# the moduli spaces that contain direct sums of line bundles

Let $$U_X(n,d)$$ be the moduli space of semistable vector bundle of rank $$n$$ and degree $$d$$ over a smooth projective curves over the complex numbers.

How do I know if $$U_X(n,d)$$ contains direct sums of line bundles? I have found on a paper that this is true when $$d=n(g-1)$$ where $$g$$ is the genus of the curve, does any of you know why is it true? When does $$U_X(n,d)$$ not contain any direct sum of line bundles?

Thanks a lot!

$$\newcommand{\L}{\mathcal L} \DeclareMathOperator{\rk}{rk}$$ Say you have some line bundles $$\L_i$$ of degree $$d_i$$. The degree is additive, so $$E = \deg (\bigoplus \L_i) = \sum_i d_i$$. So to get a vector bundle of rank $$n$$ and degree $$d$$, we whould have $$n$$ line bundles, such that $$d_1+\cdots +d_n = d$$.
Now, unless they all have the same degree the sum cannot be semistable. If they have different degree, suppose that $$d_1$$ is the maximum. Then $$d_1 = \frac{\deg \L_1}{\rk \L_1} > \frac{\deg E}{\rk E} = \frac{d_1+\cdots + d_n}{n}.$$ In other words, $$d_1$$ is strictly bigger than the average of all the degrees. This shows that in order for $$U_X(n,d)$$ to contain a direct sum of line bundles, $$d$$ has to be a multiple of $$n$$. I don't understand why you would need $$d=n(g-1)$$ specifically (for example, if $$n=1$$ then all vector bundles are sums of line bundles, in that they are just line bundles). Which paper did you see this in?
• Thank you! That is the answer I was waiting for :) I need $d=n(g-1)$ for specific reason, I am studing the generalized theta divisor. In the paper I was reading, the author claims that the result is trivial when $d=n(g-1)$ but I could not understand why, thanks again! :) Jan 29, 2020 at 9:35