Minimalist Instructions

Refer to arXiv.org:0905.2229 or here   for mathematical details.

The program does two kinds of calculations: in the tautological module, on a fixed Hilbert scheme X^[m]_B, as in § 2 of the paper; and calculations of Chern numbers of tautological bundles, as in § 3 of the paper.

At present, the program is rather slow in degrees 5 and higher. A faster version is under construction.

Throughout, we will fix a good pencil X/B and a line bundle L, and use the notations: ω = ω_X/B, g = fibre genus, d = fibre degree of L, δ = number of singular fibres, assumed 1-nodal.

Tautological module calculations

Data is as follows:

• A monoblock diagonal Γ_(n)[ω^a L^b], n ≥ 1, in X^[m]_B is represented by a 1-column matrix written as M(n,a,b). A general polyblock diagonal, which can be written as Γ_(n) [ω^aL^b] ★Γ_(n') [ω^a'L^b'] ★... is represented as M(n,a,b;n',a',b';...). Trailing zeros in a column may be omitted.

An analogous class on a single curve of genus x may be specified as M(dim_b=0;genus=x;a,b,c;...). x can be a symbol or a number. If x is the letter g, it may be omitted. The weight of M is by definition dim_b plus the sum of the first row. dim_b is by default equal to 1, unless specified to be 0 as above.

• A node scroll F ^m,n _j;n',n''... [ω^b L^c,...] is represented as F^n_j(n',b,c;...). The matrix may be empty, written as F^n_j(), which corresponds to a node scroll over a point. It always has dim_b=0 and its weight is the weight of the matrix plus n. By default, the genus of this object is taken as g-1 but may be reset by inserting genus=x; at the head of the matrix. For those node scrolls that arise in the calculation with general fibre genus x, the program itself will set the genus as x-1.

• A node section Γ^(m) F^m,n _j;n',n''... [ω^bL^c,...] is represented as GF^n_j(n',b,c;...). Its weight is again the weight of the matrix plus n.

The operation of Γ^(m) on these classes is represented by G. It is unnecessary to specify m because it is determined as the weight of the argument. A power of G, input as G^k, may be applied to any of the above classes, resulting in a linear combination of such classes. The degree of zero-dimensional class can be obtained in terms of ω², δ etc. using the operator trivialize.

Chern number calculations

These calculations take place in W^m(X/B). To compute c_i (Λ_m(L))^k/m!, one enters cc(c(m,i)^k). When the latter is zero-dimensional, its degree, i.e.

∫_W^m(X/B) c_i (Λ_m(L)) ^k/m! = ∫_X^[m]B c_i (Λ_m(L))^k

is evaluated by cc(c(m,i)^k). The Thom-Porteous polynomial Δ ₁^(m)(c(m,•)) is evaluated by cc(Delta^m_1).

Examples

(g+d)^10

trivialize(G-L)^4M(1;1;1)

cc(c(3,1)^4)

cc(G_2^2G_3L_3)

cc(c(3,1)^4-3c(3,1)^2c(3,2)+2c(3,1)c(3,3)+c(3,2)²)

cc(Delta^4_1)

sum_(i=0)^1(TⁱGTFⁿ_(j+1)(dim_b=x;genus=y²;1,i))