-- In this file we translate many examples from Schubert into Schubert2 -- We include section III (EXAMPLES) from schubertmanual.txt in comments. -- We also include assertions (with "assert()") so this file can serve as a test file for the package. needsPackage "Schubert2"; ---- III. EXAMPLES: ---- ---- # Hilbert polynomial and Todd class of projective 3-space P^3. ---- > ---- > proj(3,h,all): factor(chi(o(n*h))); ---- 1/6 (n + 3) (n + 2) (n + 1) ---- pt = base n Ph = abstractProjectiveSpace'(3,pt,VariableName=>h) -- this applies "use" to the variety, and to its intersection ring, giving h a value factor chi OO(n*h) assert( oo === new Product from {new Power from {n+1,1},new Power from {n+2,1},new Power from {n+3,1},new Power from {1/6 + 0*n,1}} ) ---- > Ph[toddclass_]; ---- 2 2 3 3 ---- 1 + 2 h t + 11/6 h t + h t todd Ph assert( oo == 1+2*h+11/6*h^2+h^3 ) ---- #------------------------------------------------------------------------- ---- # Generation of formulas: ---- > ---- > DIM:=4: ---- > A:=bundle(2,c): # a bundle with Chern classes c1,c2 and rank 2 ---- > B:=bundle(3,d): # a bundle with Chern classes d1,d2,d3 and rank 3 base(4, Bundle => (A,2,c), Bundle => (B,3,d)); ---- > chern(A); ---- 2 ---- 1 + c1 t + c2 t chern A assert( oo == 1+c_1+c_2 ) ---- > segre(B); ---- 2 2 3 3 ---- 1 + d1 t + (d1 - d2) t + (d1 - 2 d1 d2 + d3) t ---- ---- 4 2 2 4 ---- + (d1 - 3 d2 d1 + 2 d1 d3 + d2 ) t segre B assert( oo == 1+d_1+(d_1^2-d_2)+(d_1^3-2*d_1*d_2+d_3)+(d_1^4-3*d_1^2*d_2+d_2^2+2*d_1*d_3) ) ---- > chern(A&*B); # The Chern class of the tensor product ---- 2 2 2 ---- 1 + (2 d1 + 3 c1) t + (d1 + 5 c1 d1 + 3 c1 + 2 d2 + 3 c2) t + ---- ---- 3 2 2 3 ---- (6 c1 c2 + 2 d1 d2 + c1 + 2 d3 + 4 c1 d2 + 2 c1 d1 + 4 d1 c2 + 4 d1 c1 ) t ---- ---- 2 2 2 ---- + (3 c1 d1 d2 + 6 d1 c1 c2 + 3 c2 + 3 c2 c1 + 2 d1 d3 + d2 + 3 c1 d3 ---- ---- 2 2 2 2 3 4 ---- + 2 c2 d1 + 3 c1 d2 + c1 d1 + d1 c1 ) t chern(A**B) assert( oo === 1+(3*c_1+2*d_1) +(3*c_1^2+3*c_2+5*c_1*d_1+d_1^2+2*d_2) +(c_1^3+6*c_1*c_2+4*c_1^2*d_1+4*c_2*d_1+2*c_1*d_1^2+4*c_1*d_2+2*d_1*d_2+2*d_3) +(3*c_1^2*c_2+3*c_2^2+c_1^3*d_1+6*c_1*c_2*d_1+c_1^2*d_1^2+2*c_2*d_1^2+3*c_1^2*d_2+3*c_1*d_1*d_2+d_2^2+3*c_1*d_3+2*d_1*d_3)) ---- > chern(3,symm(3,dual(A))); ---- 3 ---- - 6 c1 - 30 c1 c2 chern(3,symmetricPower(3,dual(A))) assert( oo == -6*c_1^3-30*c_1*c_2 ) ---- > segre(2,Hom(wedge(2,A),wedge(2,B))); ---- 2 2 ---- 3 d1 - 8 c1 d1 + 6 c1 - d2 ---- segre(2,Hom(exteriorPower(2,A),exteriorPower(2,B))) assert( oo == 6*c_1^2-8*c_1*d_1+3*d_1^2-d_2 ) ---- #------------------------------------------------------------------------- ---- ## Grassmannian of lines in P3: ---- > ---- > grass(2,4,b,all): Gb = flagBundle({2,2}, pt, VariableNames => {,b}) (Sb,Qb) = Gb.Bundles ---- > chi(Gb,Symm(n,Qb)); ---- 2 3 ---- n + 1 + 1/6 n + 11/6 n chi symmetricPower(n,Qb) assert( oo == (1/6)*n^3+n^2+(11/6)*n+1 ) ---- > chi(Gb,o(n*b1)); ---- 4 3 23 2 ---- 1/12 n + 2/3 n + ---- n + 7/3 n + 1 ---- 12 chi OO_Gb(n*b_1) assert( 1/12*n^4+2/3*n^3+23/12*n^2+7/3*n+1 == oo ) -- some other ways to do it: chi (det Qb)^**n factor oo p = Gb.StructureMap p_* (det Qb)^**n chi (det Qb)^**n assert( 1/12*n^4+2/3*n^3+23/12*n^2+7/3*n+1 == oo ) ---- > ---- ## This should be a quadric in P5: ---- > ---- > proj(5,H,all): chi(o(n*H)-o((n-2)*H)); ---- 4 3 23 2 ---- 1/12 n + 2/3 n + ---- n + 7/3 n + 1 ---- 12 use abstractProjectiveSpace'(5,pt,VariableName=>H) chi(OO(n*H)-OO((n-2)*H)) assert( oo == 1/12*n^4+2/3*n^3+23/12*n^2+7/3*n+1 ) ---- #------------------------------------------------------------------------- ---- # Lines on a quintic threefold. This is the top Chern class of the ---- # 5th symmetric power of the universal quotient bundle on the Grassmannian ---- # of lines. ---- > ---- > grass(2,5,c): # Lines in P^4. Gc = flagBundle({3,2}, pt, VariableNames => {,c}) (Sc,Qc) = Gc.Bundles ---- > B:=symm(5,Qc): # Qc is the rank 2 quotient bundle, B its 5th ---- > # symmetric power. B = symmetricPower(5,Qc) ---- > c6:=chern(rank(B),B):# the 6th Chern class of this rank 6 bundle. c6 = chern(rank B,B) ---- > integral(c6); ---- 2875 integral c6 assert( oo == 2875 ) -- lines on a cubic surface, in a similar way Gc = flagBundle({2,2}, pt, VariableNames => {,c}) integral chern symmetricPower_3 Gc.Bundles#1 assert( oo == 27 ) ---- ---- #------------------------------------------------------------------------- ---- # Conics on a quintic threefold. This is the top Chern class of the ---- # quotient of the 5th symmetric power of the universal quotient on the ---- # Grassmannian of 2 planes in P^5 by the subbundle of quintic containing the ---- # tautological conic over the moduli space of conics. ---- > ---- > grass(3,5,c): # 2-planes in P^4. Gc = flagBundle({2,3}, pt, VariableNames => {,c}) (Sc,Qc) = Gc.Bundles ---- > B:=Symm(2,Qc): # The bundle of conics in the 2-plane. B = symmetricPower(2,Qc) ---- > Proj(X,dual(B),z): # X is the projective bundle of all conics. X = projectiveBundle'(dual B, VariableNames => {,{z}}) ---- > A:=Symm(5,Qc)-Symm(3,Qc)&*o(-z): # The rank 11 bundle of quintics ---- > # restricted to the universal conic. A = symmetricPower_5 Qc - symmetricPower_3 Qc ** OO(-z) ---- > c11:=chern(rank(A),A):# its top Chern class. c11 = chern(rank A, A) ---- > lowerstar(X,c11): # push down to G(3,5). X.StructureMap_* c11 ---- > integral(Gc,"); # and integrate there. ---- 609250 integral oo assert( oo == 609250 ) -- here's a short cut: integral chern A assert( oo == 609250 ) ---- #------------------------------------------------------------------------- ---- ## Count the number of space conics intersecting 8 given lines ---- > ---- > grass(3,4,d,all): Gd = flagBundle({1,3}, pt, VariableNames => {,d}) (Sd,Qd) = Gd.Bundles ---- > Proj(f,dual(symm(2,Qd)),e): f = projectiveBundle'(dual symmetricPower_2 Qd, VariableNames => {,{e}}) ---- > integral(Gd,lowerstar(f,(2*d1+e)^8)); ---- 92 integral f.StructureMap_* (2*d_1 + e)^8 assert( 92 == oo ) -- another way: integral (2*d_1 + e)^8 assert( 92 == oo ) ---- #------------------------------------------------------------------------- ---- # Conics tangent to 5 plane conics. Each tangency is a degree 6 ---- # condition on the P^5 of conics; but it contains the degenerate conics with ---- # multiplicity 2. ---- > ---- > proj(5,H,all): # The P^5 of conics. Tangent bundle needed for ---- > # blowup. ---- > proj(2,h,all): # The P^2 of double lines ---- > morphism(f,Ph,PH,[H=2*h]): # The Veronese embedding ---- > blowup(f): # Construct Bf, the space of complete conics ---- > integral((6*H-2*Ef)^5); # Ef is the exceptional divisor. ---- 3264 ---- ---- #------------------------------------------------------------------------- ---- # Adjunction formula. Complete intersection in P^7. ---- > ---- > proj(7,h,tan): # P^7. Tangentbundle needed. ---- > B:=o(n1)+o(n2)+o(n3): # bundle of triples of forms, of degrees n1,n2,n3. ---- > bundlesection(Z,B): # Z is the complete intersection ---- > cotbun:=-tangentbundle(Z): # the cotangent bundle ---- > chern(1,cotbun); # the Chern class of the canonical bundle. ---- - 8 h + n1 + n2 + n3 ---- ---- #------------------------------------------------------------------------- ---- # Euler characteristic of Horrocks-Mumford bundle ---- > ---- > proj(4,h,tang): # need tangentbundle for chi X = abstractProjectiveSpace'(4,pt,VariableName => h) ---- > F:=sheaf(2,[5*h,10*h^2]): # defines the Horrocks-Mumford bundle F = abstractSheaf(X, Rank => 2, ChernClass => 1 + 5*h + 10*h^2) ---- > chi(F&*o(n*h)); # computes chi of its twists ---- 4 3 125 2 ---- 1/12 n + 5/3 n + --- n + 125/6 n + 2 ---- 12 chi (F ** OO(n*h)) assert( oo == 1/12*n^4+5/3*n^3+125/12*n^2+125/6*n+2 ) ---- #------------------------------------------------------------------------- ---- # Cohomology of the universal line in P^3. Done in two ways - ---- # LL is the line as a section of a bundle on P3 x G(2,4), whereas Tf is ---- # P(universal 2-bundle on G). ---- > ---- > proj(3,h,all): # Ph=P^3 ---- > grass(2,4,b,all): # Gb=G(2,4) ---- > R:=4-Qb: # the universal subbundle on G(2,4) ---- > productvariety(PG,Ph,Gb,pp,pg): # PG=P3 x G(2,4) ---- > bundlesection(LL,dual(R)&*o(h)):# universal line ---- > lis:=NULL: for A in monomials(LL,5) do ---- > if integral(LL,A) <> 0 then lis:=lis,A=integral(LL,A) fi od: ---- > lis; ---- 2 3 3 2 4 2 2 2 ---- b1 h = 1, b1 h = 2, b1 h = 2, b2 b1 h = 1, b2 b1 h = 1, b2 h = 1 ---- ---- > ---- > setvariety(Gb): # make G(2,4) currentvariety ---- > Proj(f,Qb,h,tan): # P(universal quotient) ---- > totalspace(f,Gb,tan): ---- > lis:=NULL: for A in monomials(Tf,5) do ---- > if integral(Tf,A) <> 0 then lis:=lis,A=integral(Tf,A) fi od: ---- > lis; ---- 2 3 3 2 4 2 2 2 ---- b1 h = 1, b1 h = 2, b1 h = 2, b2 b1 h = 1, b2 b1 h = 1, b2 h = 1 ---- ---- > ---- > expand(chi(LL,o(m*h+n*b1))-chi(Tf,o(m*h+n*b1))); # chi must be the same ---- 0 ---- ---- #------------------------------------------------------------------------- ---- # We compute the Hilbert polynomial of a space curve in three ways, ---- # using the direct image and chi on P^3, and directly on C. We quote ---- # the name 'i' of the inclusion morphism, in case i has been assigned ---- # a value earlier in the session. ---- > ---- > proj(3,h,all): ---- > curve(C,g,p): ---- > morphism('i',C,Ph,[d*p]): ---- > lowerstar(i,1); # class of C in H*(P^3) ---- 2 ---- d h ---- ---- > lowershriek(i,o(n*d*p)): # Direct image of o_C(n) ---- > chi(Ph,"); ---- 1 - g + n d ---- ---- > oo:=lowershriek(i,1): setvariety(Ph): chi(oo&*o(n*h)); ---- 1 - g + n d ---- ---- > chi(C,i&^*(o(n*h))); ---- 1 - g + n d ---- ---- #------------------------------------------------------------------------- ---- # Here we test the compatibility between bundlesection ---- # and koszul. Use grass(2,5,c) as the ambient space, and ---- # consider a regular section of the rank-3 bundle symm(2,Qc). ---- > ---- > grass(2,5,c,all): ---- > bundlesection(Z,symm(2,Qc)):# Z is the zero scheme of a section of symm(2,Qc) ---- # We may compute the Hilbert polynomial of Z in two ways: ---- > setvariety(Gc): chi(o(n*c1)&*koszul(symm(2,dual(Qc)))); ---- 2 3 ---- 1 + 11/3 n + 4 n + 4/3 n ---- ---- > setvariety(Z): chi(o(n*c1)); ---- 2 3 ---- 1 + 11/3 n + 4 n + 4/3 n ---- ---- # The following should be zero: ---- > difference:=lowershriek(iZ,1)-koszul(dual(symm(2,(Qc)))); ---- difference := ---- ---- 5 2 6 3 6 4 6 2 2 5 3 ---- 32/3 t c1 c2 - 5 t c2 + 7/2 t c2 c1 - 2 t c2 c1 - 16/3 t c2 c1 ---- ---- > ---- # We check that actually it *is* zero modulo relations: ---- > ---- > grobnerbasis(Gc): ---- > normalform(Gc,difference); ---- 0 ---- ---- #------------------------------------------------------------------------- ---- # Riemann-Roch formulas. ---- # Line bundle O(D) on a threefold. ---- > ---- > variety(X,dim=3,tan=sheaf(3,[-K,c2,c3])): # traditionally, -K is ---- > # used instead of c1 ---- > chi(o(D)); ---- 3 2 2 ---- integral(X, 1/6 D - 1/4 K D + (1/12 K + 1/12 c2) D - 1/24 K c2) ---- X = base(3, Bundle=>(L,1,D), Bundle=>(T,3,b)) X.TangentBundle = T chern L chi L assert( oo === (hold integral)((1/6)*D_1^3+(1/4)*D_1^2*b_1+(1/12)*D_1*b_1^2+(1/12)*D_1*b_2+(1/24)*b_1*b_2) ) X = abstractVariety(3,QQ[K,c_2,c_3, Degrees => {1..3}][D,Join=>false]) X.TangentBundle = abstractSheaf(X,Rank=>3,ChernClass=>1-K+c_2+c_3) chi OO(D) assert( oo === (hold integral)((1/6)*D^3-(1/4)*D^2*K+(1/12)*D*K^2+(1/12)*D*c_2-(1/24)*K*c_2) ) ---- #------------------------------------------------------------------------- ---- # number of bitangents to a plane curve ---- > ---- > proj(2,h,all): # the dual projective plane ---- > proj(2,j,all): # the projective plane ---- > bundlesection(C,o(d*j)): # define a plane curve of degree d ---- > morphism(f,C,Ph,[(d-1)*j]):# the Gauss map to the dual plane ---- > multiplepoint(f,2)/2: # double points are bitangents or flexes ---- > # correct for flexes. Flexes can of ---- > # course be calculated automatically. ---- > bitangents:=expand(integral(C,")-(3*d*(d-2))); ---- 4 3 2 ---- bitangents := 1/2 d - d - 9/2 d + 9 d ---- ---- > subs(d=4,"); ---- 28 ---- ---- #------------------------------------------------------------------------- ---- # blowup 6 points in P2. Cubics through the 6 points give a linear ---- # system of degree 3 (generically an embedding in P3). ---- > ---- > proj(2,h): ---- > blowuppoints(Ph,6,e): ---- > integral(Be,(3*h-e1-e2-e3-e4-e5-e6)^2); ---- 3 ---- ---- #------------------------------------------------------------------------- ---- # Betti numbers of a product of Grassmannians ---- > ---- > grass(2,5,a,bas): grass(3,7,b,bas): # make two Grassmannians ---- > productvariety(W,Ga,Gb): # their product W ---- > betti(W); # betti numbers of W ---- [1, 2, 5, 9, 15, 21, 29, 34, 39, 40, 39, 34, 29, 21, 15, 9, 5, 2, 1] ----

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