The Geometry of the Compactification of the Hurwitz Scheme

V 3 ⊆ P consisting of divisors with three points of multiplicity two has codimension three
in P (since we assume that d>2g + 4). Thus, the subvariety of G consisting of lines in P
that pass through V 3 has codimension two, as desired.
For i = 3, here we need to show that the generic pencil does not contain two base
points. But one sees immediately that the subvariety of G corresponding to pencils with
two base points has dimension 2(d − 3) + 2 = 2d − 4, i.e., has codimension two in G, as
desired.
We now turn to considering the nonexcess divisors. Let us suppose that x : Spec(k) →
H ∼,+ is a geometric point that lands in a nonexcess divisor. One piece of information of
an admissible covering that is clearly preserved by μ is the unique nonconstant morphism,
which we shall henceforth refer as the central morphism of the admissible covering, from
the curve of genus g upstairs to a copy of P 1 downstairs. Thus, let us suppose that μ(x)
lands inside the image of κ. Then the central morphism of x must be ramified over b
distinct points downstairs; but since the curve downstairs in the admissible covering x
is stable, this could only happen if the curve downstairs in x is, in fact, smooth, which
contradicts the assumption that x lies in a nonexcess divisor. Thus, μ(x) must land inside
one of the D i . Now let us suppose that μ(x) lands in a sufficiently generic point of D i .
Then by our proof of the final statement of the Proposition, it follows that the central
morphism of x is ramified over exactly b − 1 distinct points downstairs; thus, since the
curve downstairs in the admissible covering x is stable, it follows by the same reasoning as
before, that the curve downstairs in x must be of type (b − 2, 2). On the other hand, we
gave explicit descriptions in §4 of all the divisors of type (b − 2, 2); going down the list,
one sees that the only possiblity now for the nonexcess divisor in which x sits is Species
3B with g 1 = g, g 2 =0. Butifd 2 > 1, then the central morphism of x must have at least
two base points, which contradicts what we just saw in the previous paragraph. ○
§6.6. At this point, we need to employ a result which will not be proven until the end
of §7. We state it here for ease of reference:
Theorem : Under the assumption that d>2g +4, the line bundles defined by D 1 , D 2 ,
and D 3 are linearly independent in the Picard group of G.
§6.7. Now by elementary algebraic geometry, we have an exact sequence:
ZD 1 ⊕ ZD 2 ⊕ ZD 3 −→ Pic G −→ Pic H −→ 0
If we tensor with Q, and use the fact (§5) that Pic Q G = Q 3 , then we obtain the following
(again under the assumption that d>2g +4)
Corollary : D 1 , D 2 ,andD 3 form a basis of Pic Q G.
Corollary : Pic (H) is a finite group. The homomorphism Σ of §4.8, is a surjection
when tensored with Q.
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