smoothnessWithReductions -- Check smoothness by using reductions to points

Usage:

answer = smoothnessWithReductions J
Inputs:
- J, an ideal, of a one parameter family
Optional inputs:
- BaseField => ..., default value ZZ/10007
- Verbose => ..., default value 0
Outputs:
- answer, a Boolean value, true if the fiber is smooth

Description

Given a homogeneous ideal J in S[z]=kk[x,z] which defines a flat family of affine curves we check smoothness of the general fiber at z=1. The basic idea is to compute some minors of the jacobian matrix which intersect the curve in a zero-dimensional scheme singF. Then using decompose singF to reduce the check at points of the finitely many maximal ideals p. Since Sp=S/p is a field the rank of the jacobian matrix restricted to Sp can be computed by checking the number of generators of the cokernel, since cokernel represents the Zariski tangent space at p of the original curve. This is much cheaper then computing all or enough minors of the jacobian matrix.

i1 : L={5,6,8}

o1 = {5, 6, 8}

o1 : List

i2 : genus L

o2 = 6

i3 : R=QQ[x_0, x_1, x_3, z,Degrees=>{5, 6, 8, 1}]

o3 = R

o3 : PolynomialRing

i4 : J=ideal(x_0^2*x_1-x_3^2-x_0^2*z^6-x_3*z^8-x_1*z^10+z^16,x_1^3-x_0^2*x_3-x_1^2*z^6-x_0*x_1*z^7-x_0^2*z^8-x_3*z^10-x_1*z^12+x_0*z^13,x_0^4-x_1^2*x_3+x_0*x_3*z^7+x_3*z^12-z^20)

             2      2    2 6      8      10    16   3    2      2 6        7
o4 = ideal (x x  - x  - x z  - x z  - x z   + z  , x  - x x  - x z  - x x z 
             0 1    3    0      3      1            1    0 3    1      0 1  
     ------------------------------------------------------------------------
        2 8      10      12      13   4    2          7      12    20
     - x z  - x z   - x z   + x z  , x  - x x  + x x z  + x z   - z  )
        0      3       1       0      0    1 3    0 3      3

o4 : Ideal of R

i5 : elapsedTime smoothnessWithReductions(J,Verbose=>2)
semigroup = {5, 6, 8}
dim and degree singF = (0, 4)
 -- .0163528s elapsed

o5 = true

The intermediate output dim and degree singF = (0, 4) says that after computing some minors of the jacobian matrix, we detect that the curve is smooth away from the zero dimensional scheme defined by singF of degree 4.

The function checkSmoothness takes longer, some times much longer.

Ways to use smoothnessWithReductions:

smoothnessWithReductions(Ideal)

For the programmer

The object smoothnessWithReductions is a method function with options.

The source of this document is in WeierstrassSemigroups.m2:1780:0.