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# computeBound -- compute the bound for the good types in case of k resonance

- Usage:
    `(a,b) = computeBound(a1,b1,k)`
    `c = computeBound k`
- Inputs:
  - a1, an integer,
  - b1, an integer, type of the degenerate K3
  - k, an integer, the resonance
- Outputs:
  - a, an integer,
  - b, an integer, minimal good type
  - c, an integer, lower bound for good a,b

## Description

The iterated mapping over the relative resolution of X_e(a,b) in the resonance scroll has betti numbers in a range of a general 2k-gonal canonical curve of genus a+b+1, if a,b are large enough, see ES2018. We compute the minimal type $(a,b) \equiv (a1,b1) \mod k$ where this becomes true.

In the second version c is the minimal value of a,b's for all congruence classes mod k. We conjecture that c=k^2-k.

    i1 : (a,b)=computeBound(6,4,3)

    o1 = (9, 7)

    o1 : Sequence

    i2 : computeBound 3
     -- .197777s elapsed
     -- .314516s elapsed
     -- .222266s elapsed
     -- .221123s elapsed
     -- .219611s elapsed
     -- .231724s elapsed

    o2 = 6

## See also

- relativeEquations -- compute the relative quadrics

## Ways to use computeBound:

- `computeBound(ZZ)`
- `computeBound(ZZ,ZZ,ZZ)`

## For the programmer

The object computeBound is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/K3Carpets.m2:2152:0.