127 research outputs found

    Enumerative geometry of dormant opers

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    The purpose of the present paper is to develop the enumerative geometry of dormant GG-opers for a semisimple algebraic group GG. In the present paper, we construct a compact moduli stack admitting a perfect obstruction theory by introducing the notion of a dormant faithful twisted GG-oper (or a "GG-do'per" for short. Moreover, a semisimple 22d TQFT (= 22-dimensional topological quantum field theory) counting the number of GG-do'pers is obtained by means of the resulting virtual fundamental class. This 22d TQFT gives an analogue of the Witten-Kontsevich theorem describing the intersection numbers of psi classes on the moduli stack of GG-do'pers.Comment: 64 pages, the title is changed, some mistakes are correcte

    On the cuspidalization problem for hyperbolic curves over finite fields

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    In this paper, we study some group-theoretic constructions associated to arithmetic fundamental groups of hyperbolic curves over finite fields. One of the main results of this paper asserts that any Frobenius-preserving isomorphism between the geometrically pro-ll fundamental groups of hyperbolic curves with one given point removed induces an isomorphism between the geometrically pro-ll fundamental groups of the hyperbolic curves obtained by removing other points. Finally, we apply this result to obtain results concerning certain cuspidalization problems for fundamental groups of (not necessarily proper) hyperbolic curves over finite fields.Comment: 44 pages, to appear in Kyoto Journal of Mathematic

    Duality for dormant opers

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    In the present paper, we prove that on a fixed pointed stable curve of characteristic p>0p>0, there exists a duality between dormant sln\mathfrak{sl}_n-opers (1<n<pβˆ’11 < n <p-1) and dormant sl(pβˆ’n)\mathfrak{sl}_{(p-n)}-opers. Also, we prove that there exists a unique (up to isomorphism) dormant sl(pβˆ’1)\mathfrak{sl}_{(p-1)}-oper on a fixed pointed stable curve of characteristic p>0p>0.Comment: 41 page

    Dormant opers and Gauss maps in positive characteristic

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    The Gauss map of a given projective variety is the rational map that sends a smooth point to the tangent space at that point, considered as a point of the Grassmann variety. The present paper aims to generalize a result by H. Kaji on Gauss maps in positive characteristic and establish an interaction with the study of dormant opers, as well as Frobenius-projective structures. We first prove a correspondence between dormant opers on a smooth projective variety XX and closed immersions from XX into a projective space with purely inseparable Gauss map. By using this, we determine the subfields of the function field of a smooth curve in positive characteristic induced by Gauss maps. Moreover, the correspondence gives us a Frobenius-projective structure on a Fermat hypersurface. This example embodies an exotic phenomenon of algebraic geometry in positive characteristic.Comment: 22 page
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