# The existence of a bound for degrees of subsheaves of a coherent sheaf.

Let $X$ be a smooth projective curve over $\mathbb{C}$. Then I want to show that for any coherent sheaf $\mathcal{F}$ on $X$ there exists an integer $d$ such that for any subsheaf $\mathcal{F}'\subset \mathcal{F}$ we have $$\operatorname{deg}(\mathcal{F}') \leq d.$$ Is there a reference for that?

• What definition of degree are you using? – Mohan Sep 17 '15 at 16:19
• for a line bundle $\mathcal{L}\cong \mathcal{O}_X(\sum a_i P_i)$ the degree is $\sum a_i$, for higher rank bundles it is determined by additivity of short exact sequences – Jimmy R Sep 17 '15 at 16:37
• This definition makes sense only for curves. – Mohan Sep 17 '15 at 20:19
• sorry, you're right! I've edited my question accordingly – Jimmy R Sep 17 '15 at 20:25

## 1 Answer

For your question, you may assume that $F$ is a vector bundle (why?) and then it has a filtration by line bundles, say $L_1,L_2,\ldots L_n$. Choose $d$ to be larger than sum of the absolute values of degrees of all the $L_i$s.