The Geometry of the Compactification of the Hurwitz Scheme

f : C → P 1 of the projective line by smooth curves. Thus, from the very outset, one sees
that there are essentially two ways to approach the Hurwitz scheme:
(1) We start with P 1 and regard the objects of interest as coverings of P 1 ;
(2) We start with C and regard the objects of interest as morphisms from C to P 1 .
One finds that one can obtain the most information about the Hurwitz scheme and its
compactification by exploiting interchangeably these two points of view.
Our first main result is the following
Theorem: Let b,d, andg be integers such that b =2d +2g − 2, g ≥ 5 and d>2g +4.
Let H be the Hurwitz scheme over Z[ 1 b!
] parametrizing coverings of the projective line of
degree d with b points of ramification. Then Pic(H) is finite.
Remark: The number g in the statement of the Theorem is the genus of the “curve C
upstairs” of the coverings in question. Note, however, that the Hurwitz scheme H, and
hence also the genus g, are completely determined by b and d.
This Theorem is stated in §6.7, of the text. Note that although in the statement of the
Theorem here in the introduction, we spoke of “the” Hurwitz “scheme,” there are in fact
several different Hurwitz schemes used in the literature, some of which are, in fact, not
schemes, but stacks. For details about the particular type of Hurwitz scheme for which
the main theorem is proved, we refer the reader to the exact statement in §6.7, as well as
to §1 which explains the notation. Finally, we should address the issue of what happens if
d ≤ 2g + 4. Although our proof uses the somewhat leisurely lower bound of 2g +4, it may
be possible to prove the same result for smaller d using essentially the same techniques,
but being just a bit more careful. Since at the time this paper was written, the author
was not particularly interested in this issue, he has allowed himself the luxury of taking d
to be greater than 2g +4.
The main idea of the proof is that by combinatorially analyzing the boundary of
the compactification of the Hurwitz scheme, one realizes that there are essentially three
kinds of divisors in the boundary, which we call excess divisors (§4.7), which are “more
important” than the other divisors in the boundary in the sense that the other divisors
map to sets of codimension ≥ 2 under various natural morphisms. On the other hand, we
can also consider the moduli stack G (§6.1) of pairs consisting of a smooth curve of genus
g, together with a linear system of degree d and dimension 1. The subset of G consisting of
those pairs that arise from Hurwitz coverings is open in G, and its complement consists of
three divisors, which correspond precisely to the excess divisors. Using results of Harer on
the Picard group of M g , we show that these three divisors on G form a basis of Pic(G)⊗ Z Q,
and in fact, we even compute explicitly (§7) the matrix relating these three divisors on G
to a certain standard basis of Pic(G) ⊗ Z Q. The above Theorem then follows formally.
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