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K3Carpets :: degenerateK3BettiTables

degenerateK3BettiTables -- compute the Betti tables of a degenerate K3 over all prime fields

Synopsis

Description

We compute the equation and nonminimal resolution F of the degeneate K3 of type (a,b,e) where a ≥b over a large finite prime field, lift the complex to the integers, which is possible if the coefficients are small. Finally we study the nonminimal strands over ZZ by computing the Smith normal form. The resulting data allow us to compute the Betti tables for arbitrary primes.

a=5,b=5
e=(-1,5)
h=degenerateK3BettiTables(a,b,e)
keys h
elapsedTime T= minimalBetti degenerateK3(a,b,e,Characteristic=>5)
T-h#5

Already for fairly small values of (e1,e2) the result might be incorrect, because the lift to characteristic zero fails due to high powers of e1 and e2 in the non-minimal resolution. It would be easy to alter the program to catch these mistakes.

e=(-1,5^2)
h=degenerateK3BettiTables(a,b,e)
keys h

Caveat

Already for (e1,e2) fairly small, the algorithm might give wrong answers since the lift to characteristic zero might be incorrect. A correction is easy to implement as soon res(.,FastNonminimal=>true) allows QQ (or ZZ) as coefficient ring. Another possibily would be to use the Chinese remainder for lifting to ZZ.

See also

Ways to use degenerateK3BettiTables :