Let $X$ be a smooth connected projective curve of genus $g$ over an algebraically closed field. Let $E$ be a vector bundle on $X$ of rank $n$. Is it true that there exists a constand $N(g,n)$ such that each such $E$ has a filtration $$ 0\subset E_1\subset ...\subset E_n=E $$ by subbundles with $rank(E_i)=i$, such that $deg(E_i/E_{i1})deg(E_{i+1}/E_i)\geq N(g,n)$ ? What is the reference for this?

$\begingroup$ This reminds me the HarderNarashiman filtration, where $$\mu(E_i/E_{i1}) > \mu(E_{i+1}/E_i)$$ and $\mu(F) :=\deg(F) / \textrm{rank}(F)$ is the slope. $\endgroup$ – Francesco Polizzi Aug 16 '14 at 11:32

$\begingroup$ Well, it has something to do with HarderNarasimhan, but it is not quite that. For example, if $n=2$ then the statement is that any rank 2 bundle has a line subbundle whose degree is not too small. $\endgroup$ – Alexander Braverman Aug 16 '14 at 11:38

$\begingroup$ Do you mean to assume $n > 1$? $\endgroup$ – user54268 Aug 16 '14 at 14:26

$\begingroup$ Yes, if you wish (otherwise the question is empty) $\endgroup$ – Alexander Braverman Aug 16 '14 at 15:10

$\begingroup$ I must be missing something, we have rank two (semi)stable bundles of arbitrary negative degree. So you can not bound the degree of the line subbundles from below universally. Do you want a bound that depends on $g$, $n$, and the degree? $\endgroup$ – Tony Pantev Aug 17 '14 at 1:33
According to lemma 4 of
M.F. Atiyah, Vector bundles over an elliptic curve, Proc. London Math. Soc. (3) 7 (1957), 414452.
you could take $N(g,n) =  2 g$.

2$\begingroup$ You can think of this as an example of "bendandbreak". If you denote by $(\pi:P\to X,\pi^*E\to \mathcal{O}_P(1))$ the universal invertible quotient of the pullback of $E$, then the problem is to give an upper bound on the $\mathcal{O}_P(1)$degree of a section. Since $\omega_\pi$ is $\pi^*\text{det}(E)\otimes\mathcal{O}_P(\text{rank}E)$, and since bendandbreak breaks sections until their anticanonical degree is at most $2$, this precisely gives that the minimal $\mathcal{O}_P(1)$degree of a section is no greater than $(\text{deg}(E)+2g)/\text{rank}(E)$. $\endgroup$ – Jason Starr Aug 17 '14 at 11:05

$\begingroup$ Thanks! This was exactly the reference I was looking for. $\endgroup$ – Alexander Braverman Aug 18 '14 at 9:56