Equations of Hurwitz schemes in the infinite Grassmannian

Abstract. We obtain a characterization of theta functions of Jacobian varieties of curves with automorphisms among theta functions of principally polarized abelian varieties (p.p.a.v.). We first give a characterization in terms of finite dimensional orbits for a suitable action in the Sato Grassmannian. Secondly, the introduction of formal Baker-Akhiezer functions and formal τ -functions attached to a p.p.a.v. (for the multipuncture case) allows us to characterize, in terms of bilinear identities, those Baker-Akhiezer functions that are Baker-Akhiezer functions of Jacobians of curves with automorphisms. Further, in the case of automorphisms with fixed points, we rewrite the previous result as a hierarchy of partial differential equations for the τ -function of a p.p.a.v. Finally, since Baker-Akhiezer and τ functions are written in terms of theta functions, these results give rise to characterizations of p.p.a.v. in terms of their theta functions.

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“…Following [MP2] and [Ma], we consider the u-th Baker-Akhiezer function of a point U ∈ Gr(V ) as the V -valued function defined by…”

Section: A Formal Group Schemesmentioning

confidence: 99%

“…holds for all 1 ≤ v, u ≤ p and all t, s (for notation and results see [MP2,§3]). This hierarchy is essentially equivalent to the p-multicomponent KP hierarchy ( [KvdL]).…”

Section: Wherementioning

confidence: 99%

Characterizations of Jacobians of curves with automorphisms

González

Porras²,

Martı́n³

et al. 2010

Trans. Amer. Math. Soc.

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“…In order to give the equations describing Higgs ∞ X , we first need to adapt some results concerning Tau and Baker-Akhiezer functions given in [2,13,12].…”

Section: ((Z))[t ]/P(t ) and V + P := K[[z]][t ]/P(t )mentioning

confidence: 99%

“…Moreover, the formal Jacobian of the formal spectral curve X V , J ( X V ), is isomorphic to Γ − Vp (see [2,Theorem 4.14] and also [12]). Let A ∞ be the formal group scheme …”

Section: Formal Jacobian and Abel Morphismmentioning

confidence: 99%

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Equations of the Moduli of Higgs Pairs and Infinite Grassmannian

2009

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In this paper the moduli space of Higgs pairs over a fixed smooth projective curve with extra formal data is defined and is endowed with a scheme structure. We introduce a relative version of the Krichever map using a fibration of Sato Grassmannians and show that this map is injective. This, together with the characterization of the points of the image of the Krichever map, allows us to prove that this moduli space is a closed subscheme of the product of the moduli of vector bundles (with formal extra data) and a formal anologue of the Hitchin base. This characterization also provides us with a method for explicitly computing KP-type equations that describe the moduli space of Higgs pairs. Finally, for the case where the spectral cover is totally ramified at a fixed point of the curve, these equations are given in terms of the characteristic coefficients of the Higgs field.

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Coverings with prescribed ramification and Virasoro groups

Porras¹,

Martı́n²

2010

Kyoto J. Math.

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Equations of Hurwitz schemes in the infinite Grassmannian

Cited by 4 publications

References 16 publications

Characterizations of Jacobians of curves with automorphisms

Characterizations of Jacobians of curves with automorphisms

Equations of the Moduli of Higgs Pairs and Infinite Grassmannian

Coverings with prescribed ramification and Virasoro groups

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