-- Examples from the M2 session during the workshop. -- For general information on, downloading and the documentation of -- Macaulay2, visit http://www.math.uiuc.edu/Macaulay2/ -- (Two dashes in a line declares that the rest of the line is a comment.) kk = ZZ/32749 -- setting up notation. -- Example 1. Koszul complex on the variables S = kk[x,y,z] mS = ideal vars S K = resolution mS K.dd betti K regularity mS regularity K -- Example 2. Three points in P^2 in general position restart S = kk[x,y,z] I = ideal "xy, yz, xz" F = resolution I F.dd betti F regularity I regularity F -- Example 3. 2x3 maximal minors restart S = kk[x_(0,0)..x_(1,2)] apply(2, i -> apply(3, j -> x_(i,j))) M = matrix oo I = minors(2, M) F = resolution I F.dd betti F regularity I regularity F -- Example 4. 3x3 minors of a 4x5 matrix restart S = kk[x_(0,0)..x_(3,4)] M = matrix apply(4, i -> apply(5, j -> x_(i,j))) I = minors(3, M) F = resolution I -- F.dd (Not a good idea!) betti F regularity I regularity F -- Example 5. Hilbert functions, Hilbert series and Hilbert polynomials restart S = kk[x,y,z] I = ideal "xy, yz, xz" hilbertSeries I -- numerator is the alternating sum obtained from the Betti table HS = reduceHilbert hilbertSeries I hilbertPolynomial I -- Example 6. Mapping cones restart S = kk[x..z] I = ideal "xy, yz, xz" F = resolution I G = resolution (S^{-1}/(I:x)) phi = extend(F,G, map(F_0, G_0, matrix{{x}})) C = cone phi minPres HH C -- Example 7. A coherent sheaf on P^3 restart S = kk[w..z] P3 = Proj S F = sheaf cokernel random(S^{4:1}, S^1) -- Do you know which sheaf this is? loadPackage "BGG" -- for cohomologyTable cohomologyTable (F, -5, 10) -- Example 8. on P^3. restart S = kk[w..z] P3 = Proj S -- C = twisted cubic in P3 IC = kernel map (kk[s,t], S, matrix{{s^3, s^2*t, s*t^2, t^3}}) OC = sheaf (S/IC) J = intersect (IC, power(ideal vars S, 3)) G = sheaf (S^1/J) HH^0(OC(>=-3)) HH^0(G(>=-3)) loadPackage "BGG" -- for cohomologyTable cohomologyTable (sheaf module OC, -5, 10) -- cohomologyTable applies to only CoherentSheaf (as O_P3-module) and not to SheafOfRings cohomologyTable (G, -5, 10) regularity (S^1/IC) regularity (S^1/J) -- Example 8. Complete intersection of two quadrics in P^3 restart S = kk[w..z] P3 = Proj S OC = sheaf coker random(S^1, S^{2:-2}) -- define OC directly as a CoherentSheaf (as O_P3 module) and not as SheafOfRings. HH^0(OC(>=-3)) loadPackage "BGG" -- for cohomologyTable cohomologyTable (OC, -5, 10)