ring().gens()[4:], [1]*10))))

In [14]:

# Helper function for constructing subfamilies
def m_to_a(*m):
    a = [1]*13
    a[3] = a[7] = a[8] = m[0]^2
    a[5] = a[6] = a[9] = m[1]^2
    a[4] = a[10] = a[11] = m[2]^2
    a[0] = a[1] = a[2] = m[3]^2
    a[12] = m[0]^2 + m[1]^2 + m[2]^2 + m[3]^2 - p^2
    return a

show(m_to_a(m0,m1,m2,m3))
show(m_to_a(m0,m0,m2,m2))

In [15]:

# Fibration code predates toric varieties in Sage, so needs different steps to set up.
from fibration import *
from elliptic_fibration import *

In [16]:

Delta_polar = P.Delta_polar()
slices = reflexive_slices(Delta_polar, symmetries=True)
len(slices)

2

In [17]:

# These are normals for symmetric fibrations, 6 hexagons and 3 quadrangles
for s in slices:
    print s.symmetries

((0, 1, -1), (0, 0, 1), (1, 0, -1), (0, 1, 0), (1, -1, 0), (1, 0, 0))
((1, -1, -1), (1, 1, -1), (1, -1, 1))

In [18]:

# The first (i.e. 0-th) slice is the hexagon that seemed to be of bigger interest.
fib = EllipticFibration(Delta_polar, slices[0].normal, extra_parameters=["m0", "m1", "m2", "m3", "p"])

In [19]:

show(fib.hypersurface())

In [20]:

%%time
j = fib.j()
print(j.numerator().degree(), j.denominator().degree())
print(fib._degree_shift)

Jupyter Kernel terminated: This might be caused by running out of memory or hitting a bug in some library (e.g., forking too many processes, trying to access invalid memory, etc.). Consider restarting or upgrading your project or running the relevant code directly in a terminal to track down the cause, as explained here.

In [0]:

fib.singular_fibers_summary()

In [0]:

fib.plot3d().show(frame=False)

In [0]:

# Intersection matrix of torically induced divisors on K3, in our case there is one divisor per point,
# each edge corresponds to a simple intersection. Note that there are no edges 3-6 and 2-7.
M=intersection_matrix(Delta_polar); M

In [0]:

M.dimensions()

In [0]:

curves = [X.intersection(P.subscheme(z)) for z in P.gens()]

In [0]:

# Simplify curve equations and check that they all are given as (polynomial, monomial)
curves = [P.subscheme(c.defining_ideal().groebner_basis()) for c in curves]
for c in curves:
    I = c.defining_ideal()
    print(I.gens())
    assert(I.ngens() == 2)
    assert(len(I.gen(1).monomials()) == 1)

In [0]:

# No intersection between 0 and 1
pts = curves[0].intersection(curves[1])
pts.dimension()

In [0]:

# Some intersection between 0 and 8
pts = curves[0].intersection(curves[8])
pts.dimension()

In [0]:

# Equations for the intersection in all charts where it is visible
# (no claims that these points are the same or different)
for i in range(P.fan().ngenerating_cones()):
    ap = pts.affine_patch(i)
    if ap.dimension() != -1:
        print(i, ap.defining_ideal().groebner_basis())

In [0]:

cm = matrix(ZZ, len(curves))
for i in range(len(curves)):
    for j in range(i + 1, len(curves)):
        if curves[i].intersection(curves[j]).dimension() == 0:
            cm[i, j] = cm [j, i] = 1
cm

In [0]:

# Assuming that all intersections are 0 or 1 and all self-intersections are -2, we recover the correct intersection matrix
M - cm

In [1]:

c = curves[0]
c

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-1-f989cd57b657> in <module>()
----> 1 c = curves[Integer(0)]
      2 c
NameError: name 'curves' is not defined

In [35]:

I = c.defining_ideal()
I

Ideal (m0^2*z0^2*z1*z2*z5*z6*z7^2*z8^2*z9^2*z10*z11 + m1^2*z0*z1^2*z2*z4*z6*z7^2*z8^2*z9*z10^2*z12 + m0^2*z0^2*z1*z3*z5*z6^2*z7*z8^2*z9*z11^2*z12 + m1^2*z0*z1^2*z3*z4*z6^2*z7*z8^2*z10*z11*z12^2 + m2^2*z0*z1*z2^2*z4*z5*z7^2*z8*z9^2*z10^2*z13 + m0^2*z0^2*z2*z3*z5^2*z6*z7*z8*z9^2*z11^2*z13 + (m0^2 + m1^2 + m2^2 + m3^2 - p^2)*z0*z1*z2*z3*z4*z5*z6*z7*z8*z9*z10*z11*z12*z13 + m1^2*z1^2*z2*z3*z4^2*z6*z7*z8*z10^2*z12^2*z13 + m3^2*z0*z1*z3^2*z4*z5*z6^2*z8*z11^2*z12^2*z13 + m2^2*z0*z2^2*z3*z4*z5^2*z7*z9^2*z10*z11*z13^2 + m2^2*z1*z2^2*z3*z4^2*z5*z7*z9*z10^2*z12*z13^2 + m3^2*z0*z2*z3^2*z4*z5^2*z6*z9*z11^2*z12*z13^2 + m3^2*z1*z2*z3^2*z4^2*z5*z6*z10*z11*z12^2*z13^2, z8) of Multivariate Polynomial Ring in z0, z1, z2, z3, z4, z5, z6, z7, z8, z9, z10, z11, z12, z13 over Fraction Field of Multivariate Polynomial Ring in m0, m1, m2, m3, p over Rational Field

In [44]:

sp = SR(I.groebner_basis()[0])
sp

m3^2*z1*z10*z11*z12^2*z13^2*z2*z3^2*z4^2*z5*z6/m2^2 + m3^2*z0*z11^2*z12*z13^2*z2*z3^2*z4*z5^2*z6*z9/m2^2 + z1*z10^2*z12*z13^2*z2^2*z3*z4^2*z5*z7*z9 + z0*z10*z11*z13^2*z2^2*z3*z4*z5^2*z7*z9^2

In [46]:

P.subscheme(I.groebner_basis())

Closed subscheme of 3-d CPR-Fano toric variety covered by 24 affine patches defined by:
  z0*z2^2*z3*z4*z5^2*z7*z9^2*z10*z11*z13^2 + z1*z2^2*z3*z4^2*z5*z7*z9*z10^2*z12*z13^2 + m3^2/m2^2*z0*z2*z3^2*z4*z5^2*z6*z9*z11^2*z12*z13^2 + m3^2/m2^2*z1*z2*z3^2*z4^2*z5*z6*z10*z11*z12^2*z13^2,
  z8

In [45]:

sp.factor_list()

[(m3^2*z11*z12*z3*z6 + m2^2*z10*z2*z7*z9, 1),
 (z1*z10*z12*z4 + z0*z11*z5*z9, 1),
 (m2, -2),
 (z13, 2),
 (z2, 1),
 (z3, 1),
 (z4, 1),
 (z5, 1)]

In [22]:

im16 = diagonal_matrix(ZZ, [-2]*20)

cones = P.fan().cones(dim=1)
# The first 4 divisors are coordinate planes from P3
# The next 4 are blown up fixed points
# P1xP1 divisors correspond to the last 6 which are blown up torus invariant lines
# (on the polytope plot they correspond to vertices with 4 edges)
for i in range(6):
    # Two lines in P1xP1 intersect each other
    im16[8 + 2 * i, 9 + 2 * i] = im16[9 + 2 * i, 8 + 2 * i] = 1
    ars = list(cones[8 + i].adjacent())
    r1 = ars.pop()
    for r2 in ars:
        if r2 not in r1.adjacent():
            ars.remove(r2)
            break
    # Now r1 and r2 are "opposite"
    k = r1.ambient_ray_indices()[0]
    im16[k, 8 + 2 * i] = im16[8 + 2 * i, k] = 1
    k = r2.ambient_ray_indices()[0]
    im16[k, 8 + 2 * i] = im16[8 + 2 * i, k] = 1
    # And ars contains another "opposite pair"
    k = ars[0].ambient_ray_indices()[0]
    im16[k, 9 + 2 * i] = im16[9 + 2 * i, k] = 1
    k = ars[1].ambient_ray_indices()[0]
    im16[k, 9 + 2 * i] = im16[9 + 2 * i, k] = 1

print(im16.str())

In [23]:

im16.rank()

16

In [24]:

# Sections form 2 chains, 2-13-3 and 6-8-7, so there are 2 cases: "end" and "middle"
fib.sections()

[2 (toric), 3 (toric), 6 (toric), 7 (toric), 8 (toric), 13 (toric)]

In [25]:

s = fib.sections()[0]
s

2 (toric)

In [26]:

# F - fiber, S - section, T_n - top divisors, B_n - bottom divisors, S_n - slice divisors
print s.decorated_intersection_matrix()

        F  S  T_ T_ T_ B_ B_ B_ S_ S_ S_
              0  5  11 1  4  12 7  8  13
                                        
F      [ 0  1  0  0  0  0  0  0  1  1  1] F
S      [ 1 -2  0  0  0  0  0  0  0  0  1] S
T_0    [ 0  0 -2  0  1  0  0  0  0  1  0] T_0
T_5    [ 0  0  0 -2  1  0  0  0  0  0  1] T_5
T_11   [ 0  0  1  1 -2  0  0  0  0  0  0] T_11
B_1    [ 0  0  0  0  0 -2  0  1  0  1  0] B_1
B_4    [ 0  0  0  0  0  0 -2  1  0  0  1] B_4
B_12   [ 0  0  0  0  0  1  1 -2  0  0  0] B_12
S_7    [ 1  0  0  0  0  0  0  0 -2  1  0] S_7
S_8    [ 1  0  1  0  0  1  0  0  1 -2  0] S_8
S_13   [ 1  1  0  1  0  0  1  0  0  0 -2] S_13
        F  S  T_ T_ T_ B_ B_ B_ S_ S_ S_
              0  5  11 1  4  12 7  8  13
                                        

In [27]:

proj = s._ef.slice().normal * s._ef.polytope().points()
D_fiber = [(pt, proj[pt]) for pt in s._ef.polytope().skeleton_points() if proj[pt] > 0]
D_fiber

[(0, 1), (5, 1), (9, 1), (11, 1)]

In [28]:

s = fib.sections()[-1]
s

13 (toric)

In [29]:

# F - fiber, S - section, T_n - top divisors, B_n - bottom divisors, S_n - slice divisors
print s.decorated_intersection_matrix()

        F  S  T_ T_ T_ B_ B_ B_ S_ S_ S_
              0  9  11 1  10 12 6  7  8 
                                        
F      [ 0  1  0  0  0  0  0  0  1  1  1] F
S      [ 1 -2  0  0  0  0  0  0  0  0  0] S
T_0    [ 0  0 -2  1  1  0  0  0  0  0  1] T_0
T_9    [ 0  0  1 -2  0  0  0  0  0  1  0] T_9
T_11   [ 0  0  1  0 -2  0  0  0  1  0  0] T_11
B_1    [ 0  0  0  0  0 -2  1  1  0  0  1] B_1
B_10   [ 0  0  0  0  0  1 -2  0  0  1  0] B_10
B_12   [ 0  0  0  0  0  1  0 -2  1  0  0] B_12
S_6    [ 1  0  0  0  1  0  0  1 -2  0  1] S_6
S_7    [ 1  0  0  1  0  0  1  0  0 -2  1] S_7
S_8    [ 1  0  1  0  0  1  0  0  1  1 -2] S_8
        F  S  T_ T_ T_ B_ B_ B_ S_ S_ S_
              0  9  11 1  10 12 6  7  8 
                                        

In [30]:

# Since we have used the same Delta/Delta_polar and there are no points apart from vertices and the origin,
# coordinates and coefficients are the same and we can use the same a-matching.
fib.set_coefficients(dict(zip(fib.coefficients_ring().gens()[:13], a)))
show(fib.hypersurface())

In [31]:

#fib.hypersurface()

In [32]:

show(fib.fiber())

In [33]:

j = fib.j()

---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
<ipython-input-33-4c0091c2dd0b> in <module>()
----> 1 j = fib.j()

/ext/sage/sage-8.1/src/sage/misc/cachefunc.pyx in sage.misc.cachefunc.CachedMethodCallerNoArgs.__call__ (build/cythonized/sage/misc/cachefunc.c:13511)()
   2377         if self.cache is None:
   2378             f = self.f
-> 2379             self.cache = f(self._instance)
   2380         return self.cache
   2381 
/home/user/elliptic_fibration.pyc in j(self)
    877         y^2 = x^3 + g2*x + g3.
    878         """
--> 879         return 4 * self.g2()**3 / self.delta()
    880 
    881     def plot3d(self):
/home/user/elliptic_fibration.pyc in g2(self)
    832             return self._g2
    833         except AttributeError:
--> 834             self._compute_invariants()
    835             return self._g2
    836 
/home/user/elliptic_fibration.pyc in _compute_invariants(self, d)
    771                 else:
    772                     f0 = SingularFiber(self._top, degenerate=True)
--> 773                     d = degree_shift(f0)
    774                     self._f_0 = f0
    775 
/home/user/elliptic_fibration.pyc in degree_shift(f0)
    755             d = (evo[2] - dvo) / 6
    756             if not d.is_integral():
--> 757                 raise ValueError("degree shift %s is not integer!" % d)
    758             if (g2vo + 2 * d < evo[0]
    759                 or g3vo + 3 * d < evo[1]
ValueError: degree shift -1/3 is not integer!

In [34]:

# The normalization above (and raising an exception) is necessary for other applications of this code.
# It fails here because the subfamily has more comlicated singular fibers over zero and infinity than the full family.
# Nevertheless, the j invariant (and probably simpler g2 and g3) are accessible on subsequent calls without normalization.
# Moreover, we can try to compute invariants with the same degree shift as in the generic case above.
fib._compute_invariants(0)
# It is possible that the current code never needs this shift at all, but it is not obvious.
j = fib.j()
j.numerator().degree(), j.denominator().degree()

(48, 42)

In [35]:

fib.singular_fibers_summary()

'8x I_1, I_2 at (-m1^2)/m0^2, I_2 at -1, I_6 at +Infinity, I_6 at 0'

In [36]:

fib.show()

A fibration of

\Delta^{3}_{2355}

corresponding to 2-dimensional reflexive slice of a 3-dimensional polytope.
Points: [2, 3, 6, 7, 8, 13, 14]
Skeleton points: [2, 3, 6, 7, 8, 13]
Normal to the slice hyperplane: (1, -1, 0)
Vertices of the slice polytope:
M(-1, -1),
M(-1, 0),
M( 0, -1),
M( 1, 1),
M( 0, 1),
M( 1, 0)
in 2-d lattice M
Coordinate points:

\left(\begin{array}{rrrrrrrrrrrrrr} 1 & 0 & 0 & -1 & -1 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & -1 & -1 \\ 0 & 1 & 0 & -1 & 0 & -1 & 0 & 1 & 1 & 0 & 1 & -1 & 0 & -1 \\ 0 & 0 & 1 & -1 & 0 & 0 & -1 & 1 & 0 & 1 & 1 & -1 & -1 & 0 \end{array}\right)

Monomial points:

\left(\begin{array}{rrrrrrrrrrrrr} 0 & -1 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & -1 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 1 & 1 & -1 & 0 & 1 & 0 & -1 & 0 \\ -1 & 0 & 0 & 0 & 1 & 0 & -1 & 0 & -1 & 0 & 1 & 1 & 0 \end{array}\right)

Equation of a hypersurface:

m_{0}^{2} z_{0}^{2} z_{1} z_{2} z_{5} z_{6} z_{7}^{2} z_{8}^{2} z_{9}^{2} z_{10} z_{11} + m_{1}^{2} z_{0} z_{1}^{2} z_{2} z_{4} z_{6} z_{7}^{2} z_{8}^{2} z_{9} z_{10}^{2} z_{12} + m_{0}^{2} z_{0}^{2} z_{1} z_{3} z_{5} z_{6}^{2} z_{7} z_{8}^{2} z_{9} z_{11}^{2} z_{12} + m_{1}^{2} z_{0} z_{1}^{2} z_{3} z_{4} z_{6}^{2} z_{7} z_{8}^{2} z_{10} z_{11} z_{12}^{2} + m_{2}^{2} z_{0} z_{1} z_{2}^{2} z_{4} z_{5} z_{7}^{2} z_{8} z_{9}^{2} z_{10}^{2} z_{13} + m_{0}^{2} z_{0}^{2} z_{2} z_{3} z_{5}^{2} z_{6} z_{7} z_{8} z_{9}^{2} z_{11}^{2} z_{13} + (m_{0}^{2} + m_{1}^{2} + m_{2}^{2} + m_{3}^{2} - p^{2}) z_{0} z_{1} z_{2} z_{3} z_{4} z_{5} z_{6} z_{7} z_{8} z_{9} z_{10} z_{11} z_{12} z_{13} + m_{1}^{2} z_{1}^{2} z_{2} z_{3} z_{4}^{2} z_{6} z_{7} z_{8} z_{10}^{2} z_{12}^{2} z_{13} + m_{3}^{2} z_{0} z_{1} z_{3}^{2} z_{4} z_{5} z_{6}^{2} z_{8} z_{11}^{2} z_{12}^{2} z_{13} + m_{2}^{2} z_{0} z_{2}^{2} z_{3} z_{4} z_{5}^{2} z_{7} z_{9}^{2} z_{10} z_{11} z_{13}^{2} + m_{2}^{2} z_{1} z_{2}^{2} z_{3} z_{4}^{2} z_{5} z_{7} z_{9} z_{10}^{2} z_{12} z_{13}^{2} + m_{3}^{2} z_{0} z_{2} z_{3}^{2} z_{4} z_{5}^{2} z_{6} z_{9} z_{11}^{2} z_{12} z_{13}^{2} + m_{3}^{2} z_{1} z_{2} z_{3}^{2} z_{4}^{2} z_{5} z_{6} z_{10} z_{11} z_{12}^{2} z_{13}^{2}

Projection to the base:

\left(z_{0} z_{5} z_{9} z_{11}, z_{1} z_{4} z_{10} z_{12}\right)

Coordinate on the base:

z_{0}

Fiber over

\left(t^{ 1 }, 1\right)

(m_{0}^{2} t^{2} + m_{1}^{2} t) z_{3} z_{6}^{2} z_{7} z_{8}^{2} + (m_{0}^{2} t^{2} + m_{1}^{2} t) z_{2} z_{6} z_{7}^{2} z_{8}^{2} + (m_{3}^{2} t) z_{3}^{2} z_{6}^{2} z_{8} z_{13} + (m_{0}^{2} t^{2} + m_{0}^{2} t + m_{1}^{2} t + m_{2}^{2} t + m_{3}^{2} t - p^{2} t + m_{1}^{2}) z_{2} z_{3} z_{6} z_{7} z_{8} z_{13} + (m_{2}^{2} t) z_{2}^{2} z_{7}^{2} z_{8} z_{13} + (m_{3}^{2} t + m_{3}^{2}) z_{2} z_{3}^{2} z_{6} z_{13}^{2} + (m_{2}^{2} t + m_{2}^{2}) z_{2}^{2} z_{3} z_{7} z_{13}^{2}

Top: ExtA3, generic

F_0

I_{4}

Bottom: ExtA3, generic

F_\infty

I_{4}

In [37]:

%%time

# Singular fibers (some of them) for different subfamilies
Ms = [
    (m0, m1, m2, m3),
    (m0, m0, m2, m3),
    (m0, m0, m2, m2),
    (m0, m0, m0, m0),
]
for M in Ms:
    fib.set_coefficients(dict(zip(fib.coefficients_ring().gens()[:13], m_to_a(*M))))
    fib._compute_invariants(0)
    print fib.singular_fibers_summary()

8x I_1, I_2 at (-m1^2)/m0^2, I_2 at -1, I_6 at +Infinity, I_6 at 0
8x I_1, I_4 at -1, I_6 at +Infinity, I_6 at 0
4x I_1, I_4 at -1, I_6 at +Infinity, I_6 at 0
4x I_1, I_4 at -1, I_6 at +Infinity, I_6 at 0
CPU times: user 9.48 s, sys: 128 ms, total: 9.61 s
Wall time: 10.6 s

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SR(fib.fiber()).subs(t=-1).factor()

-(m0^2*z3^2*z6^2 + 2*m0^2*z2*z3*z6*z7 - p^2*z2*z3*z6*z7 + m0^2*z2^2*z7^2)*z13*z8

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## the euler characteristic looks wrong. Some multiplicities are coming out right and the index of the fiber at 0 should increase.
# by explicitely looking at the j-invariant we could determine the missing fibers

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# The other slice
fib = EllipticFibration(Delta_polar, slices[1].normal, extra_parameters=["m0", "m1", "m2", "m3", "p"])
fib.hypersurface()

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fib.show()

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%%time
j = fib.j()
print(j.numerator().degree(), j.denominator().degree())
print(fib._degree_shift)

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fib.singular_fibers_summary()

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fib.set_coefficients(dict(zip(fib.coefficients_ring().gens()[:13], a)))
fib.hypersurface()

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fib._compute_invariants(0)

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fib.singular_fibers_summary()

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fib.sections()

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%%time

# Singular fibers (some of them) for different subfamilies
Ms = [
    (m0, m1, m2, m3),
    (m0, m0, m2, m3),
    (m0, m0, m2, m2),
    (m0, m0, m0, m0),
]
for M in Ms:
    fib.set_coefficients(dict(zip(fib.coefficients_ring().gens()[:13], m_to_a(*M))))
    fib._compute_invariants(0)
    print fib.singular_fibers_summary()

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LatticeDiagram?

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Mnew=matrix(ZZ,[[-2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 1, 0, 0, 1, 1, 0], [0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1], [0, 0, 0, -2, 
0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 
1, 0, 1], [0, 0, 0, 0, 0, -2, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 1, 
0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0], [1, 
0, 0, 0, 0, 0, 1, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 1, 0, 0, 0, -2, 0, 0, 0, 
0, 0, 0, 0, 1, 0, 0], [1, 0, 0, 1, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 1, 0, 
0, 1, 0, 0, 0, 0, -2, 0, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, -2, 0, 0, 0, 1, 
0, 0, 0], [0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 1, 0, 
0, 0, 0, 0, 1, -2, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, -2, 0, 0, 0, 0], [0, 
0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, -2, 0, 0, 0], [0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 
0, 0, 0, -2, 0, 0], [0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -2, 0], [0, 0, 1, 0, 1, 0, 
0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2]])

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Mnew.is_symmetric()

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Mnew.rank()

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# the matrix Mnew is similar to the one constructed by Andrey. This can be seen by checking that they both have the same Jordan form.

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Jnew,Pnew=Mnew.jordan_form(transformation=True);
J16,P16=im16.jordan_form(transformation=True);

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show(Jnew-J16)