Heisenberg invariant quartics and  C (2) for a curve of genus four

Oxbury, W. M.; Pauly, Christian

doi:10.1017/s0305004198003028

Cited by 11 publications

(5 citation statements)

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“…Finally, θ is generically finite for g = 2 [4], [11] and we know its degree for r ≤ 4, [25], [35]. There are also good descriptions of the image of θ for r = 2 g = 2, 3 [28] [34], r = 3, g = 2 [31] [29], r = 2, g = 4 [33]. Moreover, it has recently been shown in [12] that if C is general and g >> r then θ is generically injective.…”

Section: The Theta Mapmentioning

confidence: 99%

COHERENT SYSTEMS AND MODULAR SUBAVRIETIES OF $\mathcal{SU}_C(r)$

Bolognesi

Brivio

2012

Int. J. Math.

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Let C be an algebraic smooth complex curve of genus g > 1. The object of this paper is the study of the birational structure of certain moduli spaces of vector bundles and of coherent systems on C and the comparison of different type of notions of stability arising in moduli theory. Notably we show that in certain cases these moduli spaces are birationally equivalent to fibrations over simple projective varieties, whose fibers are GIT quotients (P r−1 ) rg //PGL(r), where r is the rank of the considered vector bundles. This allows us to compare different definitions of (semi-)stability (slope stability, α-stability, GIT stability) for vector bundles, coherent systems and point sets, and derive relations between them. In certain cases of vector bundles of low rank when C has small genus, our construction produces families of classical modular varieties contained in the Coble hypersurfaces. a double covering of P 8 branched along a hypersurface of degree six C 6 called the Coble-Dolgachev sextic [25]. Our result is the following. Theorem 1.3. The Coble-Dolgachev sextic C 6 is birational to a fibration over P 4 whose fibers are Igusa quartics. More precisely, C 6 contains a four-dimensional family of Igusa quartics parametrized by an open subset of P 4 .We recall that an Igusa quartic is a modular quartic hypersurface in P 4 that is related to some classical GIT quotients (see e.g. [15]) and moduli spaces. Its dual variety is a cubic three-fold called the Segre cubic, that is isomorphic to the GIT quotient (P 1 ) 6 //PGL(2).If r = 2 and g = 3, then SU C (2) is embedded by θ in P 7 as a remarkable quartic hypersurface C 4 called the Coble quartic [27]. Our methods also allow us to give a quick proof of the following fact.

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Section: The Theta Mapmentioning

confidence: 99%

COHERENT SYSTEMS AND MODULAR SUBAVRIETIES OF $\mathcal{SU}_C(r)$

Bolognesi

Brivio

2012

Int. J. Math.

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“…The cubic normality is true in both of these cases. For a general curve of genus 4 without effective theta-constant, cubic normality is proved in Theorem 4.1 of [OP99].…”

Section: This Map Takes Values In Hmentioning

confidence: 99%

The space of generalized G2-theta functions

Grégoire¹

2011

Preprint

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“…From (5.4) in [6] (or see proof of Proposition 10.1 (3) in [7]), we have We have no formula to compute U(N i (ν j )) for any h ∈ Aut(X), not even in the case that h has prime order (unless it satisfies the condition that h \ {1} is contained in a conjugacy class of Aut(X)). In what follows we will explain a way to compute U(N i (ν j )) when enough information about the quotient map f : X → Y is known and for the case in which N i is the normal bundle of the curve Y in S p X under the embedding…”

Section: Lemma 37 Let H Be An Automorphism Of X Of Prime Order P Amentioning

confidence: 99%