Related papers: Exterior power of stable vector bundle destabilize…

Let $C$ be a genus 2 curve and $\su$ the moduli space of semi-stable rank 2 vector bundles on $C$ with trivial determinant. In \cite{bol:wed} we described the parameter space of non stable extension classes (invariant with respect to the…

Let $G$ be a reductive group acting on an affine scheme $V$. We study the set of principal $G$-bundles on a smooth projective curve $\mathcal C$ such that the associated $V$-bundle admits a section sending the generic point of $\mathcal C$…

Let $f : X \rightarrow Y$ be a genuinely ramified map between irreducible smooth projective curves defined over an algebraically closed field. Let $P$ be a branch data on $Y$ such that $P(y)$ and $B_f(y)$ where $B_f$ is branch data for $f$…

Let X be a geometrically connected smooth projective curve of genus one, defined over the field of real numbers, such that X does not have any real points. We classify the isomorphism classes of all stable real algebraic vector bundles over…

Let $M(n,\xi)$ be the moduli space of stable vector bundles of rank $n\geq 3$ and fixed determinant $\xi$ over a smooth projective algebraic curve $X$ over $\mathbb{C}$ of genus $g\geq 4.$ We use the gonality of the curve and $r$-Hecke…

We formulate a tropical analogue of Grothendieck's section conjecture: that for every stable graph G of genus g>2, and every field k, the generic curve with reduction type G over k satisfies the section conjecture. We prove many cases of…

We prove the existence of a projective good moduli space of principal $\mathcal{G}$-bundles under nonconnected reductive group schemes $\mathcal{G}$ over a smooth projective curve $C$. We also prove that the moduli stack of…

We study parabolic bundles on an algebraic curve in positive characteristic. Our motivation is to properly formulate Frobenius pull-backs of parabolic bundles in a way that extends various previous facts and arguments for the usual…

We prove that the kernel bundle of the evaluation morphism of global sections, namely the syzygy bundle, of a sufficiently ample line bundle on a smooth projective variety is slope stable with respect to any polarization. This settles a…

Moduli of vector bundles on stacky curves behave similarly to moduli of vector bundles on curves, except there are additional numerical invariants giving many different notions of stability. We apply the existence criterion for good moduli…

We analyze the local structure of the moduli space of semi-stable bundles on a curve. In particular, a complete description of the local structure is given in the rank 2 case. We obtain as a corollary of this analysis new results about the…

Let $C$ be a smooth projective curve of genus $g\geq 2$ over $\mathbb C$. Fix $n\geq 1$, $d\in {\mathbb Z}$. A pair $(E,\phi)$ over $C$ consists of an algebraic vector bundle $E$ of rank $n$ and degree $d$ over $C$ and a section $\phi \in…

In this paper, we proved that, for a semi-stable fibration of a proper smooth surface to a proper smooth curve over a filed of positive characteristic, if the p-rank of the generic fiber is 0, then the base change of the fibration by a…

We give restrictions on the existence of families of curves on smooth projective surfaces $S$ of nonnegative Kodaira dimension all having constant geometric genus $g \geq 2$ and hyperelliptic normalizations. In particular, we prove a…

Faltings conjectured that under the p-adic Simpson correspondence, finite dimensional p-adic representations of the geometric \'etale fundamental group of a smooth proper p-adic curve X are equivalent to semi-stable Higgs bundles of degree…

We show that principal bundles for a semisimple group on an arbitrary affine curve over an algebraically closed field are trivial, provided the order of $\pi_1$ of the group is invertible in the ground field, or if the curve has semi-normal…

Let $E$ be a vector bundle on a curve $C$ of compact type and $V \subseteq \mathrm{H}^0(C, E)$ be a linear subspace that generates $E$. In this note, we study the stability of the syzygy bundle $M_{E,V}$ associated to $(E, V )$ over certain…

We give a combinatorial criterion for the tangent bundle on a smooth toric variety to be stable with respect to a given polarisation in terms of the corresponding lattice polytope. Furthermore, we show that for a smooth toric surface and a…

Let $X$ be a smooth projective curve with genus $g\geq3$. Let $\mathcal{N}$ be the moduli space of stable rank two vector bundles on $X$ with a fixed determinant $\mathcal{O}_X(-x)$ for $x\in X$. In this paper, as a generalization of Kiem…

Let X be a smooth projective curve of genus g \textgreater{}1 defined over an algebraically closed field k of characteristic p \textgreater{}0. For p sufficiently large (explicitly given in terms of r,g) we construct an atlas for the locus…