Related papers: Codimension of jumping loci

We compute the Harder-Narasimhan filtration of vector bundles $f_*\mathcal O_Y$ for certain finite morphisms $f\,:\,Y\,\longrightarrow\, X$ and in some other cases.

Suppose that $X$ is a smooth projective variety and that $C$ is a general member of a family of free rational curves on $X$. We prove several statements showing that the Harder-Narasimhan filtration of $T_{X}|_{C}$ is approximately the same…

Fix a finite field. A hyperelliptic curve determines a measure on the discrete space of rank two bundles on the projective line: the mass of a given vector bundle is the number of line bundles whose pushforward it is. In a sequence of…

In this paper, we study the curvature estimate of the Hermitian-Yang-Mills flow on holomorphic vector bundles. In one simple case, we show that the curvature of the evolved Hermitian metric is uniformly bounded away from the analytic…

A trigonal canonical curve lies on a rational normal surface scroll $Q \subset \mathbb{P}^{g-1}$. In this note we use this fact to compute the Harder-Narasimhan filtration of the normal bundle of a general such curve $C$ in…

Let $\pi: X \longrightarrow C$ be a fibration with reduced 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 $\pi^*E$ is a $H-$stable…

Given a complex curve C of genus 2, there is a well-known relationship between the moduli space of rank 3 semistable bundles on C and a cubic hypersurface known as the Coble cubic. Some of the aspects of this is known to be related to the…

We study the dimension of loci of special line bundles on stable curves and for a fixed semistable multidegree. In case of total degree $d = g - 1$, we characterize when the effective locus gives a Theta divisor. In case of degree $g - 2$…

In the present thesis we study the geometry of the moduli spaces of Bradlow-Higgs triples on a smooth projective curve $C$. $(E,\varphi, s)$ is a Bradlow-Higgs triple if $(E,\varphi)$ is a Higgs bundle and $s$ is a non-zero global section…

The slopes of maximal subbundles of rank $s$ divided by the degree of the map under various pull backs form a bounded collection of numbers called the $s$-spectrum of the bundle. We study the supremum of the $s$-spectrum and determine it in…

We compute the Harder-Narasimhan Filtration of the normal bundle $N_{C/\mathbb{P}^{g-1}}$ where $C$ is a general tetragonal canonical curve of genus $g$.

Let C be a smooth projective curve over the field of the complex numbers. We consider Brill-Noether loci over the moduli of maps from C to the Grassmannian G(m,n) and the corresponding Quot schemes of quotients of a trivial vector bundle on…

It is shown that the singular set for the Yang-Mills flow on unstable holomorphic vector bundles over compact Kaehler manifolds is completely determined by the Harder-Narasimhan-Seshadri filtration of the initial holomorphic bundle. We…

We describe the Harder--Narasimhan filtration of the Hodge bundle for Teichm\"uller curves in the non-varying strata of quadratic differentials appearing in [CM2]. Moreover, we show that the Hodge bundle on the non-varying strata away from…

Let $(X,H)$ be a polarized K3 surface with $\mathrm{Pic}(X) = \mathbb Z H$, and let $C\in |H|$ be a smooth curve of genus $g$. We give an upper bound on the dimension of global sections of a semistable vector bundle on $C$. This allows us…

Let $X$ be a smooth projective curve of genus $g\geq 2$ over an algebraically closed field $k$ of characteristic $p>0$, $F_X:X\rightarrow X$ the absolute Frobenius morphism. Let $\M^s_X(r,d)$ be the moduli space of stable vector bundles of…

We study singular Hermitian metrics on vector bundles. There are two main results in this paper. The first one is on the coherence of the higher rank analogue of multiplier ideals for singular Hermitian metrics defined by global sections.…

We study the Yang-Mills flow on a holomorphic vector bundle E over a compact Kahler manifold X. We construct a natural barrier function along the flow, and introduce some techniques to study the blow-up of the curvature along the flow.…

We introduce and study the maximal-variation locus in families and moduli spaces of projective curves, defined via conductor-level balancing of meromorphic differentials on the normalization. This notion captures precisely when the space of…

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…