# VSP(F_Q,9) -- Computation appearing in the proof of Theorem 5.16 in [QQ]

Let S be the ring $S$ and SQ be $S_{Q}$.

 i1 : S=QQ[h_1..h_4]; i2 : SQ=QQ[x0,x1,y0,y1, Degrees=>{{1,0},{1,0},{0,1},{0,1}}];--quadric surface in PP^3 i3 : A=gens kernel matrix{{h_1,h_2,h_3,h_4}}; 4 6 o3 : Matrix S <--- S i4 : B=A++A++A++A++A++A;-- this gives a resolution of 6U 24 36 o4 : Matrix S <--- S

Let $F_Q$ be a (4,4)-form, the restriction of a quartic form $F$ of type [100]. IS23 is the matrix of generators of $F_Q^{\perp}(2,3)$. The big matrix ZX represents the multiplication map $\rho$ from the resolution of $U\otimes F^{\perp}(2,3) \to S(3,4)$.

 i5 : F_{Q}=x0^4*y0^4+x1^4*y0^4+x0^3*x1*y0*y1^3+x0^2*x1^2*y0^2*y1^2+x1^4*y1^4 4 4 4 4 2 2 2 2 3 3 4 4 o5 = x0 y0 + x1 y0 + x0 x1 y0 y1 + x0 x1*y0*y1 + x1 y1 o5 : SQ i6 : IS23=((gens intersect(inverseSystem(F_{Q}), ((ideal(x0,x1))^2*(ideal(y0,y1))^3)))); --F_{Q}^\perp(2,3) 1 6 o6 : Matrix SQ <--- SQ i7 : ZX=sub((coefficients( IS23**matrix{{x0*y0,x1*y0,x0*y1,x1*y1}} , Monomials=>gens ((ideal(x0,x1))^3*(ideal(y0,y1))^4)))_1, S)*B;-- the big matrix representing the multiplication map from the resolution of U\otimes F^{\perp}(2,3) to S(3,4) 20 36 o7 : Matrix S <--- S i8 : Z1=submatrix(ZX,{0,1,2,3,4,5,6,7,8,9,10,11,13,14,15,16,17,18},{1,2,4,6,8,10,13,15,17,19,22,23,25,28,29,32,34,35}); 18 18 o8 : Matrix S <--- S

Compute the ideal X that defines $VSP(F_{Q}, 9)$.

 i9 : D1=ideal det Z1;--rank <= 17 locus o9 : Ideal of S i10 : X1=decompose D1 o10 = {ideal h , ideal h , ideal h , ideal h , ideal(h h - h h ), 4 3 2 1 2 3 1 4 ----------------------------------------------------------------------- 4 4 3 2 2 ideal(949968h - 6713280h - 540372420h h + 481656096h h - 1 2 2 3 2 3 ----------------------------------------------------------------------- 3 4 2 214742080h h + 47738880h - 100117184265h h h + 44496901348h h h h - 2 3 3 1 2 4 1 2 3 4 ----------------------------------------------------------------------- 2 2 2 4 82880h h h - 34412740h h + 949968h )} 1 3 4 1 4 4 o10 : List i11 : X=(X1)_5--the ideal generated by X(F_{Q}) 4 4 3 2 2 o11 = ideal(949968h - 6713280h - 540372420h h + 481656096h h - 1 2 2 3 2 3 ----------------------------------------------------------------------- 3 4 2 214742080h h + 47738880h - 100117184265h h h + 44496901348h h h h - 2 3 3 1 2 4 1 2 3 4 ----------------------------------------------------------------------- 2 2 2 4 82880h h h - 34412740h h + 949968h ) 1 3 4 1 4 4 o11 : Ideal of S

We check that $VSP(F_{Q}, 9)$ is smooth.

 i12 : JX=ideal flatten jacobian X; o12 : Ideal of S i13 : JX=ideal mingens JX; o13 : Ideal of S i14 : codim JX o14 = 4

We check that the quartic form which defines $VSP(F_{Q})$ is of type [000].

 i15 : IX=inverseSystem X; o15 : Ideal of S i16 : betti res IX 0 1 2 3 4 o16 = total: 1 16 30 16 1 0: 1 . . . . 1: . . . . . 2: . 16 30 16 . 3: . . . . . 4: . . . . 1 o16 : BettiTally

There is a function that gives us the type directly too.

 i17 : quarticType X_0 o17 = [000]