Stable Vector Bundles Of Rank Research Articles

Stability of symmetric powers of vector bundles of rank two with even degree on a curve

This paper treats the strict semi-stability of the symmetric powers [Formula: see text] of a stable vector bundle [Formula: see text] of rank [Formula: see text] with even degree on a smooth projective curve [Formula: see text] of genus [Formula: see text]. The strict semi-stability of [Formula: see text] is equivalent to the orthogonality of [Formula: see text] or the existence of a bisection on the ruled surface [Formula: see text] whose self-intersection number is zero. A relation between the two interpretations is investigated in this paper through elementary transformations. This paper also gives a classification of [Formula: see text] with strictly semi-stable [Formula: see text]. Moreover, it is shown that when [Formula: see text] is stable, every symmetric power [Formula: see text] is stable for all but a finite number of [Formula: see text] in the moduli of stable vector bundles of rank [Formula: see text] with fixed determinant of even degree on [Formula: see text].

A construction of equivariant bundles on the space of symmetric forms

We construct stable vector bundles on the space $\mathbb{P}(S^d \mathbb{C}^{n+1})$ of symmetric forms of degree $d$ in $n+1$ variables which are equivariant for the action of $\text{SL}\_{n+1}(\mathbb{C})$ and admit an equivariant free resolution of length $2$. For $n=1$, we obtain new examples of stable vector bundles of rank $d-1$ on $\mathbb{P}^d$, which are moreover equivariant for $\operatorname{SL}\_2(\mathbb{C})$. The presentation matrix of these bundles attains Westwick's upper bound for the dimension of vector spaces of matrices of constant rank and fixed size.

A NOTE ON RANK TWO STABLE BUNDLES OVER SURFACES

Abstract.Let π : X → C be a fibration with integral fibers over a curve C and consider a polarization H on the surface X. Let E be a stable vector bundle of rank 2 on C. We prove that the pullback π*(E) is a H-stable bundle over X. This result allows us to relate the corresponding moduli spaces of stable bundles $${{\mathcal M}_C}(2,d)$$ and $${{\mathcal M}_{X,H}}(2,df,0)$$ through an injective morphism. We study the induced morphism at the level of Brill–Noether loci to construct examples of Brill–Noether loci on fibered surfaces. Results concerning the emptiness of Brill–Noether loci follow as a consequence of a generalization of Clifford’s Theorem for rank two bundles on surfaces.

Projective Poincaré and Picard bundles for moduli spaces of vector bundles over nodal curves

Let UL′s(n,d) be the moduli space of stable vector bundles of rank n with determinant L where L is a fixed line bundle of degree d over a nodal curve Y. We prove that the projective Poincaré bundle on Y×UL′s(n,d) and the projective Picard bundle on UL′s(n,d) are stable for suitable polarisations. For a nonsingular point x∈Y, we show that the restriction of the projective Poincaré bundle to {x}×UL′s(n,d) is stable for any polarisation. We prove that for arithmetic genus g≥3 and for g=n=2,d odd, the Picard group of the moduli space UL′(n,d) of semistable vector bundles of rank n with determinant L of degree d is isomorphic to Z.

ON MAXIMALLY FROBENIUS DESTABILISED VECTOR BUNDLES

Let $X$ be a smooth projective curve of genus $g\geq 2$ over an algebraically closed field $k$ of characteristic $p>0$. We show that for any integers $r$ and $d$ with $0<r<p$, there exists a maximally Frobenius destabilised stable vector bundle of rank $r$ and degree $d$ on $X$ if and only if $r\mid d$.

Irreducibility of the moduli space of stable vector bundles of rank two and odd degree on a very general quintic surface

The moduli space $M(c_2)$, of stable rank two vector bundles of degree one on a very general quintic surface $X\subset {\mathbb P}^3$, is irreducible for all $c_2\geq 4$ and empty otherwise.

Tangent cones to generalised theta divisors and generic injectivity of the theta map

Let $C$ be a Petri general curve of genus $g$ and $E$ a general stable vector bundle of rank $r$ and slope $g-1$ over $C$ with $h^{0}(C,E)=r+1$. For $g\geqslant (2r+2)(2r+1)$, we show how the bundle $E$ can be recovered from the tangent cone to the generalised theta divisor $\unicode[STIX]{x1D6E9}_{E}$ at ${\mathcal{O}}_{C}$. We use this to give a constructive proof and a sharpening of Brivio and Verra’s theorem that the theta map $\mathit{SU}_{C}(r){\dashrightarrow}|r\unicode[STIX]{x1D6E9}|$ is generically injective for large values of $g$.

Theta divisors and the geometry of tautological model

Let E be a stable vector bundle of rank r and slope \(2g-1\) on a smooth irreducible complex projective curve C of genus \(g \ge 3\). In this paper we show a relation between theta divisor \(\Theta _E\) and the geometry of the tautological model \(P_E\) of E. In particular, we prove that for \(r > g-1\), if C is a Petri curve and E is general in its moduli space then \(\Theta _E\) defines an irreducible component of the variety parametrizing \((g-2)\)-linear spaces which are g-secant to the tautological model \(P_E\). Conversely, for a stable, \((g-2)\)-very ample vector bundle E, the existence of an irreducible non special component of dimension \(g-1\) of the above variety implies that E admits theta divisor.

Derived categories of moduli spaces of vector bundles on curves

Let X be a smooth projective curve of genus g≥4 over C and M the moduli space of stable vector bundles of rank 2 and determinant isomorphic to a fixed line bundle of degree 1. We prove in this paper that Db(X) is embedded (via ΦE) in Db(M) as a full subcategory (if g≥4).

Brauer Group and Birational Type of Moduli Spaces of Torsionfree Sheaves on a Nodal Curve

Let be the moduli space of stable vector bundles of rank n and fixed determinant 𝕃 of degree d on a nodal curve Y. The moduli space of semistable vector bundles of rank n and degree d will be denoted by . We calculate the Brauer groups of . We study the question of rationality of and .

The morphism induced by Frobenius push-forward

Let X be a smooth projective curve of genus g ⩾ 2 over an algebraically closed field k of characteristic p > 0, and F: X → X (1) the relative Frobenius morphism. Let \(\mathfrak{M}_X^s (r,d)\) (resp. \(\mathfrak{M}_X^{ss} (r,d)\)) be the moduli space of (resp. semi-)stable vector bundles of rank r and degree d on X. We show that the set-theoretic map \(S_{Frob}^{ss} :\mathfrak{M}_X^{ss} (r,d) \to \mathfrak{M}_{X^{(1)} }^{ss} (rp,d + r(p - 1)(g - 1))\) induced by Open image in new window is a proper morphism. Moreover, the induced morphism \(S_{Frob}^s :\mathfrak{M}_X^s (r,d) \to \mathfrak{M}_{X^{(1)} }^s (rp,d + r(p - 1)(g - 1))\) is a closed immersion. As an application, we obtain that the locus of moduli space \(\mathfrak{M}_{X^{(1)} }^s (p,d)\) consisting of stable vector bundles whose Frobenius pull backs have maximal Harder-Narasimhan polygons is isomorphic to the Jacobian variety JacX of X.

On the Hilbert scheme of the moduli space of vector bundles over an algebraic curve

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.

SMALL RATIONAL CURVES ON THE MODULI SPACE OF STABLE BUNDLES

For a smooth projective curve C with genus g ≥ 2 and a degree 1 line bundle [Formula: see text] on C, let [Formula: see text] be the moduli space of stable vector bundles of rank r over C with the fixed determinant [Formula: see text]. In this paper, we study the small rational curves on M and estimate the codimension of the locus of the small rational curves. In particular, we determine all small rational curves when r = 3.

Moduli of torsionfree sheaves of rank two and odd degree on a nodal hyperelliptic curve

Let X be an irreducible nodal projective hyperelliptic curve of arithmetic genus g. We find explicit descriptions of the moduli spaces of stable vector bundles of rank 2 with a fixed determinant of odd degree on X in terms of spaces associated to the singular pencil of quadrics determined by X.

Existence of vector bundles of rank two with fixed determinant and sections

Consider the scheme $B_{2,L}^{k}$ of stable vector bundles of rank two and fixed determinant $L$ which have at least $k$ sections. Under suitable numerical conditions and for generic $L$, we show the existence of a component of the expected dimension of $B_{2,L}^{k}$.

Jumping conics on a smooth quadric in $${\mathbb {P}_3}$$

We investigate the jumping conics of stable vector bundles E of rank 2 on a smooth quadric surface Q with the first Chern class \({c_1= \mathcal{O}_Q(-1,-1)}\) with respect to the ample line bundle \({\mathcal {O}_Q(1,1)}\) . We show that the set of jumping conics of E is a hypersurface of degree c 2(E) − 1 in \({\mathbb {P}_3^{*}}\) . Using these hypersurfaces, we describe moduli spaces of stable vector bundles of rank 2 on Q in the cases of lower c 2(E).

On some moduli spaces of stable vector bundles on cubic and quartic threefolds

We study certain moduli spaces of stable vector bundles of rank 2 on cubic and quartic threefolds. In many cases under consideration, it turns out that the moduli space is complete and irreducible and a general member has vanishing intermediate cohomology. In one case, all except one component of the moduli space has such vector bundles.

Some results on special stable vector bundles of rank 3 on algebraic curves

The authors discuss the existence and classification of stable vector bundles of rank 3, with 3 or 4 linearly independent holomorphic sections. The sets of all such bundles are denoted by ω3,d2 and ω3,d2, respectively. Our argument leads to sufficient and necessary conditions for the existence of both kinds of bundles. The conclusion is very interesting because of its contradiction to the conjectured dimension formula of stable bundles. Finally, we give a preliminary classification of ω3,42 and a complete discussion on the structure of \( \omega _{3,\tfrac{2} {3}g + 2}^3 \).

On the rational homotopy type of a moduli space of vector bundles over a curve

We study the rational homotopy of the moduli space N-X that parametrizes the isomorphism classes of all stable vector bundles of rank two and fixed determinant of odd degree over a compact connected Riemann surface X of genus g, with g >= 2. The symplectic group Aut(H-1(X, Z)) congruent to Sp(2g, Z) has a natural action on the rational homotopy groups pi(n)(N-X)circle times(Z)Q. We prove that this action extends to an action of Sp(2g, C) on pi(n)(N-X)circle times C-Z. We also show that pi(n)(N-X)circle times C-Z is a non-trivial representation of Sp(2g, C) congruent to Aut (H-1(X, C)) for all n >= 2g - 1. In particular, N-X is a rationally hyperbolic space. In the special case where g = 2, for each n is an element of N, we compute the leading Sp(2g, C) representation occurring in pi(n)(N-X)circle times C-Z.

Frobenius-unstable bundles and $p$-curvature

We use the theory of p-curvature of connections to analyze stable vector bundles of rank 2 on curves of genus 2 which pull back to unstable bundles under the Frobenius morphism. We take two approaches, first using explicit formulas for p-curvature to analyze low-characteristic cases, and then using degeneration techniques to obtain an answer for a general curve by degenerating to an irreducible rational nodal curve, and applying the results of additional works by the author. We also apply our explicit formulas to give a new description of the strata of curves of genus 2 of different p-rank.