The Geometry of the Compactification of the Hurwitz Scheme

I V1 ⊆ I ⊆ P × G
⏐ ⏐
↓ ↓ ↙ ↘
V 1 ⊆ P G
Now let us consider the following diagram:
Δ 12 , Δ 13 ⊆ C × C × C
↙ π 12
⏐ ⏐↓
π 23 π 13 ↘
(Diagram 6.2)
C × C C × C C × C
where the π ij ’s are the respective projections and Δ ij = π −1
ij (Δ). Let us suppose that
i, j are such that d − i − j>2g − 2. Let F (i,j) =(π1 ∗L)(−iΔ 12 − jΔ 13 ). Let K (i,j) =
π 23,∗ F (i,j) .ThenK (i,j) is a vector bundle on C × C of rank d +1− g − i − j. Note that
we have a natural locally split injection K (2,2) →K (0,0) which gives rise to a morphism
P(K (2,2),∧ ) → P(K (0,0),∧ )=C × C × P . Composing with the projection P ,wegeta
morphism P(K (2,2),∧ ) → P which is clearly generically injective after taking the quotient
by the action of Z/2Z permuting the two factors of C, and whose image consists exactly of
the bad divisors that appear in (3). The image is an irreducible variety V 2 of codimension
2inP . Forming I V2 and σ 2 : I V2 → G as before, we see that σ 2 is generically injective,
and that its image in G is some irreducible divisor D 2 whose points are exactly the bad
lines that appear in condition (3). We shall compute the degree of D 2 in §6B.
In particular, since G ≠ D 1
⋃
D2
⋃
D3 , we see that we can always construct a map
α : C → P 1 with at most simple ramification. Thus ψ ◦κ is surjective. Note that instead of
working over k, then, we could have worked over J d . In summary, we have the following:
Proposition : The complement of κ(H) in G consists of three relative divisors D 1 ,
D 2 ,andD 3 over J d , which are also geometrically irreducible over J d . They correspond,
respectively, to the linear pencils that fail to meet one of the conditions (2),(3), or(1)
listed at the beginning of this §.
§6.5. We now wish to relate these divisors of G to the divisors of H constructed in §4.
To do this, we shall need to construct some sort of map from H (or at least part of it) into
G. This is a somewhat delicate task, which we now undertake.
We shall call divisors at infinity of H ∼,+ or H + that lie over divisors of species X in
H ∼ or H “divisors of species X”, as well. We propose to extend the map μ ′ : H + →H κ →G
to a map μ : H ∼,+ →G. For any two distinct integers i and j between 1 and b, letus
denote by U ij the open substack of H ∼,+ parametrizing admissible covers for which the
marking sections numbered i and j of the curve downstairs lie in the irreducible component
of the curve downstairs which is the image of the curve of genus g upstairs. Then it is
easy to see that, as i, j ranges over all admissible values, the union of the U ij is all of
54