next | previous | forward | backward | up | top | index | toc | Macaulay2 website
SpecialFanoFourfolds :: parameterCount(SpecialGushelMukaiFourfold)

parameterCount(SpecialGushelMukaiFourfold) -- count of parameters in the moduli space of GM fourfolds

Synopsis

Description

This method implements a parameter count explained in the paper On some families of Gushel-Mukai fourfolds.

Below, we show that the closure of the locus of GM fourfolds containing a cubic scroll has codimension at most one (hence exactly one) in the moduli space of GM fourfolds.

i1 : G = Grass(1,4,ZZ/33331);
i2 : S = schubertCycle({2,0},G) + ideal(random(1,G), random(1,G))

o2 = ideal (p    + 8480p    + 6727p    - 11656p    - 14853p    - 13522p   , p    + 8480p    - 15777p    - 11656p    + 664p    -
             1,2        1,3        2,3         1,4         2,4         3,4   0,2        0,3         2,3         0,4       2,4  
     ----------------------------------------------------------------------------------------------------------------------------
     11804p   , p    - 6727p    - 15777p    + 14853p    + 664p    - 14854p   , - 2322p    + 8971p    - 15493p    - 4330p    -
           3,4   0,1        0,3         1,3         0,4       1,4         3,4         0,1        0,2         1,2        0,3  
     ----------------------------------------------------------------------------------------------------------------------------
     1233p    - 7781p    + 8886p    - 8356p    - 1343p    + 6377p   , - 2520p    - 16661p    - 8928p    - 7368p    - 4492p    -
          1,3        2,3        0,4        1,4        2,4        3,4         0,1         0,2        1,2        0,3        1,3  
     ----------------------------------------------------------------------------------------------------------------------------
     4888p    - 7955p    + 4700p    + 5969p    - 6811p   )
          2,3        0,4        1,4        2,4        3,4

o2 : Ideal of G
i3 : X = specialGushelMukaiFourfold S;

o3 : SpecialGushelMukaiFourfold (Gushel-Mukai fourfold containing a surface of degree 3 and sectional genus 0)
i4 : time parameterCount X
S: cubic surface in PP^8 cut out by 7 hypersurfaces of degrees (1,1,1,1,2,2,2)
X: GM fourfold containing S
Y: del Pezzo fivefold containing X
h^1(N_{S,Y}) = 0
h^0(N_{S,Y}) = 11
h^1(O_S(2)) = 0, and h^0(I_{S,Y}(2)) = 28 = h^0(O_Y(2)) - \chi(O_S(2));
in particular, h^0(I_{S,Y}(2)) is minimal
h^0(N_{S,Y}) + 27 = 38
h^0(N_{S,X}) = 0
dim{[X] : S\subset X \subset Y} >= 38
dim P(H^0(O_Y(2))) = 39
codim{[X] : S\subset X \subset Y} <= 1
     -- used 9.31324 seconds

o4 = (1, (28, 11, 0))

o4 : Sequence
i5 : time discriminant X
     -- used 2.68774 seconds

o5 = 12

See also