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 $deg(E_i/E_{i-1})-deg(E_{i+1}/E_i)\geq N(g,n)$ ? What is the reference for this?

  • $\begingroup$ This reminds me the Harder-Narashiman filtration, where $$\mu(E_i/E_{i-1}) > \mu(E_{i+1}/E_i)$$ and $\mu(F) :=\deg(F) / \textrm{rank}(F)$ is the slope. $\endgroup$ Commented Aug 16, 2014 at 11:32
  • $\begingroup$ Well, it has something to do with Harder-Narasimhan, but it is not quite that. For example, if $n=2$ then the statement is that any rank 2 bundle has a line sub-bundle whose degree is not too small. $\endgroup$ Commented Aug 16, 2014 at 11:38
  • $\begingroup$ Do you mean to assume $n > 1$? $\endgroup$
    – user54268
    Commented Aug 16, 2014 at 14:26
  • $\begingroup$ Yes, if you wish (otherwise the question is empty) $\endgroup$ Commented Aug 16, 2014 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 sub-bundles from below universally. Do you want a bound that depends on $g$, $n$, and the degree? $\endgroup$ Commented Aug 17, 2014 at 1:33

1 Answer 1


According to lemma 4 of

M.F. Atiyah, Vector bundles over an elliptic curve, Proc. London Math. Soc. (3) 7 (1957), 414-452.

you could take $N(g,n) = - 2 g$.

  • 2
    $\begingroup$ You can think of this as an example of "bend-and-break". 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 bend-and-break 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 $(\text{deg}(E)+2g)/\text{rank}(E)$. $\endgroup$ Commented Aug 17, 2014 at 11:05
  • $\begingroup$ Thanks! This was exactly the reference I was looking for. $\endgroup$ Commented Aug 18, 2014 at 9:56

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.