degenerateK3 -- Ideal of a degenerate K3 surface X_e(a,b)

Synopsis

Usage:

I = degenerateK3(a,b,e)

I = degenerateK3(a,b,t)
Inputs:
- a, an integer,
- b, an integer, a and b should be positive
- e, a sequence, of two integers (e_1,e_2)
- t, a list, of two integers {t_1,t_2}
Optional inputs:
- Characteristic => an integer, default value 32003, the characteristic of the ground field
Outputs:
- I, an ideal,

Description

The routine computes the homogeneous ideal of the degenerate K3 surface in $\mathbb P^{a+b+1}$ associated as in HREF{y}{x} to a polynomial $$p(z)=z^2-e_1z+e_2=(z-t_1)(z-t_2)$$ In case $p(z)=(z-1)^2$ it coincides with carpet(a,b).

i1 : I=degenerateK3(5,5,{1,1});

                ZZ
o1 : Ideal of -----[y   ..y   ]
              32003  0,0   1,5

i2 : minimalBetti I

            0  1   2   3   4   5   6   7  8 9
o2 = total: 1 36 160 315 288 288 315 160 36 1
         0: 1  .   .   .   .   .   .   .  . .
         1: . 36 160 315 288   .   .   .  . .
         2: .  .   .   .   . 288 315 160 36 .
         3: .  .   .   .   .   .   .   .  . 1

o2 : BettiTally

i3 : I_10

o3 = y   y    - 2y   y    + y   y
      0,2 1,3     0,1 1,4    0,0 1,5

       ZZ
o3 : -----[y   ..y   ]
     32003  0,0   1,5

i4 : I=degenerateK3(5,5,(-1,1));

                ZZ
o4 : Ideal of -----[y   ..y   ]
              32003  0,0   1,5

i5 : I_10

o5 = y   y    + y   y    + y   y
      0,2 1,3    0,1 1,4    0,0 1,5

       ZZ
o5 : -----[y   ..y   ]
     32003  0,0   1,5

i6 : minimalBetti I

            0  1   2   3   4   5   6   7  8 9
o6 = total: 1 36 160 315 293 293 315 160 36 1
         0: 1  .   .   .   .   .   .   .  . .
         1: . 36 160 315 288   5   .   .  . .
         2: .  .   .   .   5 288 315 160 36 .
         3: .  .   .   .   .   .   .   .  . 1

o6 : BettiTally

i7 : I=degenerateK3(5,5,{1,1},Characteristic=>3);

              ZZ
o7 : Ideal of --[y   ..y   ]
               3  0,0   1,5

i8 : minimalBetti I

            0  1   2   3   4   5   6   7  8 9
o8 = total: 1 36 160 315 302 302 315 160 36 1
         0: 1  .   .   .   .   .   .   .  . .
         1: . 36 160 315 288  14   .   .  . .
         2: .  .   .   .  14 288 315 160 36 .
         3: .  .   .   .   .   .   .   .  . 1

o8 : BettiTally

Caveat

Ways to use `degenerateK3` :

"degenerateK3(ZZ,ZZ,List)"
"degenerateK3(ZZ,ZZ,Sequence)"

For the programmer

The object degenerateK3 is a method function with options.