knownUnirationalComponentOfSpaceCurves -- check whether there is a unirational construction for a component of the Hilbert scheme of space curves

Synopsis

Usage:
knownUnirationalComponentOfSpaceCurves(d,g)
Inputs:
- d, an integer
- g, an integer
Outputs:
- a Boolean value, whether there is a component of maximal rank curves of degree d and genus g in ℙ³ with O_C(2) non-special and Hartshorne-Rao module of diameter ≤3 that have a natural free resolution

Description

* diameter = 1. All modules can be constructed

* diameter = 2. The modules can be constructed if the resolution of the generic module is minimal. This is for instance not the case for (d,g) being among (2,1), (1,2), (1,1) .

* diameter = 3. The construction is possible unless the expected Betti table of the Hartshorne-Rao module has shape

a b c₁ - -

- - c₂ - -

- - c₃ d e

with both 4b-10c₁ < a and 4d-10c₃ < e.

diameter ≥ 4. he routine returns false, although we actually do know a couple of constructions which work in a few further cases.

The following example prints an overview table for the constructable cases:

i1 : matrix apply(toList(2..18),d-> apply(toList(0..26),g->
          if knownUnirationalComponentOfSpaceCurves(d,g) then 1 else 0))

o1 = | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
     | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
     | 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
     | 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
     | 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
     | 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
     | 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
     | 0 0 1 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
     | 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
     | 0 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
     | 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 |
     | 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 |
     | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 |
     | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 |
     | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
     | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
     | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |

              17        27
o1 : Matrix ZZ   <--- ZZ

Ways to use `knownUnirationalComponentOfSpaceCurves` :

knownUnirationalComponentOfSpaceCurves(ZZ,ZZ)

knownUnirationalComponentOfSpaceCurves -- check whether there is a unirational construction for a component of the Hilbert scheme of space curves

Synopsis

Description

See also

Ways to use knownUnirationalComponentOfSpaceCurves :

Ways to use `knownUnirationalComponentOfSpaceCurves` :