Singularities and Kodaira dimension of moduli scheme of stable sheaves on Enriques surfaces

Yamada, Kimiko

doi:10.1215/21562261-1966098

Cited by 9 publications

(17 citation statements)

References 12 publications

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“…and smooth in codimension 1, so it's normal and Gorenstein as well. That M s σ,Y (v) has canonical singularities follows precisely as in [Yam13].…”

Section: Singularities Of Bridgeland Moduli Spaces and Kodaira Dimensionmentioning

confidence: 80%

Projectivity and birational geometry of Bridgeland moduli spaces on an Enriques surface

Nuer

2016

Proc. London Math. Soc.

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Abstract. We construct moduli spaces of semistable objects on an Enriques surface for generic Bridgeland stability condition and prove their projectivity. We further generalize classical results about moduli spaces of semistable sheaves on an Enriques surface to their Bridgeland counterparts. Using Bayer and Macrì's construction of a natural nef divisor varying with the stability condition, we begin a systematic exploration of the relation between wall-crossing on the Bridgeland stability manifold and the minimal model program for these moduli spaces. We give three applications of our machinery to obtain new information about the classical moduli spaces of Gieseker-stable sheaves:1) We obtain a region in the ample cone of the moduli space of Gieseker-stable sheaves which works for all unnodal Enriques surfaces.2) We determine the nef cone of the Hilbert scheme of n points on an unnodal Enriques surface in terms of the classical geometry of its half-pencils and the Cossec-Dolgachev φ-function.3) We recover some classical results on linear systems on Enriques surfaces and obtain some new ones about n-very ample line bundles.

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“…and smooth in codimension 1, so it's normal and Gorenstein as well. That M s σ,Y (v) has canonical singularities follows precisely as in [Yam13].…”

Section: Singularities Of Bridgeland Moduli Spaces and Kodaira Dimensionmentioning

confidence: 80%

Projectivity and birational geometry of Bridgeland moduli spaces on an Enriques surface

Nuer

2016

Proc. London Math. Soc.

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“…We have dealt with the global nature of Sing(M s σ,S (v)) in Proposition 9.1, which generalizes results of [24] for stable vector bundles on Enriques surfaces. The next result, which deals with the local nature of these singularities, can be proven exactly as in [47] for the corresponding result on sheaves: Proposition 9.2. Suppose that S is a bielliptic surface, v ∈ H * alg (S, Z), and σ ∈ Stab † (S).…”

Section: Singularities and Kodaira Dimensionmentioning

confidence: 81%

“…From the proof of [47, Theorem 1.1], N ≥ 4 and codim(Sing(M s σ,S (v))) ≥ 2 if v 2 ≥ 4 ord(K S ). We nevertheless show that the original dimension condition in [47], namely v 2 ≥ 3 ord(K S ), suffices to guarantee that the singularities are not just canonical but in fact terminal. To this end, first note (25…”

Section: Singularities and Kodaira Dimensionmentioning

confidence: 82%

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Stable sheaves on bielliptic surfaces: from the classical to the modern

Nuer

2021

Preprint

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Bielliptic surfaces are the last family of Kodaira dimension zero algebraic surfaces without a classification result for the Chern characters of stable sheaves. We rectify this and prove such a classification using a combination of classical techniques, on the one hand, and derived category and Bridgeland stability techniques, on the other. Along the way, we prove the existence of projective coarse moduli spaces of objects in the derived category of a bielliptic surface that Bridgeland semistable with respect to a generic stability condition. By systematically studying the connection between Bridgeland wall-crossing and birational geometry, we show that for any two generic stability conditions τ, σ, the two moduli spaces Mτ (v) and Mσ(v) of objects of Chern character v that are semistable with respect to τ (resp. σ) are birational. As a consequence, we show that for primitive v, the moduli space of stable sheaves of class v is birational to a moduli space of stable sheaves whose Chern character has one of finitely many easily understood "shapes".

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“…Hence we assume that r is even. By [7] (see also Remark 1.2) or [18], dim M H (v, L) s sing is odd and dim M H (v, L) s sing ≤ v 2 2 + 1. Moreover if the equality holds, then 2 | c 1 and L ≡ r 2 K X mod 2, and if v is primitive, v 2 ≡ 0 mod 8 (see Remark 1.1.…”

Section: The Dimension Of Moduli Stacksmentioning

confidence: 94%

Moduli spaces of stable sheaves on Enriques surfaces

Yoshioka¹

2018

Kyoto J. Math.

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We shall study the existence condition of µ-stable sheaves on Enriques surfaces. We also give a different proof of the irreducibility of the moduli spaces of rank 2 stable sheaves.In the second part, we shall study the irreducibility of the moduli spaces M H (v, L) ss . The irreducibility of these moduli spaces on an arbitrary surfaces was proved by Gieseker and Li [4] and O'Grady [15] when the expected dimension d is larger than a constant N (r) that depends only on the rank r. We can expect a better estimate for N (r) in the Enriques case, as occurs for K3 and abelian surfaces, since an Enriques surface also has a numerically trivial canonical divisor. Let v = (r, ξ, s 2 ) be a primitive Mukai vector on an Enriques surface. If r is odd, then the irreducibility of M H (v, L) ss was proved in [24]. If r = 2, then the

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Singularities and Kodaira dimension of moduli scheme of stable sheaves on Enriques surfaces

Abstract: Let M be a moduli scheme of stable sheaves with fixed Chern classes on an Enriques surface or a hyper-elliptic surface. If its expected dimension is 7 or more, then M admits only canonical singularities. Moreover, if M is compact, then its Kodaira dimension is zero.

Cited by 9 publications

References 12 publications

Projectivity and birational geometry of Bridgeland moduli spaces on an Enriques surface

Projectivity and birational geometry of Bridgeland moduli spaces on an Enriques surface

Stable sheaves on bielliptic surfaces: from the classical to the modern

Moduli spaces of stable sheaves on Enriques surfaces

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