Hecke correspondence, stable maps, and the Kirwan desingularization

Abstract

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 ...

Mori's Program for the Moduli Space of Conics in Grassmannian

“…Thus on the fiber of M , the second blow-up π 2 : X 2 //G → X 1 //G is the partial desingularization of the fiber P(H ⊗ sl 2 )//SL 2 . In [19,Theorem 4.1], it was shown that the partial resolution is isomorphic to the moduli space M 0,0 (PH, 2) of degree two stable maps to PH. Note that Z 2 acts trivially on the projectivized normal cone.…”

Stable maps of genus zero in the space of stable vector bundles on a curve

“…Definition-Proposition 2.1. ( [16]) Let E be a rank two vector bundle on X and p ∈ X. Let E vp be the kernel of the surjective map…”

“…Since deg(i) = 0, the closure l := i(l \ (l ∩ X)) in N is a smooth conic. By [16,Proposition 3.6], l should be a Hecke curve or a conic in PV M ∼ = P g−1 for some M ∈ Pic 0 (X). In the latter case, l ∩ X = r, l \ r ⊂ PV s L ∩ PV M for some M ∈ Pic 0 (X).…”

“…When d = 3, it is quite natural to start with finding the possible irreducible components of the moduli space M 0 (N , 3). In this direction, in [3,16], the authors found the two components parametrizing the degree 3 map f : P 1 → N . For L ∈ Pic k (X), let P L := P Ext 1 (L, L −1 (−x)) be the space of non-split extensions…”

“…Two components parametrize one of the following types of degree 3 maps f : P 1 → N . ( [3,Lemma 4.10], [16,Proposition 3.9]). i) When k = 0, f is a composition of degree 3 maps P 1 → P Ext 1 (L, L −1 (−x)) ∼ = P g−1 with the map Ψ L .…”