Hurwitz spaces of triple coverings of elliptic curves and moduli spaces of Abelian threefolds

Kanev, Vassil

doi:10.1007/s10231-003-0098-9

Cited by 13 publications

(23 citation statements)

References 28 publications

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“…The same proof applies also to the case g = 1, using the fact that regular polystability is a Zariski open condition in families of vector bundles over elliptic curves (see e.g. [28] Appendix B). We notice that by construction the variety S A = δ −1 (A) is isomorphic to H 1 (Y, Hom(A, I r−1 )).…”

Section: Lemma 12 Let a ∈ Hmentioning

confidence: 98%

“…For each η ∈ N the covering X η → {η} × Y is ramified and the degree of the discriminant divisor is n = 2e (see e.g. [28] Let the assumptions and notation be as in § 3.8. If η ∈ N , let X η → Y be the associated covering.…”

Section: 8mentioning

confidence: 99%

“…[54]). The author studied in [28] and [29] the Hurwitz spaces H is a rational variety. The proof of these results is based on the classification of the Gorenstein coverings of degree ≤ 5, by means of vector bundles over the base, due to Miranda, Casnati and Ekedahl [38], [14], [15].…”

Section: Introductionmentioning

confidence: 99%

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Unirationality of Hurwitz Spaces of Coverings of Degree ≤5

Kanev

2012

International Mathematics Research Notices

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Abstract. Let Y be a smooth, projective curve of genus g ≥ 1 over the complex numbers. Let H 0 d,A (Y ) be the Hurwitz space which parameterizes equivalence classes of coverings π : X → Y of degree d simply branched in n = 2e points, such that the monodromy group is S d and det(π * O X /O Y ) ∨ is isomorphic to a fixed line bundle A of degree e. We prove that, when d = 3, 4 or 5 and n is sufficiently large (precise bounds are given), the Hurwitz space In studying Hurwitz spaces when g(Y ) ≥ 1, it is natural to restrict ourselves to simple coverings of degree d whose monodromy group is S d . We denote the corresponding Hurwitz space by H 0 d,n (Y ). This is the principal case, since those with smaller monodromy group are reduced to coverings of smaller degree via ań etale covering of Y . There is a canonical morphismThe problem of unirationality may be posed for the fibers of this morphism. Given A ∈ P ic n 2 Y we denote the fiber overWe study the problem of irreducibility and unirationality/rationality of H is a rational variety. The proof of these results is based on the classification of the Gorenstein coverings of degree ≤ 5, by means of vector bundles over the base, due to Miranda, Casnati and Ekedahl [38], [14], [15]. The unirationality/rationality of the Hurwitz spaces is eventually a consequence of known results on unirationality/rationality of moduli spaces of vector bundles over curves with fixed determinants. Here is an outline of the content of the paper by sections.Section 1 contains some preliminaries on vector bundles. Section 2 is devoted to the explicit description of the Gorenstein coverings of degree d = 3, 4 or 5. We recall and complement some results from the papers [38], [14] and [15]. In the proofs of Theorem 0.1 and Theorem 0.2 we use certain statements about uniqueness of the representation of a Gorenstein covering by means of the vector bundle data on the base. When d = 5 this is related with a problem discussed in [13] p.457 Remark 2. Namely, given the Buchsbaum-Eisenbud resolution of a Gorenstein ideal of grade 3, to what extent is the skew-symmetric matrix of odd degree uniquely determined by the Pfaffian ideal? We focus mainly on the more difficult case d = 5 in this section and prove in Lemma 2.15, Lemma 2.16 and Proposition 2.19 the uniqueness statements we need. For reader's convenience we give in Proposition 2.5 a proof of the Buchsbaum-Eisenbud resolution as we use it.In Section 3 we construct families of coverings with rational parameter varieties, which eventually will dominate the Hurwitz spaces. Here the main result is Lemma 3.10.In Section 4 the proofs of Theorem 0.1 and Theorem 0.2 are given.In Appendix A we give a proof of a result due to Dolgachev and Libgober that we use. The reason of including this appendix is a comment in [49] p.337 regarding its validity. We give a detailed proof along the sketch in [17] p.9. UNIRATIONALITY OF HURWITZ SPACES 3Notation. We assume the base field k = C, unless otherwise specified. A scheme is always supposed to be separated of finite t...

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Section: Lemma 12 Let a ∈ Hmentioning

confidence: 98%

Section: 8mentioning

confidence: 99%

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Unirationality of Hurwitz Spaces of Coverings of Degree ≤5

Kanev

2012

International Mathematics Research Notices

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“…If one proves the irreducibility and the unirationality of some Hurwitz spaces, and the dominance of the Prym maps, this would imply the unirationality of the corresponding Siegel modular varieties. This idea was successfully realized in proving the unirationality of A 3 (1, 1, d) and A 3 (1, d, d) for d 4 [20,21] (the case d = 5 is a work in progress). We hope the method may be extended considering coverings with monodromy groups an arbitrary irreducible Weyl group.…”

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confidence: 92%

Hurwitz spaces of Galois coverings of P1, whose Galois groups are Weyl groups

Kanev

2006

Journal of Algebra

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“…Nowadays, the term Hurwitz space refers to a variety which parametrizes, up to equivalence, coverings π : F → E of curves with some geometric restrictions. In this article we will use a local version of Hurwitz spaces, namely a local family of coverings, whose seminal idea can be found in [Kan04]. Roughly, given a fixed covering π : F → E where E is an elliptic curve, one is able to construct a map p : F → E of curves over H, where H is a contractible open set.…”

Section: Introductionmentioning

confidence: 99%

Covering of elliptic curves and the kernel of the Prym map

Favale,

Torelli

2017

Preprint

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Motivated by a conjecture of Xiao, we study families of coverings of elliptic curves and their corresponding Prym map Φ. More precisely, we describe the codifferential of the period map P associated to Φ in terms of the residue of meromorphic 1-forms and then we use it to give a characterization for the coverings for which the dimension of Ker(dP ) is the least possibile. This is useful in order to exclude the existence of non isotrivial fibrations with maximal relative irregularity and thus also in order to give counterexamples to the Xiao's conjecture mentioned above. The first counterexample to the original conjecture, due to Pirola, is then analysed in our framework.

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Hurwitz spaces of triple coverings of elliptic curves and moduli spaces of Abelian threefolds

Cited by 13 publications

References 28 publications

Unirationality of Hurwitz Spaces of Coverings of Degree ≤5

Unirationality of Hurwitz Spaces of Coverings of Degree ≤5

Hurwitz spaces of Galois coverings of P1, whose Galois groups are Weyl groups

Covering of elliptic curves and the kernel of the Prym map

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