Secant varieties and Hirschowitz bound on vector bundles over a curve

Choe, Insong; Hitching, George H.

doi:10.1007/s00229-010-0381-1

Cited by 15 publications

(33 citation statements)

References 12 publications

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“…In [LN83], the Segre invariants of rank two bundles are interpreted geometrically in terms of secant varieties to a projective model of the curve. This interpretation is generalised to higher rank and to symplectic and orthogonal bundles in [CH10,CH12,CH16] and elsewhere.…”

Section: Introductionmentioning

confidence: 84%

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Quot schemes, Segre invariants, and inflectional loci of scrolls over curves

Hitching

2019

Geom Dedicata

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Let E be a vector bundle over a smooth curve C, and S = PE the associated projective bundle. We describe the inflectional loci of certain projective models ψ : S P n in terms of Quot schemes of E. This gives a geometric characterisation of the Segre invariant s1(E), which leads to new geometric criteria for semistability and cohomological stability of bundles over C. We also use these ideas to show that for general enough S and ψ, the inflectional loci are all of the expected dimension. An auxiliary result, valid for a general subvariety of P n , is that under mild hypotheses, the inflectional loci associated to a projection from a general centre are of the expected dimension.

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Section: Introductionmentioning

confidence: 84%

“…Taking direct sums of line bundles, it is easy to find a bundle E with s n (E) arbitrarily small. Conversely, however, Hirschowitz [Hir88] determined the following upper bound on s n (E), valid for all E (see also [CH10]).…”

Section: Segre Invariant and Dimensions Of Osculating Spacesmentioning

confidence: 99%

Quot schemes, Segre invariants, and inflectional loci of scrolls over curves

Hitching

2019

Geom Dedicata

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“…The goal of this section is to answer this question by giving a geometric construction for points in the negative slice of x u which correspond to unbroken flow lines connecting x u and x ℓ in terms of the secant varieties Sec n (N φ,φu ). For holomorphic bundles, the connection between secant varieties and Hecke modifications has been studied in [23], [4] and [12].…”

Section: Secant Varieties Associated To the Space Of Hecke Modificatimentioning

confidence: 99%

“…4 shows that the subsheaf (G, φ G ) can be resolved to form a Higgs subbundle (G ′ , φ ′ G ) of (E, φ) with slope(G ′ ) ≥ slope(G).…”

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confidence: 99%

The reverse Yang–Mills–Higgs flow in a neighbourhood of a critical point

Wilkin

2020

J. Differential Geom.

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The main result of this paper is a construction of solutions to the reverse Yang-Mills-Higgs flow converging in the C ∞ topology to a critical point. The construction uses only the complex gauge group action, which leads to an algebraic classification of the isomorphism classes of points in the unstable set of a critical point in terms of a filtration of the underlying Higgs bundle.Analysing the compatibility of this filtration with the Harder-Narasimhan-Seshadri double filtration gives an algebraic criterion for two critical points to be connected by a flow line. As an application, we can use this to construct Hecke modifications of Higgs bundles via the Yang-Mills-Higgs flow. When the Higgs field is zero (corresponding to the Yang-Mills flow), this criterion has a geometric interpretation in terms of secant varieties of the projectivisation of the underlying bundle inside the unstable manifold of a critical point, which gives a precise description of broken and unbroken flow lines connecting two critical points. For non-zero Higgs field, at generic critical points the analogous interpretation involves the secant varieties of the spectral curve of the Higgs bundle.

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“…Although the present note can be read independently of [2,3,4,5], we use several results from these articles. In particular, access to [4, §2 and §5] may be helpful for the reader.…”

Section: Introductionmentioning

confidence: 99%

Non-defectivity of Grassmannian bundles over a curve

Choe

Hitching

2016

Int. J. Math.

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Let Gr(2, E) be the Grassmann bundle of two-planes associated to a general bundle E over a curve X. We prove that an embedding of Gr(2, E) by a certain twist of the relative Plücker map is not secant defective. This yields a new and more geometric proof of the Hirschowitz-type bound on the Lagrangian Segre invariant for orthogonal bundles over X, analogous to those given for vector bundles and symplectic bundles in [2,3]. From the non-defectivity we also deduce an interesting feature of a general orthogonal bundle over X, contrasting with the classical and symplectic cases: Any maximal Lagrangian subbundle intersects at least one other maximal Lagrangian subbundle in positive rank.

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Secant varieties and Hirschowitz bound on vector bundles over a curve

Cited by 15 publications

References 12 publications

Quot schemes, Segre invariants, and inflectional loci of scrolls over curves

Quot schemes, Segre invariants, and inflectional loci of scrolls over curves

The reverse Yang–Mills–Higgs flow in a neighbourhood of a critical point

Non-defectivity of Grassmannian bundles over a curve

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