The geometry of the compactification of the Hurwitz scheme.

*(English)*Zbl 0866.14013The purpose of this paper is to study the geometry of the Harris-Mumford compactification of the Hurwitz scheme. The Hurwitz scheme parametrizes certain ramified coverings \(f:C \to\mathbb{P}^1\) 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 \(\mathbb{P}^1\) and regard the objects of interest as coverings of \(\mathbb{P}^1\).

(2) We start with \(C\) and regard the objects of interest as morphisms from \(C\) to \(\mathbb{P}^1\).

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 \(b,d\), and \(g\) be integers such that \(b=2d+ 2g-2\), \(g\geq 5\) and \(d>2g+4\). Let \({\mathcal H}\) be the Hurwitz scheme over \(\mathbb{Z} [{1\over b!}]\) parametrizing coverings of the projective line of degree \(d\) with \(b\) points of ramification. Then \(\text{Pic} ({\mathcal H})\) is finite.

The number \(g\) is the genus of the “curve \(C\) upstairs” of the coverings in question. Note, however, that the Hurwitz scheme \({\mathcal H}\), and hence also the genus \(g\), are completely determined by \(b\) and \(d\). – Note that although in the statement of the theorem 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. 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, which are “more important” than the other divisors in the boundary in the sense that the other divisors map to sets of codimension \(\geq 2\) under various natural morphisms. On the other hand, we can also consider the moduli stack \({\mathcal G}\) of pairs consisting of a smooth curve of genus \(g\), together with a linear system of degree \(d\) and dimension 1. The subset of \({\mathcal G}\) consisting of those pairs that arise from Hurwitz coverings is open in \({\mathcal G}\), and its complement consists of three divisors, which correspond precisely to the excess divisors. Using results of Harer on the Picard group of \({\mathcal M}_g\), we show that these three divisors on \({\mathcal G}\) form a basis of \(\text{Pic} ({\mathcal G}) \otimes_\mathbb{Z} \mathbb{Q}\), and in fact, we even compute explicitly the matrix relating these three divisors on \({\mathcal G}\) to a certain standard basis of \(\text{Pic} ({\mathcal G}) \otimes_\mathbb{Z} \mathbb{Q}\). The above theorem then follows formally.

Crucial to our study of the Hurwitz scheme is its compactification by means of admissible coverings and we prove a rather general theorem concerning the existence of a canonical logarithmic algebraic stack \(({\mathcal A}, M)\) parametrizing such coverings:

Fix non-negative integers \(g,r,q,s,d\) such that \(2g-2+r =d(2q-2+s) \geq 1\). Let \({\mathcal A}\) be the stack over \(\mathbb{Z}\) defined as follows: For a scheme \(S\), the objects of \({\mathcal A}(S)\) are admissible coverings \(\pi:C\to D\) of degree \(d\) from a symmetrically \(r\)-pointed stable curve \((f:C\to S\); \(\mu_f \subseteq C)\) of genus \(g\) to a symmetrically \(s\)-pointed stable curve \((h:D \to S\); \(\mu_h \subseteq D)\) of genus \(q\); and the morphisms of \({\mathcal A} (S)\) are pairs of \(S\)-isomorphisms \(\alpha: C\to C\) and \(\beta: D\to D\) that stabilize the divisors of marked points such that \(\pi\circ \alpha= \beta\circ \pi\). Then \({\mathcal A}\) is a separated algebraic stack of finite type over \(\mathbb{Z}\). Moreover, \({\mathcal A}\) is equipped with a canonical log structure \(M_{\mathcal A} \to {\mathcal O}_{\mathcal A}\), together with a logarithmic morphism \(({\mathcal A}, M_{\mathcal A}) \to \overline {{\mathcal M} {\mathcal S}}^{\log}_{q,s}\) (obtained by mapping \((C;D;\pi) \mapsto D)\) which is log étale (always) and proper over \(\mathbb{Z} [{1\over d!}]\).

(1) We start with \(\mathbb{P}^1\) and regard the objects of interest as coverings of \(\mathbb{P}^1\).

(2) We start with \(C\) and regard the objects of interest as morphisms from \(C\) to \(\mathbb{P}^1\).

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 \(b,d\), and \(g\) be integers such that \(b=2d+ 2g-2\), \(g\geq 5\) and \(d>2g+4\). Let \({\mathcal H}\) be the Hurwitz scheme over \(\mathbb{Z} [{1\over b!}]\) parametrizing coverings of the projective line of degree \(d\) with \(b\) points of ramification. Then \(\text{Pic} ({\mathcal H})\) is finite.

The number \(g\) is the genus of the “curve \(C\) upstairs” of the coverings in question. Note, however, that the Hurwitz scheme \({\mathcal H}\), and hence also the genus \(g\), are completely determined by \(b\) and \(d\). – Note that although in the statement of the theorem 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. 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, which are “more important” than the other divisors in the boundary in the sense that the other divisors map to sets of codimension \(\geq 2\) under various natural morphisms. On the other hand, we can also consider the moduli stack \({\mathcal G}\) of pairs consisting of a smooth curve of genus \(g\), together with a linear system of degree \(d\) and dimension 1. The subset of \({\mathcal G}\) consisting of those pairs that arise from Hurwitz coverings is open in \({\mathcal G}\), and its complement consists of three divisors, which correspond precisely to the excess divisors. Using results of Harer on the Picard group of \({\mathcal M}_g\), we show that these three divisors on \({\mathcal G}\) form a basis of \(\text{Pic} ({\mathcal G}) \otimes_\mathbb{Z} \mathbb{Q}\), and in fact, we even compute explicitly the matrix relating these three divisors on \({\mathcal G}\) to a certain standard basis of \(\text{Pic} ({\mathcal G}) \otimes_\mathbb{Z} \mathbb{Q}\). The above theorem then follows formally.

Crucial to our study of the Hurwitz scheme is its compactification by means of admissible coverings and we prove a rather general theorem concerning the existence of a canonical logarithmic algebraic stack \(({\mathcal A}, M)\) parametrizing such coverings:

Fix non-negative integers \(g,r,q,s,d\) such that \(2g-2+r =d(2q-2+s) \geq 1\). Let \({\mathcal A}\) be the stack over \(\mathbb{Z}\) defined as follows: For a scheme \(S\), the objects of \({\mathcal A}(S)\) are admissible coverings \(\pi:C\to D\) of degree \(d\) from a symmetrically \(r\)-pointed stable curve \((f:C\to S\); \(\mu_f \subseteq C)\) of genus \(g\) to a symmetrically \(s\)-pointed stable curve \((h:D \to S\); \(\mu_h \subseteq D)\) of genus \(q\); and the morphisms of \({\mathcal A} (S)\) are pairs of \(S\)-isomorphisms \(\alpha: C\to C\) and \(\beta: D\to D\) that stabilize the divisors of marked points such that \(\pi\circ \alpha= \beta\circ \pi\). Then \({\mathcal A}\) is a separated algebraic stack of finite type over \(\mathbb{Z}\). Moreover, \({\mathcal A}\) is equipped with a canonical log structure \(M_{\mathcal A} \to {\mathcal O}_{\mathcal A}\), together with a logarithmic morphism \(({\mathcal A}, M_{\mathcal A}) \to \overline {{\mathcal M} {\mathcal S}}^{\log}_{q,s}\) (obtained by mapping \((C;D;\pi) \mapsto D)\) which is log étale (always) and proper over \(\mathbb{Z} [{1\over d!}]\).

##### MSC:

14H10 | Families, moduli of curves (algebraic) |

14C22 | Picard groups |

14E20 | Coverings in algebraic geometry |

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\textit{S. Mochizuki}, Publ. Res. Inst. Math. Sci. 31, No. 3, 355--441 (1995; Zbl 0866.14013)

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