On the theta divisor of <i>SU</i>(<i>r</i>; 1)

Brivio, Sonia; Verra, Alessandro

doi:10.1017/s0027763000008205

Cited by 7 publications

(18 citation statements)

References 13 publications

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“…Item (2) implies that H 1 (P 1 , f * T N) = 0 and hence M 0,0 (N, 2) is nonsingular at the points of Hecke curves.…”

Section: Examples Of Stable Maps To Nmentioning

confidence: 99%

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Hecke correspondence, stable maps, and the Kirwan desingularization

Kiem

2007

Duke Math. J.

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Abstract. We prove that the moduli space M0,0(N, 2) of stable maps of degree 2 to the moduli space N of rank 2 stable bundles of fixed determinant over a smooth projective curve of genus g has two irreducible components which intersect transversely. One of them is Kirwan's partial desingularization MX of the moduli space MX of rank 2 semistable bundles with determinant isomorphic to OX (y − x) for some y ∈ X. The other component is the partial desingularization of PHom(sl (2) ∨ , W)/ /P GL(2) for a vector bundle W = R 1 π * L −2 (−x) of rank g over the Jacobian of X. We also show that the Hilbert scheme H, the Chow scheme C of conics in N and M0,0(N, 2) are related by explicit contractions. IntroductionLet X be a smooth projective curve of genus g ≥ 3 over the complex number field. We fix x ∈ X throughout this paper. The moduli space N of rank 2 stable bundles over X with determinant O X (−x) is a smooth projective Fano variety whose Picard group is generated by a very ample line bundle Θ. (See [2].) The problem that we address in this paper is the following. Obviously the degree of a stable map f : C → N is defined as the degree ofis a vector bundle of rank g over J where L is a Poincaré bundle over J × X and π J : J × X → J is the projection.The purpose of this paper is to study the d = 2 case. This is particularly interesting because of Hecke correspondence which has been one of the most powerful tools in the study of moduli spaces of bundles. Let M X be the moduli space of rank 2 semistable bundles E with det E ∼ = O X (y − x) for some y ∈ X. This is a (3g − 2)-dimensional singular projective normal where M (2, 0) is the moduli space of semistable bundles of rank 2 and degree 0 over X and the bottom map is y → O X (y − x). Hecke correspondence refers to a diagram{ { w w w w w w w w wwhere q 1 is the projectivization of a universal bundle over N × X and q 0 is a P 1 -bundle over the stable part M s X . Hence for any θ ∈ M s X , we have a stable map Kirwan's partial desingularization is a systematic way to partially resolve the singularities of GIT quotients. The partial desingularization M X of M X (see [10]) is the consequence of two blowups first along the deepest stratumwhich is the locus of bundles ξ ⊕ ξ, ξ ∈X and then along the proper transform of the middle stratumwhich is the locus of bundles ξ 1 ⊕ ξ 2 , (ξ 1 , ξ 2 ) ∈ K J where Z 2 interchanges ξ 1 and ξ 2 . The upshot is an orbifold M X which is singular along a nonsingular subvariety ∆X ∼ = P(S 2 A ∨ X ) where AX → Gr(2, W 0 ) is the universal rank 2 bundle on the relative Grassmannian of the bundleL 0 being the Poincaré line bundle overX × X and π :X × X →X being the projection. By blowing up M X along ∆X we obtain a (full) desingularization M X of M X . The exceptional divisor of the last blowup isfor some line bundle η over Gr(2, W 0 ) and the normal bundle is O(−1, −2) on the fibersAlthough the partial desingularization M X has been quite useful especially for cohomological computations, its moduli theoretic meaning has been unknown. We prove ...

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“…Item (2) implies that H 1 (P 1 , f * T N) = 0 and hence M 0,0 (N, 2) is nonsingular at the points of Hecke curves.…”

Section: Examples Of Stable Maps To Nmentioning

confidence: 99%

“…Note that M 0,0 (PV, 2) = P s /P GL(2) = Q is singular along the proper transform of the quotient Q 2 of the locus of rank ≤ 2 homomorphisms P 2 := PHom 2 (sl(2) ∨ , V ) by P GL (2). In fact the proper transform of P 2 in P s is the locus of nontrivial stabilizers which are isomorphic to Z 2 .…”

Section: Stable Maps To Projective Spacesmentioning

confidence: 99%

Hecke correspondence, stable maps, and the Kirwan desingularization

Kiem

2007

Duke Math. J.

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“…Observe that there exist smooth anticanonical hypersurfaces, by the very ampleness of Θ [3] and the Bertini theorem.…”

Section: The Canonical Strip Hypothesesmentioning

confidence: 99%

Examples Violating Golyshev’s Canonical Strip Hypotheses

Belmans

Galkin

Mukhopadhyay

2019

Experimental Mathematics

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We give the first examples of smooth Fano and Calabi-Yau varieties violating the (narrow) canonical strip hypothesis, which concerns the location of the roots of Hilbert polynomials of polarised varieties. They are given by moduli spaces of rank 2 bundles with fixed odd-degree determinant on curves of sufficiently high genus, hence our Fano examples have Picard rank 1, index 2, are rational, and have moduli. The hypotheses also fail for several other closely related varieties.

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“…(1.6) In this hypothesis, E turns out to be stable, see [BV2] Lemma 2.4.1, so it represents a point of the moduli space SU (r, rg+1) of stable bundle of rank r and fixed determinant of degree rg + 1. This method allows us to produce stable bundles of rank r having r + 1 sections, which seem to be a useful tool in studying the moduli spaces SU (r, d) and their immersions, see [BV1], [BV2] and [GI].…”

Section: Introductionmentioning

confidence: 99%

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Brivio

2002

Math. Nachr.

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On the theta divisor of SU(r; 1)

Cited by 7 publications

References 13 publications

Hecke correspondence, stable maps, and the Kirwan desingularization

Hecke correspondence, stable maps, and the Kirwan desingularization

Examples Violating Golyshev’s Canonical Strip Hypotheses

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