-- examples of determinantal 3 folds in P2xP3 studied in Lazic-Schreyer 2019. kk=QQ; -- the ground field be=3; -- the degree of the forms b on P3 S=kk[x_0..x_2,y_0..y_3,b_(0,0)..b_(1,2),Degrees=>{3:{1,0},4:{0,1},6:{0,be}}]; -- the coordinate ring of P2xP3 with in addition generic forms b_ij y23=matrix apply(2,i->apply(3,j->y_(i+j))) -* o4 = | y_0 y_1 y_2 | | y_1 y_2 y_3 | *- --the 2x3 matrix defining a rational normal curve in P3 kx=diagonalMatrix{1,-1,1}*(koszul(2,matrix{{x_0,-x_1,x_2}}))_{2,1,0} -* o5 = | 0 -x_2 x_1 | | x_2 0 -x_0 | | -x_1 x_0 0 | *- bb=matrix apply(2,i->apply(3,j->b_(i,j))) -* o6 = | b_(0,0) b_(0,1) b_(0,2) | | b_(1,0) b_(1,1) b_(1,2) | *- m=map(S^2,,(y23*kx|bb *transpose matrix{{x_0..x_2}})); -- transpose m is the generic homomorphism 2O -> G -- where G =ker(3O(1,1) ->O(2,1)) \oplus O(1,be) betti (fm=res coker m) -* 0 1 2 o8 = total: 2 4 2 0: 2 . . 1: . 3 1 2: . . . 3: . 1 . 4: . . . 5: . . 1 *- J=ann coker m; betti res J -* 0 1 o10 = total: 1 4 0: 1 . 1: . . 2: . 1 3: . . 4: . . 5: . 3 *- C=minors(2,y23); -- We check Proposition 3.1 cF=decompose (J+C); #cF==2 apply(cF,c->(codim c,betti res c)) -* 0 1 2 3 0 1 2 3 4 o14 = {(3, total: 1 6 8 3), (4, total: 1 10 20 15 4)} 0: 1 . . . 0: 1 . . . . 1: . 3 2 . 1: . 10 20 15 4 2: . . . . 3: . . . . 4: . 3 6 3 *- bby=diff(transpose basis({1,0},S), (gens cF_0)_{3..5}); bby-(transpose bb*matrix{{0,1},{-1,0}}*y23)==0 -- => the formulas in Prop. 3.1 (c) is correct trim(minors(2,bby)+minors(2,y23)) -- => bby has has rank <= 1 over C => E is a P^1-bundle over C. yx25=map(S^2,,y23|matrix apply(2,i->apply(2,j->x_(i+j)))) -* o18 = | y_0 y_1 y_2 x_0 x_1 | | y_1 y_2 y_3 x_1 x_2 | *- minors(2,yx25)==cF_1 --=> the formula in Prop. 3.1 (b) is correct -- We check proposition 4.1: P2xP3xP1=kk[x_0..x_2,y_0..y_3,b_(0,0)..b_(1,2),z_0,z_1, Degrees=>{3:{1,0,0},4:{0,1,0},6:{0,be,0},2:{0,0,1}}] -- the coordinate ring of P2xP3xP1 with 6 generic forms of degree (0,be,0) added y23=sub(y23,P2xP3xP1); kx=sub(kx,P2xP3xP1); bb=sub(bb,P2xP3xP1); J=ideal( matrix{{z_0,z_1}}*(y23*kx|bb *transpose basis({1,0,0},P2xP3xP1))); -- the defining ideal in P2xP3xP1 betti J -* 0 1 o25 = total: 1 4 0: 1 . 1: . . 2: . 3 3: . . 4: . 1 *- N=diff(transpose basis({1,0,0},P2xP3xP1),gens J) -* o26 = {1, 0, 0} | 0 y_2z_0+y_3z_1 -y_1z_0-y_2z_1 {1, 0, 0} | -y_2z_0-y_3z_1 0 y_0z_0+y_1z_1 {1, 0, 0} | y_1z_0+y_2z_1 -y_0z_0-y_1z_1 0 ------------------------------------------------------------------------- b_(0,0)z_0+b_(1,0)z_1 | b_(0,1)z_0+b_(1,1)z_1 | b_(0,2)z_0+b_(1,2)z_1 | *- C=sub(C,P2xP3xP1); cJC=decompose radical(J+C); #cJC -- need to saturate cJC1=apply(cJC,c->c:ideal basis({0,0,1},P2xP3xP1)); cJC2=apply(cJC1,c->c:ideal basis({1,0,0},P2xP3xP1)); cJC3=select(cJC2,c->not c==ideal (1_P2xP3xP1)); #cJC3==2 apply(cJC3,c->codim c)=={5,4} C1=radical minors(2,N)+C; -- C1 is the exceptional curve in P1xP3 of the map X_b -> X^1_b yz24=y23|matrix{{-z_1},{z_0}} -* o36 = | y_0 y_1 y_2 -z_1 | | y_1 y_2 y_3 z_0 | *- minors(2,yz24)==C1 -- => the formula for C1 in Proposition 4.1 (b) is correct P1xP3=kk[z_0,z_1,y_0..y_3,b_(0,0)..b_(1,2), Degrees=>{2:{0,1},4:{1,0},6:{be,0}}] N'=map(P1xP3^3,,sub(N,P1xP3)); J'=trim minors(3,N'); J1=radical J' -- J1 defines the image X_b^1 of X_b in P1xP3 betti res J1 -* 0 1 o42 = total: 1 1 0: 1 . 1: . . 2: . . 3: . . 4: . . 5: . 1 *- f=J1_0; M=map(P1xP3^2,,diff(transpose basis({0,1},P1xP3), diff(basis({0,1},P1xP3),gens J1))) -- => the formula (8) is correct C12=decompose (J1+sub(C,P1xP3)); C12_1== sub(C1,P1xP3) -- => C1 is one of the components of the preimage of Y in X^1_b C2=C12_0; apply(C12,c->betti res c) -* 0 1 2 3 0 1 2 3 o48 = {total: 1 5 5 1, total: 1 6 8 3} 0: 1 . . . 0: 1 . . . 1: . 3 2 . 1: . 6 8 3 2: . . . . 3: . . . . 4: . 2 3 . 5: . . . 1 *- (res C2).dd_2 -* o49 = {2, 0} | -y_2 y_3 z_0b_(0,1)+z_1b_(1,1) -z_0b_(0,0)-z_1b_(1,0) {2, 0} | y_1 -y_2 z_0b_(0,2)+z_1b_(1,2) 0 {2, 0} | -y_0 y_1 0 z_0b_(0,2)+z_1b_(1,2) {4, 1} | 0 0 -y_1 -y_2 {4, 1} | 0 0 y_0 y_1 ------------------------------------------------------------------------- 0 | -z_0b_(0,0)-z_1b_(1,0) | -z_0b_(0,1)-z_1b_(1,1) | -y_3 | y_2 | *- -- => the formula in Prop. 4.3 for the pfaffian is correct. -- Computing C cap det M: P3=kk[y_0..y_3,b_(0,0)..b_(1,2),Degrees=>{4:1,6:be}] I1=ideal det sub(M,P3) + sub(C,P3); I2=radical I1 degree ideal det sub(M,P3), degree I1, degree I2 -- => C intersects det M in 3(be+1) points tangentially I2_3 -* o54 = y b + y b + y b - y b - y b - y b 1 0,0 2 0,1 3 0,2 0 1,0 1 1,1 2 1,2 *- (det sub(M,P3)+(I2_3)^2 )% sub(C,P3)==0 -- => formula in proof of Prop. 4.3 is correct -- Understanding the small resolutions: Rw=kk[z_0,z_1,y_0..y_3,b_(0,0)..b_(1,2),w, Degrees=>{2:{1,0},4:{0,1},6:{0,be},{0,be+1}}] L=(sub(M,Rw)+matrix{{0,-w},{w,0}})|matrix{{z_1},{-z_0}}; I=minors(2,L); eliminate(I,{w}); eliminate(I,{w})==sub(ideal f,Rw) -- Compute the strict tranforms in X^1_b of the fibers X^2_b -> Y I1=trim saturate(ideal L_{0}+I); I2= eliminate(I1,{w}); I1'=trim saturate(ideal L_{1}+I); I2'= eliminate(I1',{w}); baseLocus=saturate(I2+I2',ideal(basis({1,0},Rw))); baseLocus==ideal sub(M,Rw) -- => |O_X^1_b(-1,b+1)| has the (be+1)^3 -- exceptional lines of X^1_b -> Y as the base locus. -- This completes the proof of all computational claims.