A stratification of the moduli space of vector bundles on curves

Abstract

Let $E$ be a vector bundle of rank $r\ge 2$ on a smooth projective curve $C$ of genus $g\ge 2$ over an algebraically closed field $K$ of arbitrary characteristic. For any integer with $1\le k\le r-1$ we define $${\se}_k(E):=k°E-r\max°F.$$ where the maximum is taken over all subbundles $F$ of rank $k$ of $E$. The $s_k$ gives a stratification of the moduli space $\mathcal{M}(r,d)$ of stable vector bundles of rank $r$ and degree on $d$ on $C$ into locally closed subsets ${\calM}(r,d,k,s)$ according to the value of $s$ and $k$. There is a component ${\mathcal{M}}^0(r,d,k,s)$ of $\mathcal{M}(r,d,k,s)$ distinguish by the fact that a general $E\in {\mathcal{M}}^0(r,d,k,s)$ admits a stable subbundle $F$ such that $E/F$ is also stable. We prove: {\it For $g\ge \frac{r+1}{2}$ and $0<s\le k(r-k)(g-1)+(r+1)$, $s\equiv kd\mathrm{mod}r,$ ${\mathcal{M}}^0(r,d,k,s)$ is non-empty,and its component ${\mathcal{M}}^0(r,d,k,s)$ is of dimension} $$\dim {\mathcal{M}}^0(r,d,k,s)=\{\begin{array}{lclcccc}(r^2+k^2-rk)(g-1)+s-1& & s<k(r-k)(g-1)& \mathrm{if}& r^2(g-1)+1& & s\ge k(r-k)(g-1)\end{array}$$

arXiv commentLatex, Permanent e-mail L. Brambila-Paz: [email protected] Classification: 14D, 14F

References in corpus (1)

• Exact sequence of semistable vector bundles on algebraic curves

Cited by in corpus (1)

• Frobenius Stratification of Moduli Spaces of Rank $3$ Vector Bundles in Characteristic $3$, I