Rank four vector bundles without theta divisor over a curve of genus two

Pauly, Christian

doi:10.1515/advgeom.2010.031

Cited by 6 publications

(3 citation statements)

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“…Notably, θ is an embedding for r = 2 [27], [10], [21] and it is a morphism when r = 3 for g ≤ 3 and for a general curve of genus g > 3, [4], [36]. 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].…”

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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“…Moreover, the indeterminacy locus of D consists of those bundles V ∈ SU C (r ) for which (1) is the whole Picard variety. This has been much studied; see for example Pauly [Pau10], Popa [Pop99] and Raynaud [Ray82]. Brivio and Verra [BV12] showed that D is generically injective for a general curve of genus g ≥ 3r r − 2r − 1, partially answering a conjecture of Beauville [Bea06,§6].…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, the indeterminacy locus of consists of those bundles for which (1) is the whole Picard variety. This has been much studied; see, for example, Pauly [Pau10], Popa [Pop99] and Raynaud [Ray82].…”

Section: Introductionmentioning

confidence: 99%

Tangent cones to generalised theta divisors and generic injectivity of the theta map

Hitching¹,

Hoff²

2017

Compositio Math.

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Let C be a Petri general curve of genus g and E a general stable vector bundle of rank r and slope g − 1 over C with h 0 (C, E) = r + 1. For g ≥ (2r + 2)(2r + 1), we show how the bundle E can be recovered from the tangent cone to the theta divisor Θ E at O C . We use this to give a constructive proof and a sharpening of Brivio and Verra's theorem that the theta map SU C (r ) |r Θ| is generically injective for large values of g .

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An interpolation problem for the normal bundle of curves of genusg ≥ 2 and high degree in ℙ^r

Ballico

2016

Communications in Algebra

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Let C ⊂ P n be a smooth curve and N C its normal bundle. N C satisfies strong interpolation if for all integers s > 0 and λ i ∈ {0, 1, . . . , n − 1}, 1 ≤ i ≤ s, there are distinct points P 1 , . . . , Ps ∈ C and linear subspaces U i ⊆ E|P i such that dim(U i ) = λ i for all i and the evaluation map H 0 (E) → ⊕ s i=1 U i has maximal rank (A. Atanasios). We prove that C satisfies strong interpolation if either C is a linearly normal elliptic curve or C is a general embedding of degree d ≥ (5n − 8)g + 2n 2 − 5n + 4 of a smooth curve X of genus g ≥ 2.

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Rank four vector bundles without theta divisor over a curve of genus two

Abstract: We show that the locus of stable rank four vector bundles without theta divisor over a smooth projective curve of genus two is in canonical bijection with the set of theta-characteristics. We give several descriptions of these bundles and compute the degree of the rational theta map.

Cited by 6 publications

References 10 publications

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

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

Tangent cones to generalised theta divisors and generic injectivity of the theta map

An interpolation problem for the normal bundle of curves of genusg ≥ 2 and high degree in ℙ^r

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Rank four vector bundles without theta divisor over a curve of genus two

Abstract: We show that the locus of stable rank four vector bundles without theta divisor over a smooth projective curve of genus two is in canonical bijection with the set of theta-characteristics. We give several descriptions of these bundles and compute the degree of the rational theta map.

Cited by 6 publications

References 10 publications

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

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

Tangent cones to generalised theta divisors and generic injectivity of the theta map

An interpolation problem for the normal bundle of curves of genusg ≥ 2 and high degree in ℙr

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Product

Resources

About

An interpolation problem for the normal bundle of curves of genusg ≥ 2 and high degree in ℙ^r