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RandomSpaceCurves :: knownUnirationalComponentOfSpaceCurves

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

Synopsis

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 c_1 - -

- - c_2 - -

- - c_3 d e

with both 4b-10c_1 < a and 4d-10c_3 < e.

diameter {\ge} 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

See also

Ways to use knownUnirationalComponentOfSpaceCurves :

For the programmer

The object knownUnirationalComponentOfSpaceCurves is a method function.