Macaulay2, version 1.18
with packages: ConwayPolynomials, Elimination, IntegralClosure, InverseSystems, LLLBases, MinimalPrimes, PrimaryDecomposition, ReesAlgebra, Saturation, TangentCone

i1 : needsPackage "ToricHigherDirectImages"
-- F_1 to P^2
--warning: symbol "Base" in Schubert2.Dictionary is shadowed by a symbol in Complexes.Dictionary
--  use the synonym Schubert2$Base

o1 = ToricHigherDirectImages

o1 : Package

i2 :      FF1 = hirzebruchSurface 1

o2 = FF1

o2 : NormalToricVariety

i3 : matrix rays FF1

o3 = | 1  0  |
     | 0  1  |
     | -1 1  |
     | 0  -1 |

              4        2
o3 : Matrix ZZ  <--- ZZ

i4 : PP2 = toricProjectiveSpace 2

o4 = PP2

o4 : NormalToricVariety

i5 : matrix rays PP2

o5 = | -1 -1 |
     | 1  0  |
     | 0  1  |

              3        2
o5 : Matrix ZZ  <--- ZZ

i6 : Pi = map(PP2,FF1,matrix{{0,-1},{1,0}})

o6 = | 0 -1 |
     | 1 0  |

o6 : ToricMap PP2 <--- FF1

i7 : isWellDefined Pi

o7 = true

i8 : -- D = -5E, where E is the exceptional divisor of FF1.
     D = {0,-5,0,0};

i9 : i = 0;

i10 : netList prepend({"tau","R^0 pi_* OO_X(D)"},
          for tau in max PP2 list {tau, affinePreimageCohomology(Pi,i,tau,D)})

      +------+--------------------------------------------+
o10 = |tau   |R^0 pi_* OO_X(D)                            |
      +------+--------------------------------------------+
      |{0, 1}|0                                           |
      +------+--------------------------------------------+
      |{0, 2}|cokernel {3, 2} | 0    0    -a_1 a_0  0    ||
      |      |         {0, 5} | a_0  0    0    0    0    ||
      |      |         {1, 4} | -a_1 a_0  0    0    0    ||
      |      |         {2, 3} | 0    -a_1 a_0  0    0    ||
      |      |         {4, 1} | 0    0    0    -a_1 a_0  ||
      |      |         {5, 0} | 0    0    0    0    -a_1 ||
      +------+--------------------------------------------+
      |{1, 2}|0                                           |
      +------+--------------------------------------------+

i11 : D = {0,5,0,0};

i12 : i = 1;

i13 : netList prepend({"tau","R^1 pi_* OO_X(D)"},
          for tau in max PP2 list {tau, affinePreimageCohomology(Pi,i,tau,D)})

      +------+------------------------------------------+
o13 = |tau   |R^1 pi_* OO_X(D)                          |
      +------+------------------------------------------+
      |{0, 1}|0                                         |
      +------+------------------------------------------+
      |{0, 2}|cokernel {3, 0} | a_0 0   0    0    -a_1 ||
      |      |         {0, 3} | 0   a_1 a_0  0    0    ||
      |      |         {1, 2} | 0   0   -a_1 a_0  0    ||
      |      |         {2, 1} | 0   0   0    -a_1 a_0  ||
      +------+------------------------------------------+
      |{1, 2}|0                                         |
      +------+------------------------------------------+

i14 : -- F1 to P1. We can recover the known formulas for higher direct images.
      PP1 = toricProjectiveSpace 1;

i15 : Pi' = map(PP1,FF1,matrix{{1,0}});

o15 : ToricMap PP1 <--- FF1

i16 : D = {0,4,0,0};

i17 : i = 0;

i18 : tau = (max PP1)_0;

i19 : affinePreimageCohomology(Pi',i,tau,D)

              5
o19 = (QQ[a ])
           0

o19 : QQ[a ]-module, free, degrees {0..4}
          0

i20 : D = {0,-4,0,0};

i21 : i = 1;

i22 : affinePreimageCohomology(Pi',i,tau,D)

              3
o22 = (QQ[a ])
           0

o22 : QQ[a ]-module, free, degrees {0..2}
          0

i23 : -- Blowup of F_1 x P^1 along a torus fixed curve, projecting to F_1.
      X = normalToricVariety({{1,0,0},{0,1,0},{-1,1,0},{0,-1,0},{0,0,1},{0,0,-1},{0,1,1}}, {{2,3,4},{2,3,5},{0,3,4},{0,3,5},{0,1,5},{1,2,5},{0,1,6},{0,4,6},{1,2,6},{2,4,6}});

i24 : Pi'' = map(FF1, X, matrix{{1,0,0},{0,1,0}});

i25 : tau = (max FF1)_0;

i26 : i = 1;

i27 : -- D = -2E' - 2H, where E' is the exceptional divisor of X
      -- and H is the pullback of a hyperplane in P^1.
      D = {0,0,0,0,0,-2,-2};

i28 : affinePreimageCohomology(Pi'',i,tau,D)
-- X projecting to P^1.

o28 = cokernel | 0   a_1^2 |
               | 0   0     |
               | a_1 0     |

                                                 3
o28 : QQ[a ..a ]-module, quotient of (QQ[a ..a ])
          0   1                           0   1

i29 :       phi = map(PP1, X, matrix{{1,0,0}});

o29 : ToricMap PP1 <--- X

i30 : tau = (max PP1)_0;

i31 : i = 2;

i32 : D = {-1,-2,-3,-4,-3,-2,-1};

i33 : affinePreimageCohomology(phi,i,tau,D)

              14
o33 = (QQ[a ])
           0

o33 : QQ[a ]-module, free, degrees {2:3..4, 4, 3, 2, 1, 0, 2, 1, 3..4, 2}
          0