A criterion for virtual global generation

Let H be the upper half plane and T a discrete subgroup of AutH.Suppose that H mod Y is of finite measure.This work stems from the question whether there is an algebraic interpretation for the moduli of unitary representations of T similar to the case whenWe show that this is indeed the case via the moduli of vector bundles on the compactification of H mod r, provided with some additional structures which we propose to call parabolic structures.The idea of parabolic structures is inspired from A. Weil's work [6,§2, Chapter I, p. 56].Let Xbe a smooth, irreducible, projective curve defined, say, over an algebraically closed field k.By vector bundles on X we understand algebraic vector bundles.DEFINITION 1.Let V be a vector bundle on X and Q G X. Then a quasiparabolic structure of Vat Q is giving a flag on the fibre VQ of Fat Ô, i.e., giving linear subspaces F*VQ of VQ,We call I = (l l9 . . .9 l r ) the type (or flag type) of the quasi-parabolic structure.Let k x = l x -/ 2 , k 2 = l 2 -/ 3 , . . ., fc fWl = /,._! -l r , k r = l r \ then k t are called the multiplicities of the quasi-parabolic structure.DEFINITION 2. Let V be a vector bundle on X and Q G X. Then a parabolic structure of V at Q is giving (i) a quasi-parabolic structure of V at Q; say I = (l t , . . ., l r ) is its type and {k.} its multiplicities, and(ii) constants a = (a 1 , . . ., a n ) called the weights of the parabolic structure such that 0 < a x < a 2 < • • • < ot n < 1 and there are r distinct elements among a, say a' = (o^, . . ., Û£), 0 < a\ < a 2 < • • • < a r < 1, such that OL\ occurs k x times, a 2 occurs k 2 times, . . ., a r occurs k r times among a.We call a. the weight ofF l V Q .Note that l x = « = rkV.Let F, H> be vector bundles on X with quasi-parabolic structures at Q.An isomorphism ƒ: F -• W of vector bundles is said to be a quasi-parabloic isomorphism if the types of V, W at Q are the same and fçiF'Vç) = F 1 WQ (/^:

Some coherent sheaves on projective varieties have a non-reduced versal deformation space; for example, this is the case for most unstable rank 2 vector bundles on ℙ2, see [18]. In particular, some moduli spaces of stable sheaves are non-reduced. We consider some sheaves on ribbons (double structures on smooth projective curves): let E be a quasi locally free sheaf of rigid type and let 𝓔 be a flat family of sheaves containing E. We find that 𝓔 is a reduced deformation of E when some canonical family associated to 𝓔 is also flat. We consider also a deformation of the ribbon to reduced projective curves with two components, and find that E can be deformed in two distinct ways to sheaves on the reduced curves. In particular some components M of the moduli spaces of stable sheaves deform to two components of the moduli spaces of sheaves on the reduced curves, and M appears as the “limit” of varieties with two components, whence the non-reduced structure of M.

Let X be a smooth projective curve, L∈Pic (X) and E a vector bundle on X. Here we study the bigraded algebra\(A(X;E,L): = \oplus _{u,t \in \mathbb{Z},t \geqslant 0} H^0 (X,Hom(E \otimes L^{ \otimes u} ,E \otimes L^{ \otimes (u + t)} ))\), mainly when E is a general stable bundle on X. We give informations on the degrees of the minimal generators and study a generalization of the notion of projective normality.

Let M(n, ξ) be the moduli space of stable vector bundles of rank n ≥ 3 and fixed determinant ξ over a complex smooth projective algebraic curve X of genus g ≥ 4. We use the gonality of the curve and r-Hecke morphisms to describe a smooth open set of an irreducible component of the Hilbert scheme of M(n, ξ), and to compute its dimension. We prove similar results for the scheme of morphisms \({M or_P (\mathbb{G}, M(n, \xi))}\) and the moduli space of stable bundles over \({X \times \mathbb{G}}\), where \({\mathbb{G}}\) is the Grassmannian \({\mathbb{G}(n - r, \mathbb{C}^n)}\). Moreover, we give sufficient conditions for \({M or_{2ns}(\mathbb{P}^1, M(n, \xi))}\) to be non-empty, when s ≥ 1.

A result of André Weil allows one to describe rank$n$vector bundles on a smooth complete algebraic curve up to isomorphism via a double quotient of the set$\text{GL}_{n}(\mathbb{A})$of regular matrices over the ring of adèles (over algebraically closed fields, this result is also known to extend to$G$-torsors for a reductive algebraic group $G$). In the present paper we develop analogous adelic descriptions for vector and principal bundles on arbitrary Noetherian schemes, by proving an adelic descent theorem for perfect complexes. We show that for Beilinson’s co-simplicial ring of adèles$\mathbb{A}_{X}^{\bullet }$, we have an equivalence$\mathsf{Perf}(X)\simeq |\mathsf{Perf}(\mathbb{A}_{X}^{\bullet })|$between perfect complexes on$X$and cartesian perfect complexes for$\mathbb{A}_{X}^{\bullet }$. Using the Tannakian formalism for symmetric monoidal$\infty$-categories, we conclude that a Noetherian scheme can be reconstructed from the co-simplicial ring of adèles. We view this statement as a scheme-theoretic analogue of Gelfand–Naimark’s reconstruction theorem for locally compact topological spaces from their ring of continuous functions. Several results for categories of perfect complexes over (a strong form of) flasque sheaves of algebras are established, which might be of independent interest.

'. Let Cg be a smooth projective irreducible curve over C of genus g ≥ 1 and let {p 1 ,..., p s } be a set of distinct points on Cg. We fix a nonnegative integer l and denote by M g (p, λ) the moduli space of parabolic semistable vector bundles of rank r on Cg with trivial determinant and fixed parabolic structure of type A = (λ 1 ,..., λ s ) at p = (p 1 ,..., p s ), where each weight λ i is in P l (SL(r)). On M g (p, λ) there is a canonical line bundle L(λ, l), whose global sections are called generalized parabolic SL(r)-theta functions of order l. In this paper we prove the existence of such nonzero nonabelian theta functions, thus establishing a part of higher genus generalizations of the celebrated saturation conjectures.

The Griffiths conjecture asserts that every ample vector bundle $E$ over a compact complex manifold $S$ admits a hermitian metric with positive curvature in the sense of Griffiths. In this article we give a sufficient condition for a positive hermitian metric on $\mathcal{O}_{\mathbb{P}(E^*)}(1)$ to induce a Griffiths positive $L^2$-metric on the vector bundle $E$. This result suggests to study the relative K\ahler-Ricci flow on $\mathcal{O}_{\mathbb{P}(E^*)}(1)$ for the fibration $\mathbb{P}(E^*)\to S$. We define a flow and give arguments for the convergence.

Algebraic Vector Bundles

In this paper, we prove that the notions of Hilbert stability and Mumford stability agree for vector bundles of arbitrary rank over smooth curves. The notion of Hilbert stability was introduced by Gieseker and Morrison in 1984, and they showed that for smooth curves and vector bundles of rank two it agrees with Mumford stability. A different proof for the rank two case was given by M. Teixidor i Bigas. Our proof uses a new approach and avoids complicated computations. Our results might serve as a first step in the construction of the Hilbert stable compactification of the universal moduli space of stable vector bundles over the moduli space of smooth curves as suggested by Teixidor.

Poincare bundles and Picard bundles on the Jacobian of a smooth curve have been studied extensively classically. Peter Newstead and his collaborators have made major contributions to the theory of Poincaré and Picard bundles over the moduli spaces of vector bundles of higher ranks. We present old and new results on the Poincare bundles and Picard bundles on the moduli spaces of vector bundles on smooth curves and their generalisations to nodal curves. Finally we consider parabolic vector bundles. We prove the parabolic stability of Poincaré bundles over the moduli spaces of parabolic vector bundles, with parabolic structure of any type, on a smooth curve.

We give an algorithm to determine whether a kernel sheaf over a smooth projective curve over an algebraically closed field is semistable. The algorithm uses symmetric powers to make destabilizing subbundles visible as global sections.

We prove a conjecture about the moduli space S U C ( 3 ) \mathcal {SU}_C(3) of semi-stable rank 3 vector bundles with trivial determinant on a genus 2 curve C C , due to I. Dolgachev. Given C C a smooth projective curve of genus 2, and the embedding of the Jacobian J C JC into | 3 Θ | |3\Theta | , A. Coble proved, at the beginning of the 20th century, that there exists a unique cubic hypersurface C \mathcal {C} in | 3 Θ | ∗ ≃ P 8 |3\Theta |^* \simeq \mathbb {P}^8 , J C [ 3 ] JC[3] -invariant and singular along J C JC . On the other hand, we have a map of degree 2 from S U C ( 3 ) \mathcal {SU}_C(3) over | 3 Θ | ≃ P 8 ∗ |3\Theta | \simeq \mathbb {P}^{8*} , ramified along a sextic hypersurface B \mathcal {B} . Dolgachev’s conjecture affirms that the sextic B \mathcal {B} is the dual variety of Coble’s cubic C \mathcal {C} .

Equivariant approach to weighted projective curves

Let $${\mathcal{E}}$$ be an ample vector bundle of rank r ≥ 2 on a smooth complex projective variety X of dimension n such that there exists a global section of $${\mathcal{E}}$$ whose zero locus Z is a smooth subvariety of dimension n − r ≥ 3 of X. Let H be an ample line bundle on X such that its restriction H Z to Z is very ample. Triplets $$(X,{\mathcal{E}},H)$$ are classified under the assumption that (Z,H Z ) has a smooth bielliptic curve section of genus ≥ 3 with $$H^{n-r}c_r({\mathcal{E}}) \leq 8$$ .

We construct Abel maps for a stable curve X. Namely, for each oneparameter deformation of X to a smooth curve, having regular total space, and each d ≥ 1, we construct by specialization a map αd X : Ẋd → P d X , where Ẋ ⊆ X is the smooth locus, and P d X is the coarse moduli scheme for equivalence classes of degree-d “semibalanced” line bundles on semistable curves having X as a stable model. For d = 1, we show that α1 X extends to a map α1 X : X → P 1 X , and does not depend on the choice of the deformation. Finally, we give a precise description of when α1 X is injective. The theory of Abel maps for smooth curves goes back to the 19th century. In the modern language, let C be a smooth projective curve, and PicC its degree-d Picard variety parametrizing line bundles of degree d on C. For each d > 0 there exists a remarkable morphism, often called the d-th Abel map: Cd −→ PicC (p1, . . . , pd) 7→ OC( ∑ pi). This map has been extensively studied and used in the literature. For d = 1, after the choice of a “base” point on C, it gives the Abel–Jacobi embedding C ↪→ PicC ∼= PicC (unless C ∼= P1). For an interesting historic survey see [K04] or [K05]. What about Abel maps for singular curves? Abel maps were constructed for all integral curves in [AK], and further studied in [EGK00], [EGK02] and [EK05]. In [AK], it is shown that the first Abel map of an integral singular curve is an embedding into its compactified Picard scheme. However almost nothing is known for reducible curves, not even when they are stable. This lack of knowledge appears all the more regrettable because of the importance of stable curves in moduli theory. In the present paper we construct Abel maps for stable curves. As we see it, Abel maps should satisfy the following natural properties. First, they should have a geometric meaning. More explicitly, recall that for a smooth