Computer calculations for "Some singular curves in Mukai's model of \(\overline{M}_7\)", Section 3

Code 3.2: Checking the automorphisms and singular points of the 7-cuspidal curve with heptagonal symmetry

Macaulay2, version 1.20
with packages: ConwayPolynomials, Elimination, IntegralClosure, InverseSystems,
               Isomorphism, LLLBases, MinimalPrimes, OnlineLookup,
               PrimaryDecomposition, ReesAlgebra, Saturation, TangentCone

i1 : loadPackage("Cyclotomic");

i2 : K=cyclotomicField(7);

i3 : z=K_0;

i4 : R=K[y_0..y_6];

i5 : I=ideal {3*y_5^2-4*y_4*y_6+y_3*y_0, 2*y_4*y_5-3*y_3*y_6+y_2*y_0, 5*y_3*y_5-8*y_2*y_6+3*y_1*y_0, 3*y_2*y_5-5*y_1*y_6+2*y_0*y_0, 5*y_4^2-9*y_2*y_6+4*y_1*y_0, y_3*y_4-2*y_1*y_6+y_0*y_0, 5*y_2*y_4-8*y_1*y_5+3*y_0*y_6, 5*y_3^2-9*y_1*y_5+4*y_0*y_6, 2*y_2*y_3-3*y_1*y_4+y_0*y_5, 3*y_2^2-4*y_1*y_3+y_0*y_4};

o5 : Ideal of R

i6 : F1 = map(R,R,apply(7, i -> z^i*y_i))

                              2      3      4      5         5     4     3     2
o6 = map (R, R, {y , ww y , ww y , ww y , ww y , ww y , (- ww  - ww  - ww  - ww  - ww  - 1)y , ww })
                  0    7 1    7 2    7 3    7 4    7 5       7     7     7     7     7      6    7

o6 : RingMap R <--- R

i7 : F1(I)==I

o7 = true

i8 : F2 = map(R,R,{y_0,y_6,y_5,y_4,y_3,y_2,y_1});

o8 : RingMap R <--- R

i9 : F2(I)==I

o9 = true

Magma V2.27-8     Mon Apr 10 2023 09:52:18 on MAC-M26AQ05N [Seed = 3247550512]

+-------------------------------------------------------------------+
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|                         Simons Foundation                         |
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| covering U.S. Colleges, Universities, Nonprofit Research entities,|
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+-------------------------------------------------------------------+

Type ? for help.  Type -D to quit.
> K:=CyclotomicField(7);
> P6:=ProjectiveSpace(K,6);
> X:=Scheme(P6,[3*y_5^2-4*y_4*y_6+y_3*y_0, 2*y_4*y_5-3*y_3*y_6+y_2*y_0, 5*y_3*y_5-8*y_2*\
y_6+3*y_1*y_0, 3*y_2*y_5-5*y_1*y_6+2*y_0*y_0, 5*y_4^2-9*y_2*y_6+4*y_1*y_0, y_3*y_4-2*y_1\
*y_6+y_0*y_0, 5*y_2*y_4-8*y_1*y_5+3*y_0*y_6, 5*y_3^2-9*y_1*y_5+4*y_0*y_6, 2*y_2*y_3-3*y_\
1*y_4+y_0*y_5, 3*y_2^2-4*y_1*y_3+y_0*y_4]);
> phi:=function(s,t,u,v)
function>     return [7*s^6*u, 6*s^5*t*u+s^6*v, 5*s^4*t^2*u+2*s^5*t*v, 4*s^3*t^3*u+3*s^4\
*t^2*v, 3*s^2*t^4*u+4*s^3*t^3*v, 2*s*t^5*u+5*s^2*t^4*v, t^6*u+6*s*t^5*v];
function> end function;
> S:={@ X!phi(z_7^i,1,z_7^i,1): i in [0..6] @};
> SingularPoints(X) eq S;
true
> [IsCusp(X!phi(z_7^i,1,z_7^i,1)): i in [0..6]];
[ true, true, true, true, true, true, true ]