Cosection localization and the Quot scheme QuotSl(E)

Stark, Samuel

doi:10.1098/rspa.2022.0419

Cited by 4 publications

(4 citation statements)

References 31 publications

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“…Relatively defective subschemes. In contrast to the situation studied in[Sta21a], the map α is not a morphism when e ≥ 3. The indeterminacy locus is the set of Z ∈ Hilb e (S) such thatE * → π * (O Z ⊗O PE (1)) is not surjective; equivalently, deg(E * Z ) > deg E * − e.For example, if Z consists of three collinear points in a fibre of PE, then deg(E *…”

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confidence: 78%

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Secant loci of scrolls over curves

Hitching¹

2023

Preprint

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Given a curve C and a linear system ℓ on C, the secant locus V e−f e (ℓ) parametrises effective divisors of degree e which impose at most e − f conditions on ℓ. For E → C a vector bundle of rank r, we define determinantal subschemes H e−f e (ℓ) ⊆ Hilb e (PE) and Q e−f e (V ) ⊆ Quot 0,e (E * ) which generalise V e−f e (ℓ), giving several examples. We describe the Zariski tangent spaces of Q e−f e (V ), and give examples showing that smoothness of Q e−f e (V )is not necessarily controlled by injectivity of a Petri map. We generalise the Abel-Jacobi map and the notion of linear series to the context of Quot schemes.We give some sufficient conditions for nonemptiness of generalised secant loci, and a criterion in the complete case when f = 1 in terms of the Segre invariant s 1 (E). This leads to a geometric characterisation of semistability similar to that in [Hit19]. Using these ideas, we also give a partial answer to a question of Lange on very ampleness of O PE (1), and show that for any curve, Q e−1 e (V ) is either empty or of the expected dimension for sufficiently general E and V . When Q e−1 e (V ) has and attains expected dimension zero, we use formulas of Oprea-Pandharipande and Stark to enumerate Q e−1 e (V ). We mention several possible avenues of further investigation. e (ℓ), let E → C be a vector bundle of rank r and degree d. The scheme Quot 0,e (E * ) parametrises torsion quotients of E * of length e; equivalently, subsheaves F * ⊂ E * of rank r and degree −d − e. Let M be a line bundle, and suppose that V ⊆ H 0 (C, E * ⊗ M ) is a subspace of dimension n + 1. We define

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confidence: 78%

“…The link between H e−f e (V ) and Q e−f e (V ). Following [Hit20] and [Sta21a], for S = PE we now describe the relation between the two generalised secant loci above. This will generalise the fact that if L is a line bundle, then…”

Section: Secant Loci On Hilbert Schemesmentioning

confidence: 99%

Secant loci of scrolls over curves

Hitching¹

2023

Preprint

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“…For X = C, S and a fixed torsion-free sheaf V on X, the Quot scheme Quot X (V , n) for X = C, S parameterizes equivalence classes of surjective morphisms V → F from V to a zero-dimensional sheaf F . When V is a vector bundle, the virtual fundamental classes Quot X (V , n) vir were constructed by Marian-Oprea-Pandharipande [42, Lemma 1.1] (see Stark [59,Proposition 5] for a more detailed proof). It was remarked in [7, Sect.…”

Section: Rank Reduction Of Virasoromentioning

confidence: 99%

Virasoro constraints for moduli of sheaves and vertex algebras

Bojko,

Lim,

Moreira

2024

Invent. math.

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In enumerative geometry, Virasoro constraints were first conjectured in Gromov-Witten theory with many new recent developments in the sheaf theoretic context. In this paper, we rephrase the sheaf theoretic Virasoro constraints in terms of primary states coming from a natural conformal vector in Joyce’s vertex algebra. This shows that Virasoro constraints are preserved under wall-crossing. As an application, we prove the conjectural Virasoro constraints for moduli spaces of torsion-free sheaves on any curve and on surfaces with only $(p,p)$ ( p , p ) cohomology classes by reducing the statements to the rank 1 case.

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“…As far as the author knows, it is not yet generally accepted but neither any particular issue is identified. The preprint [126] conjectures that not only Quot r d (A 2 ) is reduced, but it has rational singularities. For r = 1 the answer is affirmative, as Hilb d (A 2 ) is smooth [48].…”

Section: Open Problemsmentioning

confidence: 99%