alexanderSurface -- construct an Alexander surface

Synopsis

• Usage:

  alexanderSurface S

• Inputs:

  • S, a ring, the coordinate ring of P⁴
• Outputs:

  • a sequence, a pair (Y, pts) where Y is the ideal of an Alexander surface in P⁴, i.e., the blow-up of P² along 10 points embedded in P⁴ via |13L-4(p₁+...+p₁₀)|, and pts is the list of the 10 blown-up points in P², saved for future use

Description

i1 : S = ZZ/32009[x_0..x_4];

i2 : time (Y, pts) = alexanderSurface S;
     -- used 8.47809 seconds

i3 : betti res Y

            0  1  2  3 4
o3 = total: 1 16 29 18 4
         0: 1  .  .  . .
         1: .  .  .  . .
         2: .  .  .  . .
         3: .  .  .  . .
         4: . 15 26 15 3
         5: .  1  3  3 1

o3 : BettiTally

See also

• linSysAlexander -- linear systems giving rise to curves of genus >= 16 on the Alexander surface
• properTransformAlexander -- proper transform on the Alexander surface