S=ZZ/11[z_1..z_10]
R=S[x_1,x_2,x_3,y_1,y_2,y_3] --the ring of P2 x P2
T=R[e_1,e_2,e_3,f_1,f_2,f_3, SkewCommutative=>true]-- the space
wedge3 V
E1=(e_1+(x_1*y_1)*f_1+(x_1*y_2)*f_2+(x_1*y_3)*f_3)
E2=(e_2+(x_2*y_1)*f_1+(x_2*y_2)*f_2+(x_2*y_3)*f_3)
E3=(e_3+(x_3*y_1)*f_1+(x_3*y_2)*f_2+(x_3*y_3)*f_3)
--E1*E2*E3 represent the image of the point in the cone over
--  P^2 x P^2 with coordinates (1,((x_1,x_2,x_3),(y_1,y_2,y_3)))
B0=E1*E2*E3
B1=E1*E2*f_1
B2=E1*E2*f_2
B3=E1*E2*f_3
B4=E1*E3*f_1
B5=E1*E3*f_2
B6=E1*E3*f_3
B7=E2*E3*f_1
B8=E2*E3*f_2
B9=E2*E3*f_3
--Bi span the tangent to the Grassmannian in E1*E2*E3.
V1=e_1*e_2*e_3
V2=e_1*e_2*f_1
V3=e_1*e_2*f_2
V4=e_1*e_2*f_3
V5=e_1*e_3*f_1
V6=e_1*e_3*f_2
V7=e_1*e_3*f_3
V8=e_2*e_3*f_1
V9=e_2*e_3*f_2
V10=e_2*e_3*f_3
--Vi span the fixed Lagrangian space being the tangent to the
Grassmannian in
$e_1*e_2*e_3$.
W1=f_1*f_2*f_3
W2=f_1*f_2*e_1
W3=f_1*f_2*e_2
W4=f_1*f_2*e_3
W5=f_1*f_3*e_1
W6=f_1*f_3*e_2
W7=f_1*f_3*e_3
W8=f_2*f_3*e_1
W9=f_2*f_3*e_2
W10=f_2*f_3*e_3
--Wi span the Lagrangian space being the tangent to the Grassmannian
in f_1*f_2*f_3,
--together Vi and Fi span the whole space wedge^3 V.

VV=matrix{{V1,V2,V3,V4,V5,V6,V7,V8,V9,V10}}
WW=matrix{{W1,W2,W3,W4,W5,W6,W7,W8,W9,W10}}
WV=WW*(sub((transpose (VV))*WW, {e_1=>1, e_2=>1, e_3=>1, f_1=>1,
f_2=>1, f_3=>1}))
-- this is a sign adjustment making the pairing given by the wedge
product a
--duality between the two bases of the Lagrangians VV and  WW

RP=ZZ/11[ed_1..ed_45]
L=genericSymmetricMatrix(RP, 9)
G=matrix{
{0,0,0,0,0,0,0,0,1},
{0,0,0,0,0,-1,0,0,0},
{0,0,1,0,0,0,0,0,0},
{0,0,0,0,0,0,0,-1,0},
{0,0,0,0,1,0,0,0,0},
{0,-1,0,0,0,0,0,0,0},
{0,0,0,0,0,0,1,0,0},
{0,0,0,-1,0,0,0,0,0},
{1,0,0,0,0,0,0,0,0}}
--G represents a symmetry induced on  wedge^3 V by the symmetry of
-- P^2 x P^2 exchanging x coordinates with y coordinates

FG=sub(L,transpose(mingens kernel transpose (coefficients (mingens
ideal (L*G-G*L),
Monomials=>vars RP))_1 *random(RP^27,RP^1)))
MM=(map(T,RP)) FG
M0=matrix{{0,0,0,0,0,0,0,0,0}}
MMM=(0|M0)||((transpose M0)|MM)
-- MMM represents a symmetric linear map between the two Lagrangians
with bases given by
--VV and WV i.e., a Lagrangian subspace in wedge^3 V passing through
e_1*e_2*e_3
-- its relation with G means that it is invariant under the symmetry
corresponding
--to G WV*MMM
KK=VV+WV*MMM
--KK is a basis of the Lagrangian space defined as the graph of MMM
P=(KK*transpose((map(T,S)) (matrix{{z_1..z_10}})))_0_0
KOP=coefficients(matrix{{P*B0, P*B1, P*B2, P*B3, P*B4, P*B5, P*B6,
P*B7, P*B8, P*B9}},
Monomials=>{e_1*e_2*e_3*f_1*f_2*f_3})
-- KOP gives condition on elements of the Lagrangian spanned by KK
given by zi in the
--basis KK to be elements of the tangent in E1*E2*E3 represented by
the span of B
--the rank of these conditions will give the codimension of the
intersection locus
KPP=KOP_1
FRT=diff(  (transpose matrix{{z_1..z_10}}), KPP)
GFD=minors(9,FRT);
--GFD describes the locus of tangents to $E1*E2*E3$ meeting the
chosen symmetric
--Lagrangian spanned by $KK$

degree GFD
D=ZZ/11[x_1,x_2,x_3,y_1,y_2,y_3]
GFO=(map(D,T)) GFD;
SB=ZZ/11[m,q_1..q_9]
fv=map(D,SB,matrix{{1,x_1*y_1,x_1*y_2,x_1*y_3,x_2*y_1,x_2*y_2,x_2*y_3,x_3*y_1,x_3*y_2,x
_3*y_3}})
man=preimage(fv, GFO);
--we see the EPW quartic GFD in the corresponding affine part of the
cone over the Segre
--embedding of P^2 x P^2
dim man
degree man
EPW=ideal (homogenize(gens man, m))
--we take the projective closure and get EPW the EPW quartic in P^9
that is
--contained in the cone over
--P^2x P^2 and is symmetric with respect to
--the chosen involution
degree EPW
dim EPW
SFB=ZZ/11[s_1,s_2,s_3,s_4,s_5,s_6,tdt]
EPWsym=preimage(map(SB,SFB,matrix{{q_1,q_2+q_4,q_3+q_7,q_5,q_6+q_8,q_9,m}}),
EPW);
--EPWsym is the projection from the antiinvariant locus given by
the space of
--skew-matrices
degree EPWsym
mingens EPWsym
--we get a complete intersection of a cubic and a quartic in P^6
S=singularLocus EPWsym;
IS=ideal S;
--IS represents the singular locus  of EPWsym
dim IS
degree IS
--the singular locus is a surface of degree 52 as expected, here we
possibly need to repeat the whole program
-- to get a general enough choice that will give the right number

R=QQ[x_1..x_4,y_1..y_4]
G=QQ[a_1..a_10]
F= (transpose matrix{{x_1..x_4}})* matrix{{y_1..y_4}}
P=(F+transpose F)
W=P^{0}|P^{1}_{1,2,3}|P^{2}_{2,3}|P^{3}_{3}
--W represents the map from P^3 x P^3$ to the space of symmetric
--matrices which commutes with the exchange of variables.
--its image is the symmetric square of P^3
BU=preimage(map(R,G, W),
ideal(x_1^2+x_2^2+x_3^2+x_4^2,y_1^2+y_2^2+y_3^2+y_4^2))
--BU is the image of the symmetric square of the Fermat quadric in
P^3 in the chosen coordinates
dim BU
saturate BU
GH=QQ[a_1,a_5..a_10]
mingens preimage(map(G,GH), BU)
--we project the symmetric square of the quadric from the
antiinvariant
-- locus of a chosen symmetry preserving the quadric