-- This file does the computations for the example -- in Section 6 of -- -- UNRAMIFIED BRAUER GROUPS OF CONIC BUNDLES OVER -- RATIONAL SURFACES IN CHARACTERISTIC TWO -- -- By ASHER AUEL, ALESSANDRO BIGAZZI, -- CHRISTIAN BOEHNING, AND HANS-CHRISTIAN GRAF VON BOTHMER -- -- this example was found using Jakob Kroekers Packages -- "BlackBoxIdeals" and "FiniteFieldExperiments" restart -- we construct a conic bundle over the -- integers as a divisor of bidegree (2,2) in -- IP^2 \times IP^2 -- the base IP^2 Rbase = ZZ[u,v,w] -- the fiber IP^2 Rfiber = ZZ[x,y,z] -- the coordinate ring of IP^2 x IP^2 R = Rbase**Rfiber -------------------- -- Definition 6.1 -- -------------------- -- the symmetric matrix defining the conic bundle S = matrix { {4*v^2+2*v*u+2*u*w+2*w^2,u^2+u*w+w^2 ,v*u }, {u^2+u*w+w^2 ,2*u^2+2*v*w+2*w^2,u^2+v*w+w^2 }, {v*u ,u^2+v*w+w^2 ,2*v^2+2*u*w+2*w^2} } -- texMath symMatZZ -- TEST: is the matrix indeed symmetric? assert (0 == S - transpose S) -- the variables of the fiber IP^2 xyz = sub(vars Rfiber,R) -- the naive equation of the conic bundle xyzSxyz = (xyz*S*transpose(xyz)) -- TEST: is this equation indeed divisible by 2 assert (xyzSxyz % sub(2,R) == 0) -- the conic bundle X = ideal(xyzSxyz // sub(2,R)) -- the determinant of S detS = sub(det S,Rbase) -- TEST: is the determinant indeed divisible by 2? assert (detS % 2 == 0) -- TEST: is the determinant not divisible by 4? assert (detS % 4 != 0) -- the equation of the disciminant D = detS // 2 -- the ideal of the discriminant Delta = ideal D ----------------- -- Section 6.a -- ----------------- -- the singular locus of the discriminant over spec ZZ singDelta = ideal singularLocus Delta -- TEST: is the discriminant smooth over the --- generic point of spec ZZ? assert (3 == codim(singDelta,Generic=>true)) ---------------------------- -- the computation in the -- -- statement of Lemma 6.3 -- ---------------------------- -- the locus over spec ZZ (outside of char 2) -- where an ideal has less then -- 2 linearily independent linear polynomials possiblyMoreThanOnePoint = (I) -> ( ideal mingens minors(2, sub(jacobian super basis(1,I),ZZ)) ) ------------------------------ -- Proof of proposition 6.5 -- ------------------------------ -- saturate the jacobi ideal over ZZ time singDeltaSat = saturate(singDelta,ideal vars Rbase) -- used 334.877 seconds -- apply Lemma 6.3 assert (ideal 2^10 == possiblyMoreThanOnePoint(singDeltaSat)) -- this shows that outside of char 2 the discriminant has at -- most one singular point (counted with multiplicity). Therefore -- the discriminant is irreduceble for char != 2 ---------------------------- -- additional information -- ---------------------------- -- the prime divisors of the constant generator -- are those, where the discriminant might be singular factor sub((mingens singDeltaSat)_0_0,ZZ) ----------------- -- Section 6.b -- ----------------- -- the finite field of char 2 F2 = ZZ/2 -- the coordinate rings over F2 -- the base IP^2 Rbase2 = F2[u,v,w] -- the fiber IP^2 Rfiber2 = F2[x,y,z] -- the coordinate ring of IP^2 x IP^2 R2 = Rbase2**Rfiber2 --------------------- -- Proposition 6.6 -- --------------------- -- the discriminant over F2 D2 = sub(D,Rbase2) -- the cubic component of the discriminant over F2 use Rbase2 gamma = v^2+v*u+v*w+w^2 E = u*gamma + v^3 -- the components of the discriminant over F2 componentsD2 = {u,w,u+w,E} -- TEST: is the discriminant indeed a union of these component assert (D2 == product componentsD2) -- TEST: is gamma indeed nonzero at (0:0:1) ? assert (0 != sub(gamma,matrix{{0,0,1_F2}})) -- TEST: is the cubic smooth? assert (3 <= codim singularLocus ideal E) -- TEST: is the intersection point (0:1:0) of the -- lines indeed outside of E assert (0!=sub(E,matrix{{0,1,0}})) -- TEST: is the intersection of the cubic with w=0 transversal? assert (3 == degree radical(E + ideal w)) -- TEST: is the intersection of the cubic with u=w transversal? assert (3 == degree radical(E + ideal(u+w))) -- TEST: is the intersection of the cubic with u=0 a triple point? assert (1 == degree radical(E + ideal u)) --------------------- -- Proposition 6.8 -- --------------------- -- the conic bundle over F2 X2 = sub(X,R2) -- the rational points of IP^2 rationalPoints = { matrix{{0,1,0_F2}}, matrix{{0,1,1_F2}}, matrix{{1,0,0_F2}}, matrix{{1,0,1_F2}}, matrix{{0,0,1_F2}}, matrix{{1,1,0_F2}}, matrix{{1,1,1_F2}} } -- in the following matrix the numbers mean: -- -- 0 : the point does not lie on the component -- 1 : 2 conjugate line -- 2 : 2 rational lines -- -1 : a double line -- (matrix {apply(componentsD2,i-> sub(i,R))}|| matrix apply(rationalPoints,point->( apply(componentsD2,curve -> ( if sub(curve,point) != 0 then return 0; -- the fiber fiberOverPoint = sub(X,point|vars Rfiber2); -- the number of irreducible components -- over F2 numComponents = #decompose fiberOverPoint; -- the degree of the radical ideal degRadical = degree radical fiberOverPoint; -- two rational lines? if numComponents == 2 then return 2; -- to conjugate lines? if numComponents == 1 and degRadical==2 then return 1; -- a double line? if numComponents == 1 and degRadical==1 then return -1; ))) ) ) -- | u w u+w u2v+uv2+v3+uvw+uw2 | -- | -1 -1 -1 0 | -- | 2 0 0 0 | -- | 0 2 0 2 | -- | 0 0 2 0 | -- | 1 0 0 1 | -- | 0 1 0 0 | -- | 0 0 1 0 | ---------------------- -- Proposition 6.10 -- ---------------------- -- check the condition that the -- double line locus has reduced scheme -- structure on each line. -- this locus is defined by the -- coefficients of the mixed term x*y,x*z,y*z -- in the equation for X_(2). doubleLineLocus = saturate ideal sub( contract(matrix{{x*y,x*z,y*z}},gens X2), Rbase2) -- TEST: is this indeed a reduced point? assert (doubleLineLocus == ideal(u,w)) ----------------- -- Section 6.c -- ----------------- -- The support of the singular locus of X2 X2sing = radical saturate(saturate( ideal singularLocus X2, ideal sub(vars Rbase2,R2)), ideal sub(vars Rfiber2,R2) ) -- TEST: is the singular locus finite? assert (4==codim X2sing) -- project the singular locus to the pase X2singBase = sub(eliminate({x,y,z},X2sing),Rbase2) -- (i), (ii) & (iii) -- TEST: do the singularities indeed lie over the intersection points -- of the cubic with the 3 lines? assert (X2singBase == radical ideal(E,u*w*(u+w))) -- TEST: is X_(2) indeed smooth over (0:1:0)? assert (not 0 == sub(X2singBase,matrix{{0,1,0_F2}}))