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Effective Divisors on Hurwitz Spaces 3

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Effective Divisors on Hurwitz Spaces 3 EFFECTIVE DIVISORS ON HURWITZ SPACES GAVRIL FARKAS To Bill Fulton on the occasion of his 80th birthday. ABSTRACT. We prove the effectiveness of the canonical bundle of a partial compactifi- 1 cation of several Hurwitz spaces Hg,k of degree k admissible covers of P from curves of genus 14 ≤ g ≤ 19. This suggests that these moduli spaces are of general type. Following a principle due to Mumford, most moduli spaces that appear in alge- braic geometry (classifying curves, abelian varieties, K3 surfaces) are of general type, with a finite number of exceptions, which are unirational, or at least uniruled. Under- standing the transition from negative Kodaira dimension to being of general type is usually quite difficult. The aim of this paper is to investigate to some extent this princi- ple in the case of a prominent parameter space of curves, namely the Hurwitz space Hg,k classifying degree k covers C → P1 with only simple ramification from a smooth curve C of genus g. Hurwitz spaces provide an interesting bridge between the much more accessible moduli spaces of pointed rational curves and the moduli space of curves. The study of covers of the projective line emerges directly from Riemann’s Existence Theorem and Hurwitz spaces viewed as classifying spaces for covers of curves have been classically used by Clebsch [Cl] and Hurwitz [Hu] to establish the irreducibility of the moduli space of curves. Fulton was the first to pick up the problem in the modern age and his influential paper [Ful] initiated the treatment of Hurwitz spaces in arbi- trary characteristic. Harris and Mumford [HM] compactified Hurwitz spaces in order to show that Mg is of general type for large g, see also [H]. Their work led the way to the theory of admissible covers which has many current ramifications in Gromov- Witten theory. Later, Harris and Morrison [HMo] bound from below the slope of all effective divisors on Mg using the geometry of compactified Hurwitz spaces. We denote by Hg,k the moduli space of admissible covers constructed by Har- ris and Mumford [HM] and studied further in [ACV], or in the book [BR]. It comes arXiv:1804.01898v4 [math.AG] 21 Aug 2020 equipped with a finite branch map b: Hg,k → M0,2g+2k−2/S2g+2k−2, where the target is the moduli space of unordered (2g + 2k − 2)-pointed stable rational curves, as well as with a map σ : Hg,k → Mg, obtained by assigning to an admissible cover the stable model of its source. The ef- fective cone of divisors of the quotient M0,2g+2k−2/S2g+2k−2 is spanned by the divi- sor classes B2,..., Bg+k−1, where Bi denotes the closure of the locus of stable rational curves consisting of two components such that one of them contains precisely i of the 2g + 2k − 2 emarkede points. e 1 2 G. FARKAS It is a fundamental question to describe the birational nature of Hg,k and one knows much less than in the case of other moduli spaces like Mg or Ag. Recall that in a series of landmark papers [HM], [H], [EH] published in the 1980s, Harris, Mumford and Eisenbud proved that Mg is a variety of general type for g > 23. This contrasts with the classical result of Severi [Sev] that Mg is unirational for g ≤ 10. Later it has been shown that the Kodaira dimension of Mg is negative for g ≤ 15. Precisely, Mg is unirational for g ≤ 14, see [CR1], [Ser], [Ve], whereas M15 is rationally connected [BV]. It has been recently established in [FJP] that both M22 and M23 are of general type. g+2 From Brill-Noether theory it follows that when 2 ≤ k ≤ g, every curve of 1 genus g can be represented as a k-sheeted cover of P , that is, the map σ : Hg,k → Mg is dominant. Using [HT] it is easy to see that the generic fibre of the map σ is of gen- eral type, thus in this range Hg,k is of general type whenever Mg is. When k ≥ g + 1, the Hurwitz space is obviously uniruled being a Grassmannian bundle. Classically it has been known that Hg,k is unirational for k ≤ 5, see [AC] and references therein for a modern treatment (or [Pe] and [Sch1] for an alternative treatment using the Buchsbaum- Eisenbud structure theorem of Gorenstein rings of codimension 3). Geiss [G] using liai- son techniques showed that most Hurwitz spaces Hg,6 with g ≤ 45 are unirational. Schreyer and Tanturri [ST] put forward the hypothesis that there exist only finitely many pairs (g, k), with k ≥ 6, such that Hg,k is not unirational. At the other end, when k = g, the Hurwitz space Hg,g is birationally isomorphic to the universal symmetric product of degree g − 2 over Mg and it has been showed in [FV2] that Hg,g is of general type for g ≥ 12 and uniruled in all the other cases. Further results on the unirationality of some particular Hurwitz spaces are obtained in [KT] and [DS]. g+2 In this paper, we are particularly interested in the range 2 ≤ k ≤ g − 1 (when Hg,k dominates Mg) and 14 ≤ g ≤ 19, so that Mg is either unirational/uniruled in the cases g = 14, 15, or its Kodaira dimension is unknown, when g = 16, 17, 18, 19. We summarize our results showing the positivity of the canonical bundle of Hg,k and we begin with the cases when Mg is known to be (or thought to be) uniruled. Theorem 0.1. Suppose H is one of the spaces H14,9 or H16,9. Then there exists an effective b∗ Q-divisor class E on H supported on the union i≥3 (Bi), such that the twisted canonical class KH + E is effective. P e A few comments are in order. We expect that the boundary divisor E is empty. Carrying this out requires an extension of the determinantal structure defining the Koszul divisors considered in this paper to the entire space Hg,k. This technically cum- bersome step will be carried out in future work. Secondly, Theorem 0.1 is optimal. In genus 14, Verra [Ve] established the unirationality of the Hurwitz space H14,8, which is a finite cover of M14, and concluded in this way that M14 itself is unirational. In genus 16, the map σ : H16,9 → M16 is generically finite. Chang and Ran [CR3] claimed that M16 is uniruled (more precisely, that KM16 is not pseudo-effective, which by now, one knows that it implies uniruled- ness). However, it has become recently clear [Ts] that their proof contains a fatal error, so the Kodaira dimension of M16 is currently unknown. EFFECTIVE DIVISORS ON HURWITZ SPACES 3 Theorem 0.2. Suppose H is one of the spaces H17,11 or H19,13. Then there exists an effective b∗ Q-divisor class E on H supported on the union i≥3 (Bi), such that the twisted canonical class KH + E is big. P e We observe that the maps H17,11 → M17 and H19,13 → M19 have generically 3 and respectively 5-dimensional fibres. As in the case of Theorem 0.1, here too we expect the divisor E to be empty. 1 BothTheorems 0.1 and 0.2 are proven at the level of a partial compactification Gg,k (described in detail in Section 2) and which incorporates only admissible covers with a source curve whose stable model is irreducible. In each relevant genus we producee an 1 explicit effective divisor Dg on G such that the canonical class K e1 can be expressed g,k Gg,k as a positive combination of a multiple of [D ] and the pull-back under the map σ of e e g an effective divisor on Mg. Theorems 0.1 and 0.2 offer a very strong indication that the moduli spaces H14,9, H16,9, H17,11 and He19,13 are all of general type. To complete such a proof one would need to show that the singularities of Hg,k impose no adjunction conditions, that is, every pluricanonical form on Hg,k lifts to a pluricanonical form of a resolution of singularities of Hg,k. Establishing such a result, which holds at the level of Mg, see [HM], or at the level of the moduli space of ℓ-torsion points Rg,ℓ for small levels [CF], remains a significant challenge. We describe the construction of these divisors in two instances and refer to Section 2 for the remaining cases and further details. First we consider the space H16,9 which is a generically finite cover of M16. ∨ 7 Let us choose a general pair [C, A] ∈ H16,9. We set L := KC ⊗ A ∈ W21(C). k 0 0 ⊗k Denoting by IC,L(k) := Ker Sym H (C,L) → H (C,L ) , by Riemann-Roch one computes that dim I (2) = 9 and dim I (3) = 72. Therefore the locus where there C,L C,L exists a non-trivial syzygy, that is, the map 0 µC,L : IC,L(2) ⊗ H (C,L) → IC,L(3) is not an isomorphism is expected to be a divisor on H16,9. This indeed is the case and we define the syzygy divisor 0 ∨ ≇ ∨ ∨ D16 := [C, A] ∈H16,9 : IC,KC ⊗A (2) ⊗ H (C, KC ⊗ A ) → IC,KC ⊗A (3) . n o The ramification divisor of the map σ : H16,9 → M16, viewed as the Gieseker-Petri divisor 0 0 ∨ ≇ 0 GP := [C, A] ∈H16,9 : H (C, A) ⊗ H (C, KC ⊗ A ) → H (C, KC ) n o also plays an important role.
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