Construction de la Variete de Modules des Fibres Vectoriels Stables sur une Courbe Algebrique Lisse

Oesterlé, Joseph

doi:10.1007/978-1-4684-7603-3_2

Cited by 5 publications

(3 citation statements)

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“…So it is semistable of the same slope (see e.g. [45] If d = 3 let E be a locally free sheaf of rank 2, which is stable in the case g ≥ 2, and regular polystable in the case g = 1. Let R 3 = S 3 E ⊗ (det E) −1 .…”

Section: Construction Of Families Of Coverings With Rational Parametementioning

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: Construction Of Families Of Coverings With Rational Parametementioning

confidence: 99%

“…Therefore R d is globally generated (cf. [45] p.39). Using the Bertini type theorems of [14] and [15] (cf.…”

Section: Construction Of Families Of Coverings With Rational Parametementioning

confidence: 99%

Unirationality of Hurwitz Spaces of Coverings of Degree ≤5

Kanev

2012

International Mathematics Research Notices

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“…Therefore R d is globally generated (cf. [45] p.39). Using the Bertini type theorems of [14] and [15] (cf.…”

Section: Construction Of Families Of Coverings With Rational Paramete...mentioning

confidence: 99%

Unirationality of Hurwitz spaces of coverings of degree <= 5

Kanev

2011

Preprint

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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 H 0 d,A (Y ) is unirational. If in addition (e, 2) = 1 (when d = 3), (e, 6) = 1 (when d = 4) and (e, 10) = 1 (when d = 5), then H 0 d,A (Y ) is rational.

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Hurwitz spaces of quadruple coverings of elliptic curves and the modulispace of abelian threefolds 𝒜₃(1, 1, 4)

Kanev

2004

Mathematische Nachrichten

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Abstract. We prove that the moduli space A3(1, 1, 4) of polarized abelian threefolds with polarization of type (1, 1, 4) is unirational. By a result of Birkenhake and Lange this implies the unirationality of the isomorphic moduli space A3(1, 4, 4). The result is based on the study the Hurwitz space H4,n(Y ) of quadruple coverings of an elliptic curve Y simply branched in n ≥ 2 points. We prove the unirationality of its codimension one subvariety H 0 4,A (Y ) which parametrizes quadruple coverings π : X → Y with Tschirnhausen modules isomorphic to A −1 , where A ∈ P ic n/2 Y , and for which π * : J(Y ) → J(X) is injective. This is an analog of the result of Arbarello and Cornalba that the Hurwitz space H4,n(P 1 ) is unirational.

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Construction de la Variete de Modules des Fibres Vectoriels Stables sur une Courbe Algebrique Lisse

Cited by 5 publications

References 2 publications

Unirationality of Hurwitz Spaces of Coverings of Degree ≤5

Unirationality of Hurwitz Spaces of Coverings of Degree ≤5

Unirationality of Hurwitz spaces of coverings of degree <= 5

Hurwitz spaces of quadruple coverings of elliptic curves and the modulispace of abelian threefolds 𝒜₃(1, 1, 4)

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Construction de la Variete de Modules des Fibres Vectoriels Stables sur une Courbe Algebrique Lisse

Cited by 5 publications

References 2 publications

Unirationality of Hurwitz Spaces of Coverings of Degree ≤5

Unirationality of Hurwitz Spaces of Coverings of Degree ≤5

Unirationality of Hurwitz spaces of coverings of degree <= 5

Hurwitz spaces of quadruple coverings of elliptic curves and the modulispace of abelian threefolds 𝒜3(1, 1, 4)

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Hurwitz spaces of quadruple coverings of elliptic curves and the modulispace of abelian threefolds 𝒜₃(1, 1, 4)