The Horrocks-Mumford bundle on projective 4-space can be constructed with the following code. We first produce a base point whose intersection ring contains a variable named n, in terms of which we can compute the Hilbert polynomial.
i1 : pt = base(n) o1 = pt o1 : an abstract variety of dimension 0 |
Then we create the projective space of dimension 4 over the base point.
i2 : X = abstractProjectiveSpace'_4 pt o2 = X o2 : a flag bundle with subquotient ranks {4, 1} |
Note that we use abstractProjectiveSpace' to get Grothendieck-style notation. This has the advantage that the first Chern class of the tautological line bundle is assigned to the variable h:
i3 : chern_1 OO_X(1) o3 = h QQ[n][H ..H , h] 1,1 1,4 o3 : -------------------------------------------------------------------- (- H - h, - H - H h, - H - H h, - H - H h, -H h) 1,1 1,2 1,1 1,3 1,2 1,4 1,3 1,4 |
Now we create an abstract sheaf of rank 2 with $1 + 5 h + 10 h^2$ as its total Chern class:
i4 : F = abstractSheaf(X, Rank => 2, ChernClass => 1 + 5*h + 10*h^2) o4 = F o4 : an abstract sheaf of rank 2 on X |
Alternatively, we can use the representation of the Horrocks-Mumford bundle as the cohomology of the monad $$0 \rightarrow{} O_X(-1)^5 \rightarrow{} \Omega_X^2(2)^2 \rightarrow{} O_X^5 \rightarrow{} 0$$ to produce a construction:
i5 : F' = 2 * (exteriorPower_2 cotangentBundle X)(2) - 5 * OO_X(-1) - 5 * OO_X o5 = F' o5 : an abstract sheaf of rank 2 on X |
i6 : chern F' 2 o6 = 1 - h + 4h QQ[n][H ..H , h] 1,1 1,4 o6 : -------------------------------------------------------------------- (- H - h, - H - H h, - H - H h, - H - H h, -H h) 1,1 1,2 1,1 1,3 1,2 1,4 1,3 1,4 |
i7 : rank F' o7 = 2 |
Here is the relationship between the two bundles:
i8 : F === dual F'(-2) o8 = true |
Now we compute the Hilbert polynomial of $F$. This computation makes use of the Riemann-Roch Theorem.
i9 : chi F(n*h) 1 4 5 3 125 2 125 o9 = --n + -n + ---n + ---n + 2 12 3 12 6 o9 : QQ[n] |